text:
@testset "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p" begin
    (a, b, c, d, m, p, x, ) = @variables a b c d m p x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*of*the*form*(a+b*x^2+c*x^4)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^2+c*x^4)^p*with*b^2-4*a*c=0=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(a^2+2*a*b*x^2+b^2*x^4)^p=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(a^2+2*a*b*x^2+b^2*x^4)^(p/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a^2+2*a*b*x^2+b^2*x^4)^(p/4)=#

    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4), x, 4, (1/4)*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4) + (3*a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4))/(8*(a + b*x^2)) + (3*sqrt(a)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)*asinh((sqrt(b)*x)/sqrt(a)))/(8*sqrt(b)*(1 + (b*x^2)/a)^(3/2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4), x, 3, (1/2)*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4) + (sqrt(a)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)*asinh((sqrt(b)*x)/sqrt(a)))/(2*sqrt(b)*sqrt(1 + (b*x^2)/a))]
    @test_int [1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4), x, 2, (sqrt(a)*sqrt(1 + (b*x^2)/a)*asinh((sqrt(b)*x)/sqrt(a)))/(sqrt(b)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))]
    @test_int [1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4), x, 2, (x*(a + b*x^2))/(a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4))]
    @test_int [1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/4), x, 3, (x*(a + b*x^2))/(3*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/4)) + (2*x)/(3*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)), (2*x)/(3*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)) + x/(3*a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))]
    @test_int [1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(7/4), x, 4, (x*(a + b*x^2))/(5*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(7/4)) + (4*x)/(15*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)) + (8*x*(a + b*x^2))/(15*a^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)), (4*x)/(15*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)) + x/(5*a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)) + (8*x*(a + b*x^2))/(15*a^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4))]
    @test_int [1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(9/4), x, 5, (x*(a + b*x^2))/(7*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(9/4)) + (6*x)/(35*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/4)) + (8*x*(a + b*x^2))/(35*a^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/4)) + (16*x)/(35*a^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)), (16*x)/(35*a^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)) + x/(7*a*(a + b*x^2)^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)) + (6*x)/(35*a^2*(a + b*x^2)^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4)) + (8*x)/(35*a^3*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^2+c*x^4)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^2+c*x^4)^p=#

    @test_int [1/(a^2 + b + 2*a*x^2 + x^4), x, 9, -(atan((sqrt(-a + sqrt(a^2 + b)) - sqrt(2)*x)/sqrt(a + sqrt(a^2 + b)))/(2*sqrt(2)*sqrt(a^2 + b)*sqrt(a + sqrt(a^2 + b)))) + atan((sqrt(-a + sqrt(a^2 + b)) + sqrt(2)*x)/sqrt(a + sqrt(a^2 + b)))/(2*sqrt(2)*sqrt(a^2 + b)*sqrt(a + sqrt(a^2 + b))) - log(sqrt(a^2 + b) - sqrt(2)*sqrt(-a + sqrt(a^2 + b))*x + x^2)/(4*sqrt(2)*sqrt(a^2 + b)*sqrt(-a + sqrt(a^2 + b))) + log(sqrt(a^2 + b) + sqrt(2)*sqrt(-a + sqrt(a^2 + b))*x + x^2)/(4*sqrt(2)*sqrt(a^2 + b)*sqrt(-a + sqrt(a^2 + b)))]
    @test_int [1/(-1 + a^2 + 2*a*x^2 + x^4), x, 3, -(atan(x/sqrt(1 + a))/(2*sqrt(1 + a))) - atanh(x/sqrt(1 - a))/(2*sqrt(1 - a))]
    #= Tests*that*NegativeQ(b^2-4*a*c)*returns*True*to*avoid*im*in*antiderivative! =#
    @test_int [1/(1 + a^2 + 2*a*x^2 + x^4), x, 9, -(atan((sqrt(-a + sqrt(1 + a^2)) - sqrt(2)*x)/sqrt(a + sqrt(1 + a^2)))/(2*sqrt(2)*sqrt(1 + a^2)*sqrt(a + sqrt(1 + a^2)))) + atan((sqrt(-a + sqrt(1 + a^2)) + sqrt(2)*x)/sqrt(a + sqrt(1 + a^2)))/(2*sqrt(2)*sqrt(1 + a^2)*sqrt(a + sqrt(1 + a^2))) - log(sqrt(1 + a^2) - sqrt(2)*sqrt(-a + sqrt(1 + a^2))*x + x^2)/(4*sqrt(2)*sqrt(1 + a^2)*sqrt(-a + sqrt(1 + a^2))) + log(sqrt(1 + a^2) + sqrt(2)*sqrt(-a + sqrt(1 + a^2))*x + x^2)/(4*sqrt(2)*sqrt(1 + a^2)*sqrt(-a + sqrt(1 + a^2)))]

    @test_int [1/(4 - 5*x^2 + x^4), x, 3, (-(1/6))*atanh(x/2) + atanh(x)/3]
    @test_int [1/(3 + 4*x^2 + x^4), x, 3, atan(x)/2 - atan(x/sqrt(3))/(2*sqrt(3))]
    @test_int [1/(9 + 5*x^2 + x^4), x, 9, -(atan((1 - 2*x)/sqrt(11))/(6*sqrt(11))) + atan((1 + 2*x)/sqrt(11))/(6*sqrt(11)) - (1/12)*log(3 - x + x^2) + (1/12)*log(3 + x + x^2)]
    @test_int [1/(1 - x^2 + x^4), x, 9, (-(1/2))*atan(sqrt(3) - 2*x) + (1/2)*atan(sqrt(3) + 2*x) - log(1 - sqrt(3)*x + x^2)/(4*sqrt(3)) + log(1 + sqrt(3)*x + x^2)/(4*sqrt(3))]
    @test_int [1/(2 + 2*x^2 + x^4), x, 9, (-(1/4))*sqrt(-1 + sqrt(2))*atan((sqrt(2*(-1 + sqrt(2))) - 2*x)/sqrt(2*(1 + sqrt(2)))) + (1/4)*sqrt(-1 + sqrt(2))*atan((sqrt(2*(-1 + sqrt(2))) + 2*x)/sqrt(2*(1 + sqrt(2)))) - log(sqrt(2) - sqrt(2*(-1 + sqrt(2)))*x + x^2)/(8*sqrt(-1 + sqrt(2))) + log(sqrt(2) + sqrt(2*(-1 + sqrt(2)))*x + x^2)/(8*sqrt(-1 + sqrt(2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^2+c*x^4)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [1/sqrt(2 + 5*x^2 - 3*x^4), x, 2, Elliptic.F(asin(x/sqrt(2)), -6)]
    @test_int [1/sqrt(2 + 4*x^2 - 3*x^4), x, 2, sqrt((1/6)*(2 + sqrt(10)))*Elliptic.F(asin(sqrt((1/2)*(-2 + sqrt(10)))*x), (1/3)*(-7 - 2*sqrt(10)))]
    @test_int [1/sqrt(2 + 3*x^2 - 3*x^4), x, 2, sqrt(2/(-3 + sqrt(33)))*Elliptic.F(asin(sqrt(6/(3 + sqrt(33)))*x), (1/4)*(-7 - sqrt(33)))]
    @test_int [1/sqrt(2 + 2*x^2 - 3*x^4), x, 2, Elliptic.F(asin(sqrt(3/(1 + sqrt(7)))*x), (1/3)*(-4 - sqrt(7)))/sqrt(-1 + sqrt(7))]
    @test_int [1/sqrt(2 + 1*x^2 - 3*x^4), x, 2, Elliptic.F(asin(x), -(3/2))/sqrt(2)]
    @test_int [1/sqrt(2 + 0*x^2 - 3*x^4), x, 1, Elliptic.F(asin((3/2)^(1/4)*x), -1)/6^(1/4)]
    @test_int [1/sqrt(2 - 1*x^2 - 3*x^4), x, 2, Elliptic.F(asin(sqrt(3/2)*x), -(2/3))/sqrt(3)]
    @test_int [1/sqrt(2 - 2*x^2 - 3*x^4), x, 2, Elliptic.F(asin(sqrt(3/(-1 + sqrt(7)))*x), (1/3)*(-4 + sqrt(7)))/sqrt(1 + sqrt(7))]
    @test_int [1/sqrt(2 - 3*x^2 - 3*x^4), x, 2, sqrt(2/(3 + sqrt(33)))*Elliptic.F(asin(sqrt(6/(-3 + sqrt(33)))*x), (1/4)*(-7 + sqrt(33)))]
    @test_int [1/sqrt(2 - 4*x^2 - 3*x^4), x, 2, sqrt((1/6)*(-2 + sqrt(10)))*Elliptic.F(asin(sqrt((1/2)*(2 + sqrt(10)))*x), (1/3)*(-7 + 2*sqrt(10)))]
    @test_int [1/sqrt(2 - 5*x^2 - 3*x^4), x, 2, Elliptic.F(asin(sqrt(3)*x), -(1/6))/sqrt(6)]

    @test_int [1/sqrt(3 + 7*x^2 - 2*x^4), x, 2, sqrt(2/(-7 + sqrt(73)))*Elliptic.F(asin((2*x)/sqrt(7 + sqrt(73))), (1/12)*(-61 - 7*sqrt(73)))]
    @test_int [1/sqrt(3 + 6*x^2 - 2*x^4), x, 2, sqrt((1/6)*(3 + sqrt(15)))*Elliptic.F(asin(sqrt((1/3)*(-3 + sqrt(15)))*x), -4 - sqrt(15))]
    @test_int [1/sqrt(3 + 5*x^2 - 2*x^4), x, 2, Elliptic.F(asin(x/sqrt(3)), -6)]
    @test_int [1/sqrt(3 + 4*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2/(2 + sqrt(10)))*x), (1/3)*(-7 - 2*sqrt(10)))/sqrt(-2 + sqrt(10))]
    @test_int [1/sqrt(3 + 3*x^2 - 2*x^4), x, 2, sqrt(2/(-3 + sqrt(33)))*Elliptic.F(asin((2*x)/sqrt(3 + sqrt(33))), (1/4)*(-7 - sqrt(33)))]
    @test_int [1/sqrt(3 + 2*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2/(1 + sqrt(7)))*x), (1/3)*(-4 - sqrt(7)))/sqrt(-1 + sqrt(7))]
    @test_int [1/sqrt(3 + 1*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2/3)*x), -3/2)/sqrt(2)]
    @test_int [1/sqrt(3 + 0*x^2 - 2*x^4), x, 1, Elliptic.F(asin((2/3)^(1/4)*x), -1)/6^(1/4)]
    @test_int [1/sqrt(3 - 1*x^2 - 2*x^4), x, 2, Elliptic.F(asin(x), -2/3)/sqrt(3)]
    @test_int [1/sqrt(3 - 2*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2/(-1 + sqrt(7)))*x), (1/3)*(-4 + sqrt(7)))/sqrt(1 + sqrt(7))]
    @test_int [1/sqrt(3 - 3*x^2 - 2*x^4), x, 2, sqrt(2/(3 + sqrt(33)))*Elliptic.F(asin((2*x)/sqrt(-3 + sqrt(33))), (1/4)*(-7 + sqrt(33)))]
    @test_int [1/sqrt(3 - 4*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2/(-2 + sqrt(10)))*x), (1/3)*(-7 + 2*sqrt(10)))/sqrt(2 + sqrt(10))]
    @test_int [1/sqrt(3 - 5*x^2 - 2*x^4), x, 2, Elliptic.F(asin(sqrt(2)*x), -1/6)/sqrt(6)]
    @test_int [1/sqrt(3 - 6*x^2 - 2*x^4), x, 2, sqrt((1/6)*(-3 + sqrt(15)))*Elliptic.F(asin(sqrt((1/3)*(3 + sqrt(15)))*x), -4 + sqrt(15))]
    @test_int [1/sqrt(3 - 7*x^2 - 2*x^4), x, 2, sqrt(2/(7 + sqrt(73)))*Elliptic.F(asin((2*x)/sqrt(-7 + sqrt(73))), (1/12)*(-61 + 7*sqrt(73)))]

    @test_int [1/sqrt(-2 + 5*x^2 + 3*x^4), x, 1, (sqrt(2 + x^2)*sqrt(-1 + 3*x^2)*Elliptic.F(asin((sqrt(7/2)*x)/sqrt(-1 + 3*x^2)), 6/7))/(sqrt(7)*sqrt(-2 + 5*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 + 4*x^2 + 3*x^4), x, 1, (sqrt((2 - (2 - sqrt(10))*x^2)/(2 - (2 + sqrt(10))*x^2))*sqrt(-2 + (2 + sqrt(10))*x^2)*Elliptic.F(asin((2^(3/4)*5^(1/4)*x)/sqrt(-2 + (2 + sqrt(10))*x^2)), (1/10)*(5 + sqrt(10))))/(2*10^(1/4)*sqrt(1/(2 - (2 + sqrt(10))*x^2))*sqrt(-2 + 4*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 + 3*x^2 + 3*x^4), x, 1, (sqrt((4 - (3 - sqrt(33))*x^2)/(4 - (3 + sqrt(33))*x^2))*sqrt(-4 + (3 + sqrt(33))*x^2)*Elliptic.F(asin((sqrt(2)*33^(1/4)*x)/sqrt(-4 + (3 + sqrt(33))*x^2)), (1/22)*(11 + sqrt(33))))/(2*sqrt(2)*33^(1/4)*sqrt(1/(4 - (3 + sqrt(33))*x^2))*sqrt(-2 + 3*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 + 2*x^2 + 3*x^4), x, 1, (sqrt((2 - (1 - sqrt(7))*x^2)/(2 - (1 + sqrt(7))*x^2))*sqrt(-2 + (1 + sqrt(7))*x^2)*Elliptic.F(asin((sqrt(2)*7^(1/4)*x)/sqrt(-2 + (1 + sqrt(7))*x^2)), (1/14)*(7 + sqrt(7))))/(2*7^(1/4)*sqrt(1/(2 - (1 + sqrt(7))*x^2))*sqrt(-2 + 2*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 + 1*x^2 + 3*x^4), x, 1, (sqrt(1 + x^2)*sqrt(-2 + 3*x^2)*Elliptic.F(asin((sqrt(5)*x)/sqrt(-2 + 3*x^2)), 3/5))/(sqrt(5)*sqrt(-2 + x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 + 0*x^2 + 3*x^4), x, 1, (sqrt(-2 + sqrt(6)*x^2)*sqrt((2 + sqrt(6)*x^2)/(2 - sqrt(6)*x^2))*Elliptic.F(asin((2^(3/4)*3^(1/4)*x)/sqrt(-2 + sqrt(6)*x^2)), 1/2))/(2*6^(1/4)*sqrt(1/(2 - sqrt(6)*x^2))*sqrt(-2 + 3*x^4))]
    @test_int [1/sqrt(-2 - 1*x^2 + 3*x^4), x, 1, (sqrt(-1 + x^2)*sqrt(2 + 3*x^2)*Elliptic.F(asin((sqrt(5/2)*x)/sqrt(-1 + x^2)), 2/5))/(sqrt(5)*sqrt(-2 - x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 - 2*x^2 + 3*x^4), x, 1, (sqrt(-2 - (1 - sqrt(7))*x^2)*sqrt((2 + (1 + sqrt(7))*x^2)/(2 + (1 - sqrt(7))*x^2))*Elliptic.F(asin((sqrt(2)*7^(1/4)*x)/sqrt(-2 - (1 - sqrt(7))*x^2)), (1/14)*(7 - sqrt(7))))/(2*7^(1/4)*sqrt(1/(2 + (1 - sqrt(7))*x^2))*sqrt(-2 - 2*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 - 3*x^2 + 3*x^4), x, 1, (sqrt(-4 - (3 - sqrt(33))*x^2)*sqrt((4 + (3 + sqrt(33))*x^2)/(4 + (3 - sqrt(33))*x^2))*Elliptic.F(asin((sqrt(2)*33^(1/4)*x)/sqrt(-4 - (3 - sqrt(33))*x^2)), (1/22)*(11 - sqrt(33))))/(2*sqrt(2)*33^(1/4)*sqrt(1/(4 + (3 - sqrt(33))*x^2))*sqrt(-2 - 3*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 - 4*x^2 + 3*x^4), x, 1, (sqrt(-2 - (2 - sqrt(10))*x^2)*sqrt((2 + (2 + sqrt(10))*x^2)/(2 + (2 - sqrt(10))*x^2))*Elliptic.F(asin((2^(3/4)*5^(1/4)*x)/sqrt(-2 - (2 - sqrt(10))*x^2)), (1/10)*(5 - sqrt(10))))/(2*10^(1/4)*sqrt(1/(2 + (2 - sqrt(10))*x^2))*sqrt(-2 - 4*x^2 + 3*x^4))]
    @test_int [1/sqrt(-2 - 5*x^2 + 3*x^4), x, 1, (sqrt(-2 + x^2)*sqrt(1 + 3*x^2)*Elliptic.F(asin((sqrt(7)*x)/sqrt(-2 + x^2)), 1/7))/(sqrt(7)*sqrt(-2 - 5*x^2 + 3*x^4))]

    @test_int [1/sqrt(-3 + 7*x^2 + 2*x^4), x, 1, (sqrt((6 - (7 - sqrt(73))*x^2)/(6 - (7 + sqrt(73))*x^2))*sqrt(-6 + (7 + sqrt(73))*x^2)*Elliptic.F(asin((sqrt(2)*73^(1/4)*x)/sqrt(-6 + (7 + sqrt(73))*x^2)), (1/146)*(73 + 7*sqrt(73))))/(2*sqrt(3)*73^(1/4)*sqrt(1/(6 - (7 + sqrt(73))*x^2))*sqrt(-3 + 7*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 6*x^2 + 2*x^4), x, 1, (sqrt((3 - (3 - sqrt(15))*x^2)/(3 - (3 + sqrt(15))*x^2))*sqrt(-3 + (3 + sqrt(15))*x^2)*Elliptic.F(asin((sqrt(2)*15^(1/4)*x)/sqrt(-3 + (3 + sqrt(15))*x^2)), (1/10)*(5 + sqrt(15))))/(sqrt(2)*3^(3/4)*5^(1/4)*sqrt(1/(3 - (3 + sqrt(15))*x^2))*sqrt(-3 + 6*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 5*x^2 + 2*x^4), x, 1, (sqrt(3 + x^2)*sqrt(-1 + 2*x^2)*Elliptic.F(asin((sqrt(7/3)*x)/sqrt(-1 + 2*x^2)), 6/7))/(sqrt(7)*sqrt(-3 + 5*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 4*x^2 + 2*x^4), x, 1, (sqrt((3 - (2 - sqrt(10))*x^2)/(3 - (2 + sqrt(10))*x^2))*sqrt(-3 + (2 + sqrt(10))*x^2)*Elliptic.F(asin((2^(3/4)*5^(1/4)*x)/sqrt(-3 + (2 + sqrt(10))*x^2)), (1/10)*(5 + sqrt(10))))/(2^(3/4)*sqrt(3)*5^(1/4)*sqrt(1/(3 - (2 + sqrt(10))*x^2))*sqrt(-3 + 4*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 3*x^2 + 2*x^4), x, 1, (sqrt((6 - (3 - sqrt(33))*x^2)/(6 - (3 + sqrt(33))*x^2))*sqrt(-6 + (3 + sqrt(33))*x^2)*Elliptic.F(asin((sqrt(2)*33^(1/4)*x)/sqrt(-6 + (3 + sqrt(33))*x^2)), (1/22)*(11 + sqrt(33))))/(2*3^(3/4)*11^(1/4)*sqrt(1/(6 - (3 + sqrt(33))*x^2))*sqrt(-3 + 3*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 2*x^2 + 2*x^4), x, 1, (sqrt((3 - (1 - sqrt(7))*x^2)/(3 - (1 + sqrt(7))*x^2))*sqrt(-3 + (1 + sqrt(7))*x^2)*Elliptic.F(asin((sqrt(2)*7^(1/4)*x)/sqrt(-3 + (1 + sqrt(7))*x^2)), (1/14)*(7 + sqrt(7))))/(sqrt(6)*7^(1/4)*sqrt(1/(3 - (1 + sqrt(7))*x^2))*sqrt(-3 + 2*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 1*x^2 + 2*x^4), x, 1, (sqrt(-1 + x^2)*sqrt(3 + 2*x^2)*Elliptic.F(asin((sqrt(5/3)*x)/sqrt(-1 + x^2)), 3/5))/(sqrt(5)*sqrt(-3 + x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 + 0*x^2 + 2*x^4), x, 1, (sqrt(-3 + sqrt(6)*x^2)*sqrt((3 + sqrt(6)*x^2)/(3 - sqrt(6)*x^2))*Elliptic.F(asin((2^(3/4)*3^(1/4)*x)/sqrt(-3 + sqrt(6)*x^2)), 1/2))/(6^(3/4)*sqrt(1/(3 - sqrt(6)*x^2))*sqrt(-3 + 2*x^4))]
    @test_int [1/sqrt(-3 - 1*x^2 + 2*x^4), x, 1, (sqrt(1 + x^2)*sqrt(-3 + 2*x^2)*Elliptic.F(asin((sqrt(5)*x)/sqrt(-3 + 2*x^2)), 2/5))/(sqrt(5)*sqrt(-3 - x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 - 2*x^2 + 2*x^4), x, 1, (sqrt(-3 - (1 - sqrt(7))*x^2)*sqrt((3 + (1 + sqrt(7))*x^2)/(3 + (1 - sqrt(7))*x^2))*Elliptic.F(asin((sqrt(2)*7^(1/4)*x)/sqrt(-3 - (1 - sqrt(7))*x^2)), (1/14)*(7 - sqrt(7))))/(sqrt(6)*7^(1/4)*sqrt(1/(3 + (1 - sqrt(7))*x^2))*sqrt(-3 - 2*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 - 3*x^2 + 2*x^4), x, 1, (sqrt(-6 - (3 - sqrt(33))*x^2)*sqrt((6 + (3 + sqrt(33))*x^2)/(6 + (3 - sqrt(33))*x^2))*Elliptic.F(asin((sqrt(2)*33^(1/4)*x)/sqrt(-6 - (3 - sqrt(33))*x^2)), (1/22)*(11 - sqrt(33))))/(2*3^(3/4)*11^(1/4)*sqrt(1/(6 + (3 - sqrt(33))*x^2))*sqrt(-3 - 3*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 - 4*x^2 + 2*x^4), x, 1, (sqrt(-3 - (2 - sqrt(10))*x^2)*sqrt((3 + (2 + sqrt(10))*x^2)/(3 + (2 - sqrt(10))*x^2))*Elliptic.F(asin((2^(3/4)*5^(1/4)*x)/sqrt(-3 - (2 - sqrt(10))*x^2)), (1/10)*(5 - sqrt(10))))/(2^(3/4)*sqrt(3)*5^(1/4)*sqrt(1/(3 + (2 - sqrt(10))*x^2))*sqrt(-3 - 4*x^2 + 2*x^4))]
    @test_int [1/sqrt(-3 - 5*x^2 + 2*x^4), x, 1, (sqrt(-3 + x^2)*sqrt(1 + 2*x^2)*Elliptic.F(asin((sqrt(7)*x)/sqrt(-3 + x^2)), 1/7))/(sqrt(7)*sqrt(-3 - 5*x^2 + 2*x^4))]

    @test_int [1/sqrt(2 + 5*x^2 + 3*x^4), x, 1, ((1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2))*Elliptic.F(atan(x), -(1/2)))/(sqrt(2)*sqrt(2 + 5*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 4*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 4*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2 - 1/sqrt(6)))/(2*6^(1/4)*sqrt(2 + 4*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 3*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 3*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/8)*(4 - sqrt(6))))/(2*6^(1/4)*sqrt(2 + 3*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 2*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 2*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/12)*(6 - sqrt(6))))/(2*6^(1/4)*sqrt(2 + 2*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 1*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/24)*(12 - sqrt(6))))/(2*6^(1/4)*sqrt(2 + x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 0*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2))/(2*6^(1/4)*sqrt(2 + 3*x^4))]
    @test_int [1/sqrt(2 - 1*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/24)*(12 + sqrt(6))))/(2*6^(1/4)*sqrt(2 - x^2 + 3*x^4))]
    @test_int [1/sqrt(2 - 2*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 2*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/12)*(6 + sqrt(6))))/(2*6^(1/4)*sqrt(2 - 2*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 - 3*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 3*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/8)*(4 + sqrt(6))))/(2*6^(1/4)*sqrt(2 - 3*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 - 4*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 4*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2 + 1/sqrt(6)))/(2*6^(1/4)*sqrt(2 - 4*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 - 5*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 5*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/24)*(12 + 5*sqrt(6))))/(2*6^(1/4)*sqrt(2 - 5*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 - 6*x^2 + 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 6*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/4)*(2 + sqrt(6))))/(2*6^(1/4)*sqrt(2 - 6*x^2 + 3*x^4))]

    @test_int [1/sqrt(3 + 9*x^2 + 2*x^4), x, 1, (sqrt((6 + (9 - sqrt(57))*x^2)/(6 + (9 + sqrt(57))*x^2))*(6 + (9 + sqrt(57))*x^2)*Elliptic.F(atan(sqrt((1/6)*(9 + sqrt(57)))*x), (1/4)*(-19 + 3*sqrt(57))))/(sqrt(6*(9 + sqrt(57)))*sqrt(3 + 9*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 8*x^2 + 2*x^4), x, 1, (sqrt((3 + (4 - sqrt(10))*x^2)/(3 + (4 + sqrt(10))*x^2))*(3 + (4 + sqrt(10))*x^2)*Elliptic.F(atan(sqrt((1/3)*(4 + sqrt(10)))*x), (-(2/3))*(5 - 2*sqrt(10))))/(sqrt(3*(4 + sqrt(10)))*sqrt(3 + 8*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 7*x^2 + 2*x^4), x, 1, (sqrt((3 + x^2)/(1 + 2*x^2))*(1 + 2*x^2)*Elliptic.F(atan(sqrt(2)*x), 5/6))/(sqrt(6)*sqrt(3 + 7*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 6*x^2 + 2*x^4), x, 1, (sqrt((3 + (3 - sqrt(3))*x^2)/(3 + (3 + sqrt(3))*x^2))*(3 + (3 + sqrt(3))*x^2)*Elliptic.F(atan(sqrt((1/3)*(3 + sqrt(3)))*x), -1 + sqrt(3)))/(sqrt(3*(3 + sqrt(3)))*sqrt(3 + 6*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 5*x^2 + 2*x^4), x, 1, ((1 + x^2)*sqrt((3 + 2*x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/3))/(sqrt(3)*sqrt(3 + 5*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 4*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 4*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2 - 1/sqrt(6)))/(2*6^(1/4)*sqrt(3 + 4*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 3*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 3*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (4 - sqrt(6))/8))/(2*6^(1/4)*sqrt(3 + 3*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 2*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 2*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (6 - sqrt(6))/12))/(2*6^(1/4)*sqrt(3 + 2*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 1*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (12 - sqrt(6))/24))/(2*6^(1/4)*sqrt(3 + x^2 + 2*x^4))]
    @test_int [1/sqrt(3 + 0*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2))/(2*6^(1/4)*sqrt(3 + 2*x^4))]
    @test_int [1/sqrt(3 - 1*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (12 + sqrt(6))/24))/(2*6^(1/4)*sqrt(3 - x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 2*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 2*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (6 + sqrt(6))/12))/(2*6^(1/4)*sqrt(3 - 2*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 3*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 3*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (4 + sqrt(6))/8))/(2*6^(1/4)*sqrt(3 - 3*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 4*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 4*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2 + 1/sqrt(6)))/(2*6^(1/4)*sqrt(3 - 4*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 5*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 5*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/24)*(12 + 5*sqrt(6))))/(2*6^(1/4)*sqrt(3 - 5*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 6*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 6*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/4)*(2 + sqrt(6))))/(2*6^(1/4)*sqrt(3 - 6*x^2 + 2*x^4))]
    @test_int [1/sqrt(3 - 7*x^2 + 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 7*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/24)*(12 + 7*sqrt(6))))/(2*6^(1/4)*sqrt(3 - 7*x^2 + 2*x^4))]

    @test_int [1/sqrt(-3 + 7*x^2 - 2*x^4), x, 2, -(Elliptic.F(acos(x/sqrt(3)), 6/5)/sqrt(5))]
    @test_int [1/sqrt(-3 + 6*x^2 - 2*x^4), x, 2, -(Elliptic.F(acos(sqrt((1/3)*(3 - sqrt(3)))*x), (1/2)*(1 + sqrt(3)))/(sqrt(2)*3^(1/4)))]
    @test_int [1/sqrt(-3 + 5*x^2 - 2*x^4), x, 2, -Elliptic.F(acos(sqrt(2/3)*x), 3)]
    @test_int [1/sqrt(-3 + 4*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 4*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2 + 1/sqrt(6)))/(2*6^(1/4)*sqrt(-3 + 4*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 + 3*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 3*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/8)*(4 + sqrt(6))))/(2*6^(1/4)*sqrt(-3 + 3*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 + 2*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - 2*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/12)*(6 + sqrt(6))))/(2*6^(1/4)*sqrt(-3 + 2*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 + 1*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 - x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/24)*(12 + sqrt(6))))/(2*6^(1/4)*sqrt(-3 + x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 + 0*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2))/(2*6^(1/4)*sqrt(-3 - 2*x^4))]
    @test_int [1/sqrt(-3 - 1*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/24)*(12 - sqrt(6))))/(2*6^(1/4)*sqrt(-3 - x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 - 2*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 2*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/12)*(6 - sqrt(6))))/(2*6^(1/4)*sqrt(-3 - 2*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 - 3*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 3*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), (1/8)*(4 - sqrt(6))))/(2*6^(1/4)*sqrt(-3 - 3*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 - 4*x^2 - 2*x^4), x, 1, ((3 + sqrt(6)*x^2)*sqrt((3 + 4*x^2 + 2*x^4)/(3 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((2/3)^(1/4)*x), 1/2 - 1/sqrt(6)))/(2*6^(1/4)*sqrt(-3 - 4*x^2 - 2*x^4))]
    @test_int [1/sqrt(-3 - 5*x^2 - 2*x^4), x, 2, (sqrt(3 + 2*x^2)*Elliptic.F(atan(x), 1/3))/(sqrt(3)*sqrt(-1 - x^2)*sqrt((3 + 2*x^2)/(1 + x^2)))]

    @test_int [1/sqrt(-2 + 6*x^2 - 3*x^4), x, 2, -(Elliptic.F(acos(sqrt(3/(3 + sqrt(3)))*x), (1/2)*(1 + sqrt(3)))/(sqrt(2)*3^(1/4)))]
    @test_int [1/sqrt(-2 + 5*x^2 - 3*x^4), x, 2, -Elliptic.F(acos(x), 3)]
    @test_int [1/sqrt(-2 + 4*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 4*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2 + 1/sqrt(6)))/(2*6^(1/4)*sqrt(-2 + 4*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 + 3*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 3*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/8)*(4 + sqrt(6))))/(2*6^(1/4)*sqrt(-2 + 3*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 + 2*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - 2*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/12)*(6 + sqrt(6))))/(2*6^(1/4)*sqrt(-2 + 2*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 + 1*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 - x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/24)*(12 + sqrt(6))))/(2*6^(1/4)*sqrt(-2 + x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 + 0*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2))/(2*6^(1/4)*sqrt(-2 - 3*x^4))]
    @test_int [1/sqrt(-2 - 1*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/24)*(12 - sqrt(6))))/(2*6^(1/4)*sqrt(-2 - x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 - 2*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 2*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/12)*(6 - sqrt(6))))/(2*6^(1/4)*sqrt(-2 - 2*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 - 3*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 3*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), (1/8)*(4 - sqrt(6))))/(2*6^(1/4)*sqrt(-2 - 3*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 - 4*x^2 - 3*x^4), x, 1, ((2 + sqrt(6)*x^2)*sqrt((2 + 4*x^2 + 3*x^4)/(2 + sqrt(6)*x^2)^2)*Elliptic.F(2*atan((3/2)^(1/4)*x), 1/2 - 1/sqrt(6)))/(2*6^(1/4)*sqrt(-2 - 4*x^2 - 3*x^4))]
    @test_int [1/sqrt(-2 - 5*x^2 - 3*x^4), x, 2, -((sqrt(-2 - 3*x^2)*Elliptic.F(atan(x), -(1/2)))/(sqrt(2)*sqrt(1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2))))]

    @test_int [1/sqrt(2 + 5*x^2 + 5*x^4), x, 1, ((2 + sqrt(10)*x^2)*sqrt((2 + 5*x^2 + 5*x^4)/(2 + sqrt(10)*x^2)^2)*Elliptic.F(2*atan((5/2)^(1/4)*x), (1/8)*(4 - sqrt(10))))/(2*10^(1/4)*sqrt(2 + 5*x^2 + 5*x^4))]
    @test_int [1/sqrt(2 + 5*x^2 + 4*x^4), x, 1, ((1 + sqrt(2)*x^2)*sqrt((2 + 5*x^2 + 4*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/16)*(8 - 5*sqrt(2))))/(2*2^(3/4)*sqrt(2 + 5*x^2 + 4*x^4))]
    @test_int [1/sqrt(2 + 5*x^2 + 3*x^4), x, 1, ((1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2))*Elliptic.F(atan(x), -(1/2)))/(sqrt(2)*sqrt(2 + 5*x^2 + 3*x^4))]
    @test_int [1/sqrt(2 + 5*x^2 + 2*x^4), x, 1, (sqrt((2 + x^2)/(1 + 2*x^2))*(1 + 2*x^2)*Elliptic.F(atan(sqrt(2)*x), 3/4))/(2*sqrt(2 + 5*x^2 + 2*x^4))]
    @test_int [1/sqrt(2 + 5*x^2 + 1*x^4), x, 1, (sqrt((4 + (5 - sqrt(17))*x^2)/(4 + (5 + sqrt(17))*x^2))*(4 + (5 + sqrt(17))*x^2)*Elliptic.F(atan((1/2)*sqrt(5 + sqrt(17))*x), (1/4)*(-17 + 5*sqrt(17))))/(2*sqrt(5 + sqrt(17))*sqrt(2 + 5*x^2 + x^4))]
    @test_int [1/sqrt(2 + 5*x^2 - 1*x^4), x, 2, sqrt(2/(-5 + sqrt(33)))*Elliptic.F(asin(sqrt(2/(5 + sqrt(33)))*x), (1/4)*(-29 - 5*sqrt(33)))]
    @test_int [1/sqrt(2 + 5*x^2 - 2*x^4), x, 2, sqrt(2/(-5 + sqrt(41)))*Elliptic.F(asin((2*x)/sqrt(5 + sqrt(41))), (1/8)*(-33 - 5*sqrt(41)))]
    @test_int [1/sqrt(2 + 5*x^2 - 3*x^4), x, 2, Elliptic.F(asin(x/sqrt(2)), -6)]
    @test_int [1/sqrt(2 + 5*x^2 - 4*x^4), x, 2, sqrt(2/(-5 + sqrt(57)))*Elliptic.F(asin(2*sqrt(2/(5 + sqrt(57)))*x), (1/16)*(-41 - 5*sqrt(57)))]
    @test_int [1/sqrt(2 + 5*x^2 - 5*x^4), x, 2, sqrt(2/(-5 + sqrt(65)))*Elliptic.F(asin(sqrt(10/(5 + sqrt(65)))*x), (1/4)*(-9 - sqrt(65)))]
    @test_int [1/sqrt(2 + 5*x^2 - 6*x^4), x, 2, sqrt(2/(-5 + sqrt(73)))*Elliptic.F(asin(2*sqrt(3/(5 + sqrt(73)))*x), (1/24)*(-49 - 5*sqrt(73)))]
    @test_int [1/sqrt(2 + 5*x^2 - 7*x^4), x, 2, Elliptic.F(asin(x), -(7/2))/sqrt(2)]
    @test_int [1/sqrt(2 + 5*x^2 - 8*x^4), x, 2, sqrt(2/(-5 + sqrt(89)))*Elliptic.F(asin((4*x)/sqrt(5 + sqrt(89))), (1/32)*(-57 - 5*sqrt(89)))]
    @test_int [1/sqrt(2 + 5*x^2 - 9*x^4), x, 2, sqrt(2/(-5 + sqrt(97)))*Elliptic.F(asin(3*sqrt(2/(5 + sqrt(97)))*x), (1/36)*(-61 - 5*sqrt(97)))]

    #= ::Title:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p*with*a=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(b*x^2+c*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(b*x^2 + c*x^4), x, 2, (b*x^5)/5 + (c*x^7)/7]
    @test_int [x^1*(b*x^2 + c*x^4), x, 2, (b*x^4)/4 + (c*x^6)/6]
    @test_int [x^0*(b*x^2 + c*x^4), x, 1, (b*x^3)/3 + (c*x^5)/5]
    @test_int [(b*x^2 + c*x^4)/x^1, x, 2, (b*x^2)/2 + (c*x^4)/4]
    @test_int [(b*x^2 + c*x^4)/x^2, x, 2, b*x + (c*x^3)/3]
    @test_int [(b*x^2 + c*x^4)/x^3, x, 2, (c*x^2)/2 + b*log(x)]
    @test_int [(b*x^2 + c*x^4)/x^4, x, 2, -(b/x) + c*x]
    @test_int [(b*x^2 + c*x^4)/x^5, x, 2, -(b/(2*x^2)) + c*log(x)]
    @test_int [(b*x^2 + c*x^4)/x^6, x, 2, -(b/(3*x^3)) - c/x]
    @test_int [(b*x^2 + c*x^4)/x^7, x, 2, -(b/(4*x^4)) - c/(2*x^2)]
    @test_int [(b*x^2 + c*x^4)/x^8, x, 2, -(b/(5*x^5)) - c/(3*x^3)]

    @test_int [x^0*(b*x^2 + c*x^4)^2, x, 3, (b^2*x^5)/5 + (2/7)*b*c*x^7 + (c^2*x^9)/9]
    @test_int [(b*x^2 + c*x^4)^2/x^1, x, 4, (b^2*x^4)/4 + (1/3)*b*c*x^6 + (c^2*x^8)/8]
    @test_int [(b*x^2 + c*x^4)^2/x^2, x, 3, (b^2*x^3)/3 + (2/5)*b*c*x^5 + (c^2*x^7)/7]
    @test_int [(b*x^2 + c*x^4)^2/x^3, x, 2, (b + c*x^2)^3/(6*c)]
    @test_int [(b*x^2 + c*x^4)^2/x^4, x, 3, b^2*x + (2/3)*b*c*x^3 + (c^2*x^5)/5]
    @test_int [(b*x^2 + c*x^4)^2/x^5, x, 4, b*c*x^2 + (c^2*x^4)/4 + b^2*log(x)]
    @test_int [(b*x^2 + c*x^4)^2/x^6, x, 3, -(b^2/x) + 2*b*c*x + (c^2*x^3)/3]
    @test_int [(b*x^2 + c*x^4)^2/x^7, x, 4, -(b^2/(2*x^2)) + (c^2*x^2)/2 + 2*b*c*log(x)]
    @test_int [(b*x^2 + c*x^4)^2/x^8, x, 3, -(b^2/(3*x^3)) - (2*b*c)/x + c^2*x]
    @test_int [(b*x^2 + c*x^4)^2/x^9, x, 4, -(b^2/(4*x^4)) - (b*c)/x^2 + c^2*log(x)]
    @test_int [(b*x^2 + c*x^4)^2/x^10, x, 3, -(b^2/(5*x^5)) - (2*b*c)/(3*x^3) - c^2/x]
    @test_int [(b*x^2 + c*x^4)^2/x^11, x, 2, -((b + c*x^2)^3/(6*b*x^6))]
    @test_int [(b*x^2 + c*x^4)^2/x^12, x, 3, -(b^2/(7*x^7)) - (2*b*c)/(5*x^5) - c^2/(3*x^3)]

    @test_int [(b*x^2 + c*x^4)^3/x^2, x, 3, (b^3*x^5)/5 + (3/7)*b^2*c*x^7 + (1/3)*b*c^2*x^9 + (c^3*x^11)/11]
    @test_int [(b*x^2 + c*x^4)^3/x^3, x, 4, -((b*(b + c*x^2)^4)/(8*c^2)) + (b + c*x^2)^5/(10*c^2)]
    @test_int [(b*x^2 + c*x^4)^3/x^4, x, 3, (b^3*x^3)/3 + (3/5)*b^2*c*x^5 + (3/7)*b*c^2*x^7 + (c^3*x^9)/9]
    @test_int [(b*x^2 + c*x^4)^3/x^5, x, 2, (b + c*x^2)^4/(8*c)]
    @test_int [(b*x^2 + c*x^4)^3/x^6, x, 3, b^3*x + b^2*c*x^3 + (3/5)*b*c^2*x^5 + (c^3*x^7)/7]
    @test_int [(b*x^2 + c*x^4)^3/x^7, x, 4, (3/2)*b^2*c*x^2 + (3/4)*b*c^2*x^4 + (c^3*x^6)/6 + b^3*log(x)]
    @test_int [(b*x^2 + c*x^4)^3/x^8, x, 3, -(b^3/x) + 3*b^2*c*x + b*c^2*x^3 + (c^3*x^5)/5]
    @test_int [(b*x^2 + c*x^4)^3/x^9, x, 4, -(b^3/(2*x^2)) + (3/2)*b*c^2*x^2 + (c^3*x^4)/4 + 3*b^2*c*log(x)]
    @test_int [(b*x^2 + c*x^4)^3/x^10, x, 3, -(b^3/(3*x^3)) - (3*b^2*c)/x + 3*b*c^2*x + (c^3*x^3)/3]
    @test_int [(b*x^2 + c*x^4)^3/x^11, x, 4, -(b^3/(4*x^4)) - (3*b^2*c)/(2*x^2) + (c^3*x^2)/2 + 3*b*c^2*log(x)]
    @test_int [(b*x^2 + c*x^4)^3/x^12, x, 3, -(b^3/(5*x^5)) - (b^2*c)/x^3 - (3*b*c^2)/x + c^3*x]
    @test_int [(b*x^2 + c*x^4)^3/x^13, x, 4, -(b^3/(6*x^6)) - (3*b^2*c)/(4*x^4) - (3*b*c^2)/(2*x^2) + c^3*log(x)]
    @test_int [(b*x^2 + c*x^4)^3/x^14, x, 3, -(b^3/(7*x^7)) - (3*b^2*c)/(5*x^5) - (b*c^2)/x^3 - c^3/x]
    @test_int [(b*x^2 + c*x^4)^3/x^15, x, 2, -((b + c*x^2)^4/(8*b*x^8))]
    @test_int [(b*x^2 + c*x^4)^3/x^16, x, 3, -(b^3/(9*x^9)) - (3*b^2*c)/(7*x^7) - (3*b*c^2)/(5*x^5) - c^3/(3*x^3)]
    @test_int [(b*x^2 + c*x^4)^3/x^17, x, 4, -((b + c*x^2)^4/(10*b*x^10)) + (c*(b + c*x^2)^4)/(40*b^2*x^8)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^10/(b*x^2 + c*x^4), x, 4, -((b^3*x)/c^4) + (b^2*x^3)/(3*c^3) - (b*x^5)/(5*c^2) + x^7/(7*c) + (b^(7/2)*atan((sqrt(c)*x)/sqrt(b)))/c^(9/2)]
    @test_int [x^9/(b*x^2 + c*x^4), x, 4, (b^2*x^2)/(2*c^3) - (b*x^4)/(4*c^2) + x^6/(6*c) - (b^3*log(b + c*x^2))/(2*c^4)]
    @test_int [x^8/(b*x^2 + c*x^4), x, 4, (b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*atan((sqrt(c)*x)/sqrt(b)))/c^(7/2)]
    @test_int [x^7/(b*x^2 + c*x^4), x, 4, -((b*x^2)/(2*c^2)) + x^4/(4*c) + (b^2*log(b + c*x^2))/(2*c^3)]
    @test_int [x^6/(b*x^2 + c*x^4), x, 4, -((b*x)/c^2) + x^3/(3*c) + (b^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/c^(5/2)]
    @test_int [x^5/(b*x^2 + c*x^4), x, 4, x^2/(2*c) - (b*log(b + c*x^2))/(2*c^2)]
    @test_int [x^4/(b*x^2 + c*x^4), x, 3, x/c - (sqrt(b)*atan((sqrt(c)*x)/sqrt(b)))/c^(3/2)]
    @test_int [x^3/(b*x^2 + c*x^4), x, 2, log(b + c*x^2)/(2*c)]
    @test_int [x^2/(b*x^2 + c*x^4), x, 2, atan((sqrt(c)*x)/sqrt(b))/(sqrt(b)*sqrt(c))]
    @test_int [x^1/(b*x^2 + c*x^4), x, 5, log(x)/b - log(b + c*x^2)/(2*b)]
    @test_int [x^0/(b*x^2 + c*x^4), x, 3, -(1/(b*x)) - (sqrt(c)*atan((sqrt(c)*x)/sqrt(b)))/b^(3/2)]
    @test_int [1/(x^1*(b*x^2 + c*x^4)), x, 4, -(1/(2*b*x^2)) - (c*log(x))/b^2 + (c*log(b + c*x^2))/(2*b^2)]
    @test_int [1/(x^2*(b*x^2 + c*x^4)), x, 4, -(1/(3*b*x^3)) + c/(b^2*x) + (c^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/b^(5/2)]
    @test_int [1/(x^3*(b*x^2 + c*x^4)), x, 4, -(1/(4*b*x^4)) + c/(2*b^2*x^2) + (c^2*log(x))/b^3 - (c^2*log(b + c*x^2))/(2*b^3)]
    @test_int [1/(x^4*(b*x^2 + c*x^4)), x, 5, -(1/(5*b*x^5)) + c/(3*b^2*x^3) - c^2/(b^3*x) - (c^(5/2)*atan((sqrt(c)*x)/sqrt(b)))/b^(7/2)]
    @test_int [1/(x^5*(b*x^2 + c*x^4)), x, 4, -(1/(6*b*x^6)) + c/(4*b^2*x^4) - c^2/(2*b^3*x^2) - (c^3*log(x))/b^4 + (c^3*log(b + c*x^2))/(2*b^4)]

    @test_int [x^12/(b*x^2 + c*x^4)^2, x, 5, (7*b^2*x)/(2*c^4) - (7*b*x^3)/(6*c^3) + (7*x^5)/(10*c^2) - x^7/(2*c*(b + c*x^2)) - (7*b^(5/2)*atan((sqrt(c)*x)/sqrt(b)))/(2*c^(9/2))]
    @test_int [x^11/(b*x^2 + c*x^4)^2, x, 4, -((b*x^2)/c^3) + x^4/(4*c^2) + b^3/(2*c^4*(b + c*x^2)) + (3*b^2*log(b + c*x^2))/(2*c^4)]
    @test_int [x^10/(b*x^2 + c*x^4)^2, x, 5, -((5*b*x)/(2*c^3)) + (5*x^3)/(6*c^2) - x^5/(2*c*(b + c*x^2)) + (5*b^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/(2*c^(7/2))]
    @test_int [x^9/(b*x^2 + c*x^4)^2, x, 4, x^2/(2*c^2) - b^2/(2*c^3*(b + c*x^2)) - (b*log(b + c*x^2))/c^3]
    @test_int [x^8/(b*x^2 + c*x^4)^2, x, 4, (3*x)/(2*c^2) - x^3/(2*c*(b + c*x^2)) - (3*sqrt(b)*atan((sqrt(c)*x)/sqrt(b)))/(2*c^(5/2))]
    @test_int [x^7/(b*x^2 + c*x^4)^2, x, 4, b/(2*c^2*(b + c*x^2)) + log(b + c*x^2)/(2*c^2)]
    @test_int [x^6/(b*x^2 + c*x^4)^2, x, 3, -(x/(2*c*(b + c*x^2))) + atan((sqrt(c)*x)/sqrt(b))/(2*sqrt(b)*c^(3/2))]
    @test_int [x^5/(b*x^2 + c*x^4)^2, x, 2, -(1/(2*c*(b + c*x^2)))]
    @test_int [x^4/(b*x^2 + c*x^4)^2, x, 3, x/(2*b*(b + c*x^2)) + atan((sqrt(c)*x)/sqrt(b))/(2*b^(3/2)*sqrt(c))]
    @test_int [x^3/(b*x^2 + c*x^4)^2, x, 4, 1/(2*b*(b + c*x^2)) + log(x)/b^2 - log(b + c*x^2)/(2*b^2)]
    @test_int [x^2/(b*x^2 + c*x^4)^2, x, 4, -(3/(2*b^2*x)) + 1/(2*b*x*(b + c*x^2)) - (3*sqrt(c)*atan((sqrt(c)*x)/sqrt(b)))/(2*b^(5/2))]
    @test_int [x^1/(b*x^2 + c*x^4)^2, x, 4, -(1/(2*b^2*x^2)) - c/(2*b^2*(b + c*x^2)) - (2*c*log(x))/b^3 + (c*log(b + c*x^2))/b^3]
    @test_int [x^0/(b*x^2 + c*x^4)^2, x, 5, -(5/(6*b^2*x^3)) + (5*c)/(2*b^3*x) + 1/(2*b*x^3*(b + c*x^2)) + (5*c^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/(2*b^(7/2))]
    @test_int [1/(x^1*(b*x^2 + c*x^4)^2), x, 4, -(1/(4*b^2*x^4)) + c/(b^3*x^2) + c^2/(2*b^3*(b + c*x^2)) + (3*c^2*log(x))/b^4 - (3*c^2*log(b + c*x^2))/(2*b^4)]
    @test_int [1/(x^2*(b*x^2 + c*x^4)^2), x, 6, -(7/(10*b^2*x^5)) + (7*c)/(6*b^3*x^3) - (7*c^2)/(2*b^4*x) + 1/(2*b*x^5*(b + c*x^2)) - (7*c^(5/2)*atan((sqrt(c)*x)/sqrt(b)))/(2*b^(9/2))]

    @test_int [x^14/(b*x^2 + c*x^4)^3, x, 6, -((35*b*x)/(8*c^4)) + (35*x^3)/(24*c^3) - x^7/(4*c*(b + c*x^2)^2) - (7*x^5)/(8*c^2*(b + c*x^2)) + (35*b^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/(8*c^(9/2))]
    @test_int [x^13/(b*x^2 + c*x^4)^3, x, 4, x^2/(2*c^3) + b^3/(4*c^4*(b + c*x^2)^2) - (3*b^2)/(2*c^4*(b + c*x^2)) - (3*b*log(b + c*x^2))/(2*c^4)]
    @test_int [x^12/(b*x^2 + c*x^4)^3, x, 5, (15*x)/(8*c^3) - x^5/(4*c*(b + c*x^2)^2) - (5*x^3)/(8*c^2*(b + c*x^2)) - (15*sqrt(b)*atan((sqrt(c)*x)/sqrt(b)))/(8*c^(7/2))]
    @test_int [x^11/(b*x^2 + c*x^4)^3, x, 4, -(b^2/(4*c^3*(b + c*x^2)^2)) + b/(c^3*(b + c*x^2)) + log(b + c*x^2)/(2*c^3)]
    @test_int [x^10/(b*x^2 + c*x^4)^3, x, 4, -(x^3/(4*c*(b + c*x^2)^2)) - (3*x)/(8*c^2*(b + c*x^2)) + (3*atan((sqrt(c)*x)/sqrt(b)))/(8*sqrt(b)*c^(5/2))]
    @test_int [x^9/(b*x^2 + c*x^4)^3, x, 2, x^4/(4*b*(b + c*x^2)^2)]
    @test_int [x^8/(b*x^2 + c*x^4)^3, x, 4, -(x/(4*c*(b + c*x^2)^2)) + x/(8*b*c*(b + c*x^2)) + atan((sqrt(c)*x)/sqrt(b))/(8*b^(3/2)*c^(3/2))]
    @test_int [x^7/(b*x^2 + c*x^4)^3, x, 2, -(1/(4*c*(b + c*x^2)^2))]
    @test_int [x^6/(b*x^2 + c*x^4)^3, x, 4, x/(4*b*(b + c*x^2)^2) + (3*x)/(8*b^2*(b + c*x^2)) + (3*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(5/2)*sqrt(c))]
    @test_int [x^5/(b*x^2 + c*x^4)^3, x, 4, 1/(4*b*(b + c*x^2)^2) + 1/(2*b^2*(b + c*x^2)) + log(x)/b^3 - log(b + c*x^2)/(2*b^3)]
    @test_int [x^4/(b*x^2 + c*x^4)^3, x, 5, -(15/(8*b^3*x)) + 1/(4*b*x*(b + c*x^2)^2) + 5/(8*b^2*x*(b + c*x^2)) - (15*sqrt(c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(7/2))]
    @test_int [x^3/(b*x^2 + c*x^4)^3, x, 4, -(1/(2*b^3*x^2)) - c/(4*b^2*(b + c*x^2)^2) - c/(b^3*(b + c*x^2)) - (3*c*log(x))/b^4 + (3*c*log(b + c*x^2))/(2*b^4)]
    @test_int [x^2/(b*x^2 + c*x^4)^3, x, 6, -(35/(24*b^3*x^3)) + (35*c)/(8*b^4*x) + 1/(4*b*x^3*(b + c*x^2)^2) + 7/(8*b^2*x^3*(b + c*x^2)) + (35*c^(3/2)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(9/2))]
    @test_int [x^1/(b*x^2 + c*x^4)^3, x, 4, -(1/(4*b^3*x^4)) + (3*c)/(2*b^4*x^2) + c^2/(4*b^3*(b + c*x^2)^2) + (3*c^2)/(2*b^4*(b + c*x^2)) + (6*c^2*log(x))/b^5 - (3*c^2*log(b + c*x^2))/b^5]
    @test_int [x^0/(b*x^2 + c*x^4)^3, x, 7, -(63/(40*b^3*x^5)) + (21*c)/(8*b^4*x^3) - (63*c^2)/(8*b^5*x) + 1/(4*b*x^5*(b + c*x^2)^2) + 9/(8*b^2*x^5*(b + c*x^2)) - (63*c^(5/2)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(11/2))]
    @test_int [1/(x^1*(b*x^2 + c*x^4)^3), x, 4, -(1/(6*b^3*x^6)) + (3*c)/(4*b^4*x^4) - (3*c^2)/(b^5*x^2) - c^3/(4*b^4*(b + c*x^2)^2) - (2*c^3)/(b^5*(b + c*x^2)) - (10*c^3*log(x))/b^6 + (5*c^3*log(b + c*x^2))/b^6]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(b*x^2+c*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(b*x^2 + c*x^4), x, 6, (5*b^2*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(128*c^3) - (5*b*(b*x^2 + c*x^4)^(3/2))/(48*c^2) + (x^2*(b*x^2 + c*x^4)^(3/2))/(8*c) - (5*b^4*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(128*c^(7/2))]
    @test_int [x^3*sqrt(b*x^2 + c*x^4), x, 5, -((b*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(16*c^2)) + (b*x^2 + c*x^4)^(3/2)/(6*c) + (b^3*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(5/2))]
    @test_int [x^1*sqrt(b*x^2 + c*x^4), x, 4, ((b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(8*c) - (b^2*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(3/2))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^1, x, 4, (1/2)*sqrt(b*x^2 + c*x^4) + (b*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*sqrt(c))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^3, x, 4, -(sqrt(b*x^2 + c*x^4)/x^2) + sqrt(c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^5, x, 1, -((b*x^2 + c*x^4)^(3/2)/(3*b*x^6))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^7, x, 2, -((b*x^2 + c*x^4)^(3/2)/(5*b*x^8)) + (2*c*(b*x^2 + c*x^4)^(3/2))/(15*b^2*x^6)]
    @test_int [sqrt(b*x^2 + c*x^4)/x^9, x, 3, -((b*x^2 + c*x^4)^(3/2)/(7*b*x^10)) + (4*c*(b*x^2 + c*x^4)^(3/2))/(35*b^2*x^8) - (8*c^2*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^6)]
    @test_int [sqrt(b*x^2 + c*x^4)/x^11, x, 4, -((b*x^2 + c*x^4)^(3/2)/(9*b*x^12)) + (2*c*(b*x^2 + c*x^4)^(3/2))/(21*b^2*x^10) - (8*c^2*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^8) + (16*c^3*(b*x^2 + c*x^4)^(3/2))/(315*b^4*x^6)]
    @test_int [sqrt(b*x^2 + c*x^4)/x^13, x, 5, -((b*x^2 + c*x^4)^(3/2)/(11*b*x^14)) + (8*c*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) - (16*c^2*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) + (64*c^3*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8) - (128*c^4*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)]

    @test_int [x^4*sqrt(b*x^2 + c*x^4), x, 3, (8*b^2*(b*x^2 + c*x^4)^(3/2))/(105*c^3*x^3) - (4*b*(b*x^2 + c*x^4)^(3/2))/(35*c^2*x) + (x*(b*x^2 + c*x^4)^(3/2))/(7*c)]
    @test_int [x^2*sqrt(b*x^2 + c*x^4), x, 2, -((2*b*(b*x^2 + c*x^4)^(3/2))/(15*c^2*x^3)) + (b*x^2 + c*x^4)^(3/2)/(5*c*x)]
    @test_int [x^0*sqrt(b*x^2 + c*x^4), x, 1, (b*x^2 + c*x^4)^(3/2)/(3*c*x^3)]
    @test_int [sqrt(b*x^2 + c*x^4)/x^2, x, 3, sqrt(b*x^2 + c*x^4)/x - sqrt(b)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^4, x, 3, -(sqrt(b*x^2 + c*x^4)/(2*x^3)) - (c*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*sqrt(b))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^6, x, 4, -(sqrt(b*x^2 + c*x^4)/(4*x^5)) - (c*sqrt(b*x^2 + c*x^4))/(8*b*x^3) + (c^2*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(3/2))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^8, x, 5, -(sqrt(b*x^2 + c*x^4)/(6*x^7)) - (c*sqrt(b*x^2 + c*x^4))/(24*b*x^5) + (c^2*sqrt(b*x^2 + c*x^4))/(16*b^2*x^3) - (c^3*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(16*b^(5/2))]

    @test_int [x^3*(b*x^2 + c*x^4)^(3/2), x, 6, (3*b^3*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(256*c^3) - (b*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(32*c^2) + (b*x^2 + c*x^4)^(5/2)/(10*c) - (3*b^5*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(256*c^(7/2))]
    @test_int [x^1*(b*x^2 + c*x^4)^(3/2), x, 5, -((3*b^2*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(128*c^2)) + ((b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(16*c) + (3*b^4*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(128*c^(5/2))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^1, x, 5, (b*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(16*c) + (1/6)*(b*x^2 + c*x^4)^(3/2) - (b^3*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(3/2))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^3, x, 5, (3/8)*b*sqrt(b*x^2 + c*x^4) + (b*x^2 + c*x^4)^(3/2)/(4*x^2) + (3*b^2*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*sqrt(c))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^5, x, 5, (3/2)*c*sqrt(b*x^2 + c*x^4) - (b*x^2 + c*x^4)^(3/2)/x^4 + (3/2)*b*sqrt(c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^7, x, 5, -((c*sqrt(b*x^2 + c*x^4))/x^2) - (b*x^2 + c*x^4)^(3/2)/(3*x^6) + c^(3/2)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^9, x, 1, -((b*x^2 + c*x^4)^(5/2)/(5*b*x^10))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^11, x, 2, -((b*x^2 + c*x^4)^(5/2)/(7*b*x^12)) + (2*c*(b*x^2 + c*x^4)^(5/2))/(35*b^2*x^10)]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^13, x, 3, -((b*x^2 + c*x^4)^(5/2)/(9*b*x^14)) + (4*c*(b*x^2 + c*x^4)^(5/2))/(63*b^2*x^12) - (8*c^2*(b*x^2 + c*x^4)^(5/2))/(315*b^3*x^10)]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^15, x, 4, -((b*x^2 + c*x^4)^(5/2)/(11*b*x^16)) + (2*c*(b*x^2 + c*x^4)^(5/2))/(33*b^2*x^14) - (8*c^2*(b*x^2 + c*x^4)^(5/2))/(231*b^3*x^12) + (16*c^3*(b*x^2 + c*x^4)^(5/2))/(1155*b^4*x^10)]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^17, x, 5, -((b*x^2 + c*x^4)^(5/2)/(13*b*x^18)) + (8*c*(b*x^2 + c*x^4)^(5/2))/(143*b^2*x^16) - (16*c^2*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^14) + (64*c^3*(b*x^2 + c*x^4)^(5/2))/(3003*b^4*x^12) - (128*c^4*(b*x^2 + c*x^4)^(5/2))/(15015*b^5*x^10)]

    @test_int [x^6*(b*x^2 + c*x^4)^(3/2), x, 5, (128*b^4*(b*x^2 + c*x^4)^(5/2))/(15015*c^5*x^5) - (64*b^3*(b*x^2 + c*x^4)^(5/2))/(3003*c^4*x^3) + (16*b^2*(b*x^2 + c*x^4)^(5/2))/(429*c^3*x) - (8*b*x*(b*x^2 + c*x^4)^(5/2))/(143*c^2) + (x^3*(b*x^2 + c*x^4)^(5/2))/(13*c)]
    @test_int [x^4*(b*x^2 + c*x^4)^(3/2), x, 4, -((16*b^3*(b*x^2 + c*x^4)^(5/2))/(1155*c^4*x^5)) + (8*b^2*(b*x^2 + c*x^4)^(5/2))/(231*c^3*x^3) - (2*b*(b*x^2 + c*x^4)^(5/2))/(33*c^2*x) + (x*(b*x^2 + c*x^4)^(5/2))/(11*c)]
    @test_int [x^2*(b*x^2 + c*x^4)^(3/2), x, 3, (8*b^2*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - (4*b*(b*x^2 + c*x^4)^(5/2))/(63*c^2*x^3) + (b*x^2 + c*x^4)^(5/2)/(9*c*x)]
    @test_int [x^0*(b*x^2 + c*x^4)^(3/2), x, 2, -((2*b*(b*x^2 + c*x^4)^(5/2))/(35*c^2*x^5)) + (b*x^2 + c*x^4)^(5/2)/(7*c*x^3)]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^2, x, 1, (b*x^2 + c*x^4)^(5/2)/(5*c*x^5)]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^4, x, 4, (b*sqrt(b*x^2 + c*x^4))/x + (b*x^2 + c*x^4)^(3/2)/(3*x^3) - b^(3/2)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^6, x, 4, (3*c*sqrt(b*x^2 + c*x^4))/(2*x) - (b*x^2 + c*x^4)^(3/2)/(2*x^5) - (3/2)*sqrt(b)*c*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^8, x, 4, -((3*c*sqrt(b*x^2 + c*x^4))/(8*x^3)) - (b*x^2 + c*x^4)^(3/2)/(4*x^7) - (3*c^2*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*sqrt(b))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^10, x, 5, -((c*sqrt(b*x^2 + c*x^4))/(8*x^5)) - (c^2*sqrt(b*x^2 + c*x^4))/(16*b*x^3) - (b*x^2 + c*x^4)^(3/2)/(6*x^9) + (c^3*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(16*b^(3/2))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^12, x, 6, -((c*sqrt(b*x^2 + c*x^4))/(16*x^7)) - (c^2*sqrt(b*x^2 + c*x^4))/(64*b*x^5) + (3*c^3*sqrt(b*x^2 + c*x^4))/(128*b^2*x^3) - (b*x^2 + c*x^4)^(3/2)/(8*x^11) - (3*c^4*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(128*b^(5/2))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^14, x, 7, -((3*c*sqrt(b*x^2 + c*x^4))/(80*x^9)) - (c^2*sqrt(b*x^2 + c*x^4))/(160*b*x^7) + (c^3*sqrt(b*x^2 + c*x^4))/(128*b^2*x^5) - (3*c^4*sqrt(b*x^2 + c*x^4))/(256*b^3*x^3) - (b*x^2 + c*x^4)^(3/2)/(10*x^13) + (3*c^5*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(256*b^(7/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^7/sqrt(b*x^2 + c*x^4), x, 6, (5*b^2*sqrt(b*x^2 + c*x^4))/(16*c^3) - (5*b*x^2*sqrt(b*x^2 + c*x^4))/(24*c^2) + (x^4*sqrt(b*x^2 + c*x^4))/(6*c) - (5*b^3*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(7/2))]
    @test_int [x^5/sqrt(b*x^2 + c*x^4), x, 5, -((3*b*sqrt(b*x^2 + c*x^4))/(8*c^2)) + (x^2*sqrt(b*x^2 + c*x^4))/(4*c) + (3*b^2*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(5/2))]
    @test_int [x^3/sqrt(b*x^2 + c*x^4), x, 4, sqrt(b*x^2 + c*x^4)/(2*c) - (b*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*c^(3/2))]
    @test_int [x^1/sqrt(b*x^2 + c*x^4), x, 3, atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))/sqrt(c)]
    @test_int [1/(x^1*sqrt(b*x^2 + c*x^4)), x, 1, -(sqrt(b*x^2 + c*x^4)/(b*x^2))]
    @test_int [1/(x^3*sqrt(b*x^2 + c*x^4)), x, 2, -(sqrt(b*x^2 + c*x^4)/(3*b*x^4)) + (2*c*sqrt(b*x^2 + c*x^4))/(3*b^2*x^2)]
    @test_int [1/(x^5*sqrt(b*x^2 + c*x^4)), x, 3, -(sqrt(b*x^2 + c*x^4)/(5*b*x^6)) + (4*c*sqrt(b*x^2 + c*x^4))/(15*b^2*x^4) - (8*c^2*sqrt(b*x^2 + c*x^4))/(15*b^3*x^2)]
    @test_int [1/(x^7*sqrt(b*x^2 + c*x^4)), x, 4, -(sqrt(b*x^2 + c*x^4)/(7*b*x^8)) + (6*c*sqrt(b*x^2 + c*x^4))/(35*b^2*x^6) - (8*c^2*sqrt(b*x^2 + c*x^4))/(35*b^3*x^4) + (16*c^3*sqrt(b*x^2 + c*x^4))/(35*b^4*x^2)]

    @test_int [x^4/sqrt(b*x^2 + c*x^4), x, 2, -((2*b*sqrt(b*x^2 + c*x^4))/(3*c^2*x)) + (x*sqrt(b*x^2 + c*x^4))/(3*c)]
    @test_int [x^2/sqrt(b*x^2 + c*x^4), x, 1, sqrt(b*x^2 + c*x^4)/(c*x)]
    @test_int [x^0/sqrt(b*x^2 + c*x^4), x, 2, -(atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))/sqrt(b))]
    @test_int [1/(x^2*sqrt(b*x^2 + c*x^4)), x, 3, -(sqrt(b*x^2 + c*x^4)/(2*b*x^3)) + (c*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*b^(3/2))]
    @test_int [1/(x^4*sqrt(b*x^2 + c*x^4)), x, 4, -(sqrt(b*x^2 + c*x^4)/(4*b*x^5)) + (3*c*sqrt(b*x^2 + c*x^4))/(8*b^2*x^3) - (3*c^2*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(5/2))]

    @test_int [x^9/(b*x^2 + c*x^4)^(3/2), x, 6, -(x^6/(c*sqrt(b*x^2 + c*x^4))) - (15*b*sqrt(b*x^2 + c*x^4))/(8*c^3) + (5*x^2*sqrt(b*x^2 + c*x^4))/(4*c^2) + (15*b^2*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(7/2))]
    @test_int [x^7/(b*x^2 + c*x^4)^(3/2), x, 5, -(x^4/(c*sqrt(b*x^2 + c*x^4))) + (3*sqrt(b*x^2 + c*x^4))/(2*c^2) - (3*b*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*c^(5/2))]
    @test_int [x^5/(b*x^2 + c*x^4)^(3/2), x, 4, -(x^2/(c*sqrt(b*x^2 + c*x^4))) + atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))/c^(3/2)]
    @test_int [x^3/(b*x^2 + c*x^4)^(3/2), x, 1, x^2/(b*sqrt(b*x^2 + c*x^4))]
    @test_int [x^1/(b*x^2 + c*x^4)^(3/2), x, 2, -((b + 2*c*x^2)/(b^2*sqrt(b*x^2 + c*x^4)))]
    @test_int [1/(x^1*(b*x^2 + c*x^4)^(3/2)), x, 3, 1/(b*x^2*sqrt(b*x^2 + c*x^4)) - (4*sqrt(b*x^2 + c*x^4))/(3*b^2*x^4) + (8*c*sqrt(b*x^2 + c*x^4))/(3*b^3*x^2)]
    @test_int [1/(x^3*(b*x^2 + c*x^4)^(3/2)), x, 4, 1/(b*x^4*sqrt(b*x^2 + c*x^4)) - (6*sqrt(b*x^2 + c*x^4))/(5*b^2*x^6) + (8*c*sqrt(b*x^2 + c*x^4))/(5*b^3*x^4) - (16*c^2*sqrt(b*x^2 + c*x^4))/(5*b^4*x^2)]
    @test_int [1/(x^5*(b*x^2 + c*x^4)^(3/2)), x, 5, 1/(b*x^6*sqrt(b*x^2 + c*x^4)) - (8*sqrt(b*x^2 + c*x^4))/(7*b^2*x^8) + (48*c*sqrt(b*x^2 + c*x^4))/(35*b^3*x^6) - (64*c^2*sqrt(b*x^2 + c*x^4))/(35*b^4*x^4) + (128*c^3*sqrt(b*x^2 + c*x^4))/(35*b^5*x^2)]

    @test_int [x^6/(b*x^2 + c*x^4)^(3/2), x, 2, -(x^3/(c*sqrt(b*x^2 + c*x^4))) + (2*sqrt(b*x^2 + c*x^4))/(c^2*x)]
    @test_int [x^4/(b*x^2 + c*x^4)^(3/2), x, 1, -(x/(c*sqrt(b*x^2 + c*x^4)))]
    @test_int [x^2/(b*x^2 + c*x^4)^(3/2), x, 3, x/(b*sqrt(b*x^2 + c*x^4)) - atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))/b^(3/2)]
    @test_int [x^0/(b*x^2 + c*x^4)^(3/2), x, 4, 1/(b*x*sqrt(b*x^2 + c*x^4)) - (3*sqrt(b*x^2 + c*x^4))/(2*b^2*x^3) + (3*c*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*b^(5/2))]
    @test_int [1/(x^2*(b*x^2 + c*x^4)^(3/2)), x, 5, 1/(b*x^3*sqrt(b*x^2 + c*x^4)) - (5*sqrt(b*x^2 + c*x^4))/(4*b^2*x^5) + (15*c*sqrt(b*x^2 + c*x^4))/(8*b^3*x^3) - (15*c^2*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(7/2))]

    @test_int [x^3/sqrt(3*x^2 - 4*x^4), x, 4, (-(1/8))*sqrt(3*x^2 - 4*x^4) - (3/32)*asin(1 - (8*x^2)/3)]
    @test_int [x^3/sqrt(-3*x^2 - 4*x^4), x, 4, (-(1/8))*sqrt(-3*x^2 - 4*x^4) - (3/32)*asin(1 + (8*x^2)/3)]
    @test_int [x^3/sqrt(3*x^2 + 4*x^4), x, 4, (1/8)*sqrt(3*x^2 + 4*x^4) - (3/16)*atanh((2*x^2)/sqrt(3*x^2 + 4*x^4))]
    @test_int [x^3/sqrt(-3*x^2 + 4*x^4), x, 4, (1/8)*sqrt(-3*x^2 + 4*x^4) + (3/16)*atanh((2*x^2)/sqrt(-3*x^2 + 4*x^4))]
    @test_int [x^3/sqrt(a*x^2 + b*x^4), x, 4, sqrt(a*x^2 + b*x^4)/(2*b) - (a*atanh((sqrt(b)*x^2)/sqrt(a*x^2 + b*x^4)))/(2*b^(3/2))]
    @test_int [x^3/sqrt(a*x^2 - b*x^4), x, 4, -(sqrt(a*x^2 - b*x^4)/(2*b)) + (a*atan((sqrt(b)*x^2)/sqrt(a*x^2 - b*x^4)))/(2*b^(3/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(b*x^2+c*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^(7/2)*(b*x^2 + c*x^4), x, 2, (2/13)*b*x^(13/2) + (2/17)*c*x^(17/2)]
    @test_int [x^(5/2)*(b*x^2 + c*x^4), x, 2, (2/11)*b*x^(11/2) + (2/15)*c*x^(15/2)]
    @test_int [x^(3/2)*(b*x^2 + c*x^4), x, 2, (2/9)*b*x^(9/2) + (2/13)*c*x^(13/2)]
    @test_int [x^(1/2)*(b*x^2 + c*x^4), x, 2, (2/7)*b*x^(7/2) + (2/11)*c*x^(11/2)]
    @test_int [(b*x^2 + c*x^4)/x^(1/2), x, 2, (2/5)*b*x^(5/2) + (2/9)*c*x^(9/2)]
    @test_int [(b*x^2 + c*x^4)/x^(3/2), x, 2, (2/3)*b*x^(3/2) + (2/7)*c*x^(7/2)]
    @test_int [(b*x^2 + c*x^4)/x^(5/2), x, 2, 2*b*sqrt(x) + (2/5)*c*x^(5/2)]
    @test_int [(b*x^2 + c*x^4)/x^(7/2), x, 2, -((2*b)/sqrt(x)) + (2/3)*c*x^(3/2)]

    @test_int [x^(7/2)*(b*x^2 + c*x^4)^2, x, 3, (2/17)*b^2*x^(17/2) + (4/21)*b*c*x^(21/2) + (2/25)*c^2*x^(25/2)]
    @test_int [x^(5/2)*(b*x^2 + c*x^4)^2, x, 3, (2/15)*b^2*x^(15/2) + (4/19)*b*c*x^(19/2) + (2/23)*c^2*x^(23/2)]
    @test_int [x^(3/2)*(b*x^2 + c*x^4)^2, x, 3, (2/13)*b^2*x^(13/2) + (4/17)*b*c*x^(17/2) + (2/21)*c^2*x^(21/2)]
    @test_int [x^(1/2)*(b*x^2 + c*x^4)^2, x, 3, (2/11)*b^2*x^(11/2) + (4/15)*b*c*x^(15/2) + (2/19)*c^2*x^(19/2)]
    @test_int [(b*x^2 + c*x^4)^2/x^(1/2), x, 3, (2/9)*b^2*x^(9/2) + (4/13)*b*c*x^(13/2) + (2/17)*c^2*x^(17/2)]
    @test_int [(b*x^2 + c*x^4)^2/x^(3/2), x, 3, (2/7)*b^2*x^(7/2) + (4/11)*b*c*x^(11/2) + (2/15)*c^2*x^(15/2)]
    @test_int [(b*x^2 + c*x^4)^2/x^(5/2), x, 3, (2/5)*b^2*x^(5/2) + (4/9)*b*c*x^(9/2) + (2/13)*c^2*x^(13/2)]
    @test_int [(b*x^2 + c*x^4)^2/x^(7/2), x, 3, (2/3)*b^2*x^(3/2) + (4/7)*b*c*x^(7/2) + (2/11)*c^2*x^(11/2)]

    @test_int [x^(7/2)*(b*x^2 + c*x^4)^3, x, 3, (2/21)*b^3*x^(21/2) + (6/25)*b^2*c*x^(25/2) + (6/29)*b*c^2*x^(29/2) + (2/33)*c^3*x^(33/2)]
    @test_int [x^(5/2)*(b*x^2 + c*x^4)^3, x, 3, (2/19)*b^3*x^(19/2) + (6/23)*b^2*c*x^(23/2) + (2/9)*b*c^2*x^(27/2) + (2/31)*c^3*x^(31/2)]
    @test_int [x^(3/2)*(b*x^2 + c*x^4)^3, x, 3, (2/17)*b^3*x^(17/2) + (2/7)*b^2*c*x^(21/2) + (6/25)*b*c^2*x^(25/2) + (2/29)*c^3*x^(29/2)]
    @test_int [x^(1/2)*(b*x^2 + c*x^4)^3, x, 3, (2/15)*b^3*x^(15/2) + (6/19)*b^2*c*x^(19/2) + (6/23)*b*c^2*x^(23/2) + (2/27)*c^3*x^(27/2)]
    @test_int [(b*x^2 + c*x^4)^3/x^(1/2), x, 3, (2/13)*b^3*x^(13/2) + (6/17)*b^2*c*x^(17/2) + (2/7)*b*c^2*x^(21/2) + (2/25)*c^3*x^(25/2)]
    @test_int [(b*x^2 + c*x^4)^3/x^(3/2), x, 3, (2/11)*b^3*x^(11/2) + (2/5)*b^2*c*x^(15/2) + (6/19)*b*c^2*x^(19/2) + (2/23)*c^3*x^(23/2)]
    @test_int [(b*x^2 + c*x^4)^3/x^(5/2), x, 3, (2/9)*b^3*x^(9/2) + (6/13)*b^2*c*x^(13/2) + (6/17)*b*c^2*x^(17/2) + (2/21)*c^3*x^(21/2)]
    @test_int [(b*x^2 + c*x^4)^3/x^(7/2), x, 3, (2/7)*b^3*x^(7/2) + (6/11)*b^2*c*x^(11/2) + (2/5)*b*c^2*x^(15/2) + (2/19)*c^3*x^(19/2)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^(13/2)/(b*x^2 + c*x^4), x, 13, (-2*b*x^(3/2))/(3*c^2) + (2*x^(7/2))/(7*c) - (b^(7/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(11/4)) + (b^(7/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(11/4)) + (b^(7/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(11/4)) - (b^(7/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(11/4))]
    @test_int [x^(11/2)/(b*x^2 + c*x^4), x, 13, (-2*b*sqrt(x))/c^2 + (2*x^(5/2))/(5*c) - (b^(5/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(9/4)) + (b^(5/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(9/4)) - (b^(5/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(9/4)) + (b^(5/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(9/4))]
    @test_int [x^(9/2)/(b*x^2 + c*x^4), x, 12, (2*x^(3/2))/(3*c) + (b^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(7/4)) - (b^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(7/4)) - (b^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(7/4)) + (b^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(7/4))]
    @test_int [x^(7/2)/(b*x^2 + c*x^4), x, 12, (2*sqrt(x))/c + (b^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(5/4)) - (b^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(5/4)) + (b^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(5/4)) - (b^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(5/4))]
    @test_int [x^(5/2)/(b*x^2 + c*x^4), x, 11, -(atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(sqrt(2)*b^(1/4)*c^(3/4))) + atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(sqrt(2)*b^(1/4)*c^(3/4)) + log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(2*sqrt(2)*b^(1/4)*c^(3/4)) - log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(2*sqrt(2)*b^(1/4)*c^(3/4))]
    @test_int [x^(3/2)/(b*x^2 + c*x^4), x, 11, -(atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(sqrt(2)*b^(3/4)*c^(1/4))) + atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(sqrt(2)*b^(3/4)*c^(1/4)) - log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(2*sqrt(2)*b^(3/4)*c^(1/4)) + log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(2*sqrt(2)*b^(3/4)*c^(1/4))]
    @test_int [sqrt(x)/(b*x^2 + c*x^4), x, 12, -2/(b*sqrt(x)) + (c^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(5/4)) - (c^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(5/4)) - (c^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(5/4)) + (c^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(5/4))]
    @test_int [1/(sqrt(x)*(b*x^2 + c*x^4)), x, 12, -2/(3*b*x^(3/2)) + (c^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(7/4)) - (c^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(7/4)) + (c^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(7/4)) - (c^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(7/4))]
    @test_int [1/(x^(3/2)*(b*x^2 + c*x^4)), x, 13, -2/(5*b*x^(5/2)) + (2*c)/(b^2*sqrt(x)) - (c^(5/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(9/4)) + (c^(5/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(9/4)) + (c^(5/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(9/4)) - (c^(5/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(9/4))]
    @test_int [1/(x^(5/2)*(b*x^2 + c*x^4)), x, 13, -2/(7*b*x^(7/2)) + (2*c)/(3*b^2*x^(3/2)) - (c^(7/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(11/4)) + (c^(7/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(11/4)) - (c^(7/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(11/4)) + (c^(7/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(11/4))]
    @test_int [1/(x^(7/2)*(b*x^2 + c*x^4)), x, 14, -2/(9*b*x^(9/2)) + (2*c)/(5*b^2*x^(5/2)) - (2*c^2)/(b^3*sqrt(x)) + (c^(9/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(13/4)) - (c^(9/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(13/4)) - (c^(9/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(13/4)) + (c^(9/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(13/4))]

    @test_int [x^(19/2)/(b*x^2 + c*x^4)^2, x, 14, (-9*b*sqrt(x))/(2*c^3) + (9*x^(5/2))/(10*c^2) - x^(9/2)/(2*c*(b + c*x^2)) - (9*b^(5/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(13/4)) + (9*b^(5/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(13/4)) - (9*b^(5/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(13/4)) + (9*b^(5/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(13/4))]
    @test_int [x^(17/2)/(b*x^2 + c*x^4)^2, x, 13, (7*x^(3/2))/(6*c^2) - x^(7/2)/(2*c*(b + c*x^2)) + (7*b^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(11/4)) - (7*b^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(11/4)) - (7*b^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(11/4)) + (7*b^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(11/4))]
    @test_int [x^(15/2)/(b*x^2 + c*x^4)^2, x, 13, (5*sqrt(x))/(2*c^2) - x^(5/2)/(2*c*(b + c*x^2)) + (5*b^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(9/4)) - (5*b^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(9/4)) + (5*b^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(9/4)) - (5*b^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(9/4))]
    @test_int [x^(13/2)/(b*x^2 + c*x^4)^2, x, 12, -x^(3/2)/(2*c*(b + c*x^2)) - (3*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(1/4)*c^(7/4)) + (3*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(1/4)*c^(7/4)) + (3*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(1/4)*c^(7/4)) - (3*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(1/4)*c^(7/4))]
    @test_int [x^(11/2)/(b*x^2 + c*x^4)^2, x, 12, -sqrt(x)/(2*c*(b + c*x^2)) - atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(4*sqrt(2)*b^(3/4)*c^(5/4)) + atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(4*sqrt(2)*b^(3/4)*c^(5/4)) - log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(8*sqrt(2)*b^(3/4)*c^(5/4)) + log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(8*sqrt(2)*b^(3/4)*c^(5/4))]
    @test_int [x^(9/2)/(b*x^2 + c*x^4)^2, x, 12, x^(3/2)/(2*b*(b + c*x^2)) - atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(4*sqrt(2)*b^(5/4)*c^(3/4)) + atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4))/(4*sqrt(2)*b^(5/4)*c^(3/4)) + log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(8*sqrt(2)*b^(5/4)*c^(3/4)) - log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x)/(8*sqrt(2)*b^(5/4)*c^(3/4))]
    @test_int [x^(7/2)/(b*x^2 + c*x^4)^2, x, 12, sqrt(x)/(2*b*(b + c*x^2)) - (3*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(7/4)*c^(1/4)) + (3*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(7/4)*c^(1/4)) - (3*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(7/4)*c^(1/4)) + (3*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(7/4)*c^(1/4))]
    @test_int [x^(5/2)/(b*x^2 + c*x^4)^2, x, 13, -5/(2*b^2*sqrt(x)) + 1/(2*b*sqrt(x)*(b + c*x^2)) + (5*c^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(9/4)) - (5*c^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(9/4)) - (5*c^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(9/4)) + (5*c^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(9/4))]
    @test_int [x^(3/2)/(b*x^2 + c*x^4)^2, x, 13, -7/(6*b^2*x^(3/2)) + 1/(2*b*x^(3/2)*(b + c*x^2)) + (7*c^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(11/4)) - (7*c^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(11/4)) + (7*c^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(11/4)) - (7*c^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(11/4))]
    @test_int [sqrt(x)/(b*x^2 + c*x^4)^2, x, 14, -9/(10*b^2*x^(5/2)) + (9*c)/(2*b^3*sqrt(x)) + 1/(2*b*x^(5/2)*(b + c*x^2)) - (9*c^(5/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(13/4)) + (9*c^(5/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(13/4)) + (9*c^(5/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(13/4)) - (9*c^(5/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(13/4))]
    @test_int [1/(sqrt(x)*(b*x^2 + c*x^4)^2), x, 14, -11/(14*b^2*x^(7/2)) + (11*c)/(6*b^3*x^(3/2)) + 1/(2*b*x^(7/2)*(b + c*x^2)) - (11*c^(7/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(15/4)) + (11*c^(7/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(15/4)) - (11*c^(7/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(15/4)) + (11*c^(7/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(15/4))]
    @test_int [1/(x^(3/2)*(b*x^2 + c*x^4)^2), x, 15, -13/(18*b^2*x^(9/2)) + (13*c)/(10*b^3*x^(5/2)) - (13*c^2)/(2*b^4*sqrt(x)) + 1/(2*b*x^(9/2)*(b + c*x^2)) + (13*c^(9/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(17/4)) - (13*c^(9/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(17/4)) - (13*c^(9/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(17/4)) + (13*c^(9/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(17/4))]

    @test_int [x^(23/2)/(b*x^2 + c*x^4)^3, x, 14, (45*sqrt(x))/(16*c^3) - x^(9/2)/(4*c*(b + c*x^2)^2) - (9*x^(5/2))/(16*c^2*(b + c*x^2)) + (45*b^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*c^(13/4)) - (45*b^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*c^(13/4)) + (45*b^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*c^(13/4)) - (45*b^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*c^(13/4))]
    @test_int [x^(21/2)/(b*x^2 + c*x^4)^3, x, 13, -x^(7/2)/(4*c*(b + c*x^2)^2) - (7*x^(3/2))/(16*c^2*(b + c*x^2)) - (21*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(1/4)*c^(11/4)) + (21*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(1/4)*c^(11/4)) + (21*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(1/4)*c^(11/4)) - (21*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(1/4)*c^(11/4))]
    @test_int [x^(19/2)/(b*x^2 + c*x^4)^3, x, 13, -x^(5/2)/(4*c*(b + c*x^2)^2) - (5*sqrt(x))/(16*c^2*(b + c*x^2)) - (5*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(3/4)*c^(9/4)) + (5*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(3/4)*c^(9/4)) - (5*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(3/4)*c^(9/4)) + (5*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(3/4)*c^(9/4))]
    @test_int [x^(17/2)/(b*x^2 + c*x^4)^3, x, 13, -x^(3/2)/(4*c*(b + c*x^2)^2) + (3*x^(3/2))/(16*b*c*(b + c*x^2)) - (3*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(5/4)*c^(7/4)) + (3*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(5/4)*c^(7/4)) + (3*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(5/4)*c^(7/4)) - (3*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(5/4)*c^(7/4))]
    @test_int [x^(15/2)/(b*x^2 + c*x^4)^3, x, 13, -sqrt(x)/(4*c*(b + c*x^2)^2) + sqrt(x)/(16*b*c*(b + c*x^2)) - (3*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(7/4)*c^(5/4)) + (3*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(7/4)*c^(5/4)) - (3*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(7/4)*c^(5/4)) + (3*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(7/4)*c^(5/4))]
    @test_int [x^(13/2)/(b*x^2 + c*x^4)^3, x, 13, x^(3/2)/(4*b*(b + c*x^2)^2) + (5*x^(3/2))/(16*b^2*(b + c*x^2)) - (5*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(9/4)*c^(3/4)) + (5*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(9/4)*c^(3/4)) + (5*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(9/4)*c^(3/4)) - (5*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(9/4)*c^(3/4))]
    @test_int [x^(11/2)/(b*x^2 + c*x^4)^3, x, 13, sqrt(x)/(4*b*(b + c*x^2)^2) + (7*sqrt(x))/(16*b^2*(b + c*x^2)) - (21*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(11/4)*c^(1/4)) + (21*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(11/4)*c^(1/4)) - (21*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(11/4)*c^(1/4)) + (21*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(11/4)*c^(1/4))]
    @test_int [x^(9/2)/(b*x^2 + c*x^4)^3, x, 14, -45/(16*b^3*sqrt(x)) + 1/(4*b*sqrt(x)*(b + c*x^2)^2) + 9/(16*b^2*sqrt(x)*(b + c*x^2)) + (45*c^(1/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(13/4)) - (45*c^(1/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(13/4)) - (45*c^(1/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(13/4)) + (45*c^(1/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(13/4))]
    @test_int [x^(7/2)/(b*x^2 + c*x^4)^3, x, 14, -77/(48*b^3*x^(3/2)) + 1/(4*b*x^(3/2)*(b + c*x^2)^2) + 11/(16*b^2*x^(3/2)*(b + c*x^2)) + (77*c^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(15/4)) - (77*c^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(15/4)) + (77*c^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(15/4)) - (77*c^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(15/4))]
    @test_int [x^(5/2)/(b*x^2 + c*x^4)^3, x, 15, -117/(80*b^3*x^(5/2)) + (117*c)/(16*b^4*sqrt(x)) + 1/(4*b*x^(5/2)*(b + c*x^2)^2) + 13/(16*b^2*x^(5/2)*(b + c*x^2)) - (117*c^(5/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(17/4)) + (117*c^(5/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(17/4)) + (117*c^(5/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(17/4)) - (117*c^(5/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(17/4))]
    @test_int [x^(3/2)/(b*x^2 + c*x^4)^3, x, 15, -165/(112*b^3*x^(7/2)) + (55*c)/(16*b^4*x^(3/2)) + 1/(4*b*x^(7/2)*(b + c*x^2)^2) + 15/(16*b^2*x^(7/2)*(b + c*x^2)) - (165*c^(7/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(19/4)) + (165*c^(7/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(19/4)) - (165*c^(7/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(19/4)) + (165*c^(7/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(19/4))]
    @test_int [sqrt(x)/(b*x^2 + c*x^4)^3, x, 16, -221/(144*b^3*x^(9/2)) + (221*c)/(80*b^4*x^(5/2)) - (221*c^2)/(16*b^5*sqrt(x)) + 1/(4*b*x^(9/2)*(b + c*x^2)^2) + 17/(16*b^2*x^(9/2)*(b + c*x^2)) + (221*c^(9/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(21/4)) - (221*c^(9/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(21/4)) - (221*c^(9/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(21/4)) + (221*c^(9/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(21/4))]
    @test_int [1/(sqrt(x)*(b*x^2 + c*x^4)^3), x, 16, -285/(176*b^3*x^(11/2)) + (285*c)/(112*b^4*x^(7/2)) - (95*c^2)/(16*b^5*x^(3/2)) + 1/(4*b*x^(11/2)*(b + c*x^2)^2) + 19/(16*b^2*x^(11/2)*(b + c*x^2)) + (285*c^(11/4)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(23/4)) - (285*c^(11/4)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(23/4)) + (285*c^(11/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(23/4)) - (285*c^(11/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(23/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(b*x^2+c*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^(7/2)*sqrt(b*x^2 + c*x^4), x, 8, (28*b^3*x^(3/2)*(b + c*x^2))/(195*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (28*b^2*sqrt(x)*sqrt(b*x^2 + c*x^4))/(585*c^2) + (4*b*x^(5/2)*sqrt(b*x^2 + c*x^4))/(117*c) + (2/13)*x^(9/2)*sqrt(b*x^2 + c*x^4) - (28*b^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (14*b^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(5/2)*sqrt(b*x^2 + c*x^4), x, 6, -((20*b^2*sqrt(b*x^2 + c*x^4))/(231*c^2*sqrt(x))) + (4*b*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c) + (2/11)*x^(7/2)*sqrt(b*x^2 + c*x^4) + (10*b^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(3/2)*sqrt(b*x^2 + c*x^4), x, 7, -((4*b^2*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (4*b*sqrt(x)*sqrt(b*x^2 + c*x^4))/(45*c) + (2/9)*x^(5/2)*sqrt(b*x^2 + c*x^4) + (4*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (2*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)*sqrt(b*x^2 + c*x^4), x, 5, (4*b*sqrt(b*x^2 + c*x^4))/(21*c*sqrt(x)) + (2/7)*x^(3/2)*sqrt(b*x^2 + c*x^4) - (2*b^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/sqrt(x), x, 6, (4*b*x^(3/2)*(b + c*x^2))/(5*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (2/5)*sqrt(x)*sqrt(b*x^2 + c*x^4) - (4*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(3/4)*sqrt(b*x^2 + c*x^4)) + (2*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(3/2), x, 4, (2*sqrt(b*x^2 + c*x^4))/(3*sqrt(x)) + (2*b^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(5/2), x, 6, (4*sqrt(c)*x^(3/2)*(b + c*x^2))/((sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*sqrt(b*x^2 + c*x^4))/x^(3/2) - (4*b^(1/4)*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/sqrt(b*x^2 + c*x^4) + (2*b^(1/4)*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/sqrt(b*x^2 + c*x^4)]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(7/2), x, 4, -((2*sqrt(b*x^2 + c*x^4))/(3*x^(5/2))) + (2*c^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*b^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(9/2), x, 7, (4*c^(3/2)*x^(3/2)*(b + c*x^2))/(5*b*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*sqrt(b*x^2 + c*x^4))/(5*x^(7/2)) - (4*c*sqrt(b*x^2 + c*x^4))/(5*b*x^(3/2)) - (4*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^2 + c*x^4)) + (2*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(11/2), x, 5, -((2*sqrt(b*x^2 + c*x^4))/(7*x^(9/2))) - (4*c*sqrt(b*x^2 + c*x^4))/(21*b*x^(5/2)) - (2*c^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*b^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(13/2), x, 8, -((4*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (2*sqrt(b*x^2 + c*x^4))/(9*x^(11/2)) - (4*c*sqrt(b*x^2 + c*x^4))/(45*b*x^(7/2)) + (4*c^2*sqrt(b*x^2 + c*x^4))/(15*b^2*x^(3/2)) + (4*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(7/4)*sqrt(b*x^2 + c*x^4)) - (2*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(b*x^2 + c*x^4)/x^(15/2), x, 6, -((2*sqrt(b*x^2 + c*x^4))/(11*x^(13/2))) - (4*c*sqrt(b*x^2 + c*x^4))/(77*b*x^(9/2)) + (20*c^2*sqrt(b*x^2 + c*x^4))/(231*b^2*x^(5/2)) + (10*c^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*b^(9/4)*sqrt(b*x^2 + c*x^4))]

    @test_int [x^(3/2)*(b*x^2 + c*x^4)^(3/2), x, 9, (56*b^4*x^(3/2)*(b + c*x^2))/(1105*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (56*b^3*sqrt(x)*sqrt(b*x^2 + c*x^4))/(3315*c^2) + (8*b^2*x^(5/2)*sqrt(b*x^2 + c*x^4))/(663*c) + (12/221)*b*x^(9/2)*sqrt(b*x^2 + c*x^4) + (2/17)*x^(5/2)*(b*x^2 + c*x^4)^(3/2) - (56*b^(17/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(1105*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (28*b^(17/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(1105*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)*(b*x^2 + c*x^4)^(3/2), x, 7, -((8*b^3*sqrt(b*x^2 + c*x^4))/(231*c^2*sqrt(x))) + (8*b^2*x^(3/2)*sqrt(b*x^2 + c*x^4))/(385*c) + (4/55)*b*x^(7/2)*sqrt(b*x^2 + c*x^4) + (2/15)*x^(3/2)*(b*x^2 + c*x^4)^(3/2) + (4*b^(15/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/sqrt(x), x, 8, -((8*b^3*x^(3/2)*(b + c*x^2))/(65*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (8*b^2*sqrt(x)*sqrt(b*x^2 + c*x^4))/(195*c) + (4/39)*b*x^(5/2)*sqrt(b*x^2 + c*x^4) + (2/13)*sqrt(x)*(b*x^2 + c*x^4)^(3/2) + (8*b^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(65*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (4*b^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(65*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(3/2), x, 6, (8*b^2*sqrt(b*x^2 + c*x^4))/(77*c*sqrt(x)) + (12/77)*b*x^(3/2)*sqrt(b*x^2 + c*x^4) + (2*(b*x^2 + c*x^4)^(3/2))/(11*sqrt(x)) - (4*b^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(5/2), x, 7, (8*b^2*x^(3/2)*(b + c*x^2))/(15*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (4/15)*b*sqrt(x)*sqrt(b*x^2 + c*x^4) + (2*(b*x^2 + c*x^4)^(3/2))/(9*x^(3/2)) - (8*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(3/4)*sqrt(b*x^2 + c*x^4)) + (4*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(7/2), x, 5, (4*b*sqrt(b*x^2 + c*x^4))/(7*sqrt(x)) + (2*(b*x^2 + c*x^4)^(3/2))/(7*x^(5/2)) + (4*b^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(7*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(9/2), x, 7, (24*b*sqrt(c)*x^(3/2)*(b + c*x^2))/(5*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (12/5)*c*sqrt(x)*sqrt(b*x^2 + c*x^4) - (2*(b*x^2 + c*x^4)^(3/2))/x^(7/2) - (24*b^(5/4)*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4)) + (12*b^(5/4)*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(11/2), x, 5, (4*c*sqrt(b*x^2 + c*x^4))/(3*sqrt(x)) - (2*(b*x^2 + c*x^4)^(3/2))/(3*x^(9/2)) + (4*b^(3/4)*c^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(13/2), x, 7, (24*c^(3/2)*x^(3/2)*(b + c*x^2))/(5*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (12*c*sqrt(b*x^2 + c*x^4))/(5*x^(3/2)) - (2*(b*x^2 + c*x^4)^(3/2))/(5*x^(11/2)) - (24*b^(1/4)*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4)) + (12*b^(1/4)*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(15/2), x, 5, -((4*c*sqrt(b*x^2 + c*x^4))/(7*x^(5/2))) - (2*(b*x^2 + c*x^4)^(3/2))/(7*x^(13/2)) + (4*c^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(7*b^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(17/2), x, 8, (8*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (4*c*sqrt(b*x^2 + c*x^4))/(15*x^(7/2)) - (8*c^2*sqrt(b*x^2 + c*x^4))/(15*b*x^(3/2)) - (2*(b*x^2 + c*x^4)^(3/2))/(9*x^(15/2)) - (8*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(3/4)*sqrt(b*x^2 + c*x^4)) + (4*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(19/2), x, 6, -((12*c*sqrt(b*x^2 + c*x^4))/(77*x^(9/2))) - (8*c^2*sqrt(b*x^2 + c*x^4))/(77*b*x^(5/2)) - (2*(b*x^2 + c*x^4)^(3/2))/(11*x^(17/2)) - (4*c^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*b^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(21/2), x, 9, -((8*c^(7/2)*x^(3/2)*(b + c*x^2))/(65*b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (4*c*sqrt(b*x^2 + c*x^4))/(39*x^(11/2)) - (8*c^2*sqrt(b*x^2 + c*x^4))/(195*b*x^(7/2)) + (8*c^3*sqrt(b*x^2 + c*x^4))/(65*b^2*x^(3/2)) - (2*(b*x^2 + c*x^4)^(3/2))/(13*x^(19/2)) + (8*c^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(65*b^(7/4)*sqrt(b*x^2 + c*x^4)) - (4*c^(13/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(65*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(b*x^2 + c*x^4)^(3/2)/x^(23/2), x, 7, -((4*c*sqrt(b*x^2 + c*x^4))/(55*x^(13/2))) - (8*c^2*sqrt(b*x^2 + c*x^4))/(385*b*x^(9/2)) + (8*c^3*sqrt(b*x^2 + c*x^4))/(231*b^2*x^(5/2)) - (2*(b*x^2 + c*x^4)^(3/2))/(15*x^(21/2)) + (4*c^(15/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*b^(9/4)*sqrt(b*x^2 + c*x^4))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^(13/2)/sqrt(b*x^2 + c*x^4), x, 6, (30*b^2*sqrt(b*x^2 + c*x^4))/(77*c^3*sqrt(x)) - (18*b*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c^2) + (2*x^(7/2)*sqrt(b*x^2 + c*x^4))/(11*c) - (15*b^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(11/2)/sqrt(b*x^2 + c*x^4), x, 7, (14*b^2*x^(3/2)*(b + c*x^2))/(15*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (14*b*sqrt(x)*sqrt(b*x^2 + c*x^4))/(45*c^2) + (2*x^(5/2)*sqrt(b*x^2 + c*x^4))/(9*c) - (14*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (7*b^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(9/2)/sqrt(b*x^2 + c*x^4), x, 5, -((10*b*sqrt(b*x^2 + c*x^4))/(21*c^2*sqrt(x))) + (2*x^(3/2)*sqrt(b*x^2 + c*x^4))/(7*c) + (5*b^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(7/2)/sqrt(b*x^2 + c*x^4), x, 6, -((6*b*x^(3/2)*(b + c*x^2))/(5*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (2*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*c) + (6*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (3*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(5/2)/sqrt(b*x^2 + c*x^4), x, 4, (2*sqrt(b*x^2 + c*x^4))/(3*c*sqrt(x)) - (b^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(3/2)/sqrt(b*x^2 + c*x^4), x, 5, (2*x^(3/2)*(b + c*x^2))/(sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*b^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(c^(3/4)*sqrt(b*x^2 + c*x^4)) + (b^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)/sqrt(b*x^2 + c*x^4), x, 3, (x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(1/4)*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(sqrt(x)*sqrt(b*x^2 + c*x^4)), x, 6, (2*sqrt(c)*x^(3/2)*(b + c*x^2))/(b*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*sqrt(b*x^2 + c*x^4))/(b*x^(3/2)) - (2*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*sqrt(b*x^2 + c*x^4)) + (c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(3/2)*sqrt(b*x^2 + c*x^4)), x, 4, -((2*sqrt(b*x^2 + c*x^4))/(3*b*x^(5/2))) - (c^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*b^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(5/2)*sqrt(b*x^2 + c*x^4)), x, 7, -((6*c^(3/2)*x^(3/2)*(b + c*x^2))/(5*b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (2*sqrt(b*x^2 + c*x^4))/(5*b*x^(7/2)) + (6*c*sqrt(b*x^2 + c*x^4))/(5*b^2*x^(3/2)) + (6*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(7/4)*sqrt(b*x^2 + c*x^4)) - (3*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(7/2)*sqrt(b*x^2 + c*x^4)), x, 5, -((2*sqrt(b*x^2 + c*x^4))/(7*b*x^(9/2))) + (10*c*sqrt(b*x^2 + c*x^4))/(21*b^2*x^(5/2)) + (5*c^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*b^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(9/2)*sqrt(b*x^2 + c*x^4)), x, 8, (14*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b^3*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*sqrt(b*x^2 + c*x^4))/(9*b*x^(11/2)) + (14*c*sqrt(b*x^2 + c*x^4))/(45*b^2*x^(7/2)) - (14*c^2*sqrt(b*x^2 + c*x^4))/(15*b^3*x^(3/2)) - (14*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(11/4)*sqrt(b*x^2 + c*x^4)) + (7*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(11/2)*sqrt(b*x^2 + c*x^4)), x, 6, -((2*sqrt(b*x^2 + c*x^4))/(11*b*x^(13/2))) + (18*c*sqrt(b*x^2 + c*x^4))/(77*b^2*x^(9/2)) - (30*c^2*sqrt(b*x^2 + c*x^4))/(77*b^3*x^(5/2)) - (15*c^(11/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*b^(13/4)*sqrt(b*x^2 + c*x^4))]

    @test_int [x^(17/2)/(b*x^2 + c*x^4)^(3/2), x, 6, -(x^(11/2)/(c*sqrt(b*x^2 + c*x^4))) - (15*b*sqrt(b*x^2 + c*x^4))/(7*c^3*sqrt(x)) + (9*x^(3/2)*sqrt(b*x^2 + c*x^4))/(7*c^2) + (15*b^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(14*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(15/2)/(b*x^2 + c*x^4)^(3/2), x, 7, -(x^(9/2)/(c*sqrt(b*x^2 + c*x^4))) - (21*b*x^(3/2)*(b + c*x^2))/(5*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (7*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*c^2) + (21*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(11/4)*sqrt(b*x^2 + c*x^4)) - (21*b^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(10*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(13/2)/(b*x^2 + c*x^4)^(3/2), x, 5, -(x^(7/2)/(c*sqrt(b*x^2 + c*x^4))) + (5*sqrt(b*x^2 + c*x^4))/(3*c^2*sqrt(x)) - (5*b^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(6*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(11/2)/(b*x^2 + c*x^4)^(3/2), x, 6, -(x^(5/2)/(c*sqrt(b*x^2 + c*x^4))) + (3*x^(3/2)*(b + c*x^2))/(c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (3*b^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(c^(7/4)*sqrt(b*x^2 + c*x^4)) + (3*b^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(9/2)/(b*x^2 + c*x^4)^(3/2), x, 4, -(x^(3/2)/(c*sqrt(b*x^2 + c*x^4))) + (x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(1/4)*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(7/2)/(b*x^2 + c*x^4)^(3/2), x, 6, x^(5/2)/(b*sqrt(b*x^2 + c*x^4)) - (x^(3/2)*(b + c*x^2))/(b*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*c^(3/4)*sqrt(b*x^2 + c*x^4)) - (x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(3/4)*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(5/2)/(b*x^2 + c*x^4)^(3/2), x, 4, x^(3/2)/(b*sqrt(b*x^2 + c*x^4)) + (x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(5/4)*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(3/2)/(b*x^2 + c*x^4)^(3/2), x, 7, sqrt(x)/(b*sqrt(b*x^2 + c*x^4)) + (3*sqrt(c)*x^(3/2)*(b + c*x^2))/(b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (3*sqrt(b*x^2 + c*x^4))/(b^2*x^(3/2)) - (3*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(7/4)*sqrt(b*x^2 + c*x^4)) + (3*c^(1/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)/(b*x^2 + c*x^4)^(3/2), x, 5, 1/(b*sqrt(x)*sqrt(b*x^2 + c*x^4)) - (5*sqrt(b*x^2 + c*x^4))/(3*b^2*x^(5/2)) - (5*c^(3/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(6*b^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(sqrt(x)*(b*x^2 + c*x^4)^(3/2)), x, 8, 1/(b*x^(3/2)*sqrt(b*x^2 + c*x^4)) - (21*c^(3/2)*x^(3/2)*(b + c*x^2))/(5*b^3*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (7*sqrt(b*x^2 + c*x^4))/(5*b^2*x^(7/2)) + (21*c*sqrt(b*x^2 + c*x^4))/(5*b^3*x^(3/2)) + (21*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(11/4)*sqrt(b*x^2 + c*x^4)) - (21*c^(5/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(10*b^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(3/2)*(b*x^2 + c*x^4)^(3/2)), x, 6, 1/(b*x^(5/2)*sqrt(b*x^2 + c*x^4)) - (9*sqrt(b*x^2 + c*x^4))/(7*b^2*x^(9/2)) + (15*c*sqrt(b*x^2 + c*x^4))/(7*b^3*x^(5/2)) + (15*c^(7/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(14*b^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [1/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)), x, 9, 1/(b*x^(7/2)*sqrt(b*x^2 + c*x^4)) + (77*c^(5/2)*x^(3/2)*(b + c*x^2))/(15*b^4*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (11*sqrt(b*x^2 + c*x^4))/(9*b^2*x^(11/2)) + (77*c*sqrt(b*x^2 + c*x^4))/(45*b^3*x^(7/2)) - (77*c^2*sqrt(b*x^2 + c*x^4))/(15*b^4*x^(3/2)) - (77*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(15/4)*sqrt(b*x^2 + c*x^4)) + (77*c^(9/4)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(30*b^(15/4)*sqrt(b*x^2 + c*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(b*x^2+c*x^4)^p*with*m*symbolic=#

    @test_int [(c*x)^m*(b*x^2 + c*x^4)^3, x, 4, (b^3*x^7*(c*x)^m)/(7 + m) + (3*b^2*c*x^9*(c*x)^m)/(9 + m) + (3*b*c^2*x^11*(c*x)^m)/(11 + m) + (c^3*x^13*(c*x)^m)/(13 + m)]
    @test_int [(c*x)^m*(b*x^2 + c*x^4)^2, x, 4, (b^2*x^5*(c*x)^m)/(5 + m) + (2*b*c*x^7*(c*x)^m)/(7 + m) + (c^2*x^9*(c*x)^m)/(9 + m)]
    @test_int [(c*x)^m*(b*x^2 + c*x^4)^1, x, 2, (b*(c*x)^(3 + m))/(c^3*(3 + m)) + (c*x)^(5 + m)/(c^4*(5 + m))]
    @test_int [(c*x)^m/(b*x^2 + c*x^4)^1, x, 3, -(((c*x)^m*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + m), (1 + m)/2, -((c*x^2)/b)))/(b*(1 - m)*x))]
    @test_int [(c*x)^m/(b*x^2 + c*x^4)^2, x, 3, -(((c*x)^m*HypergeometricFunctions._₂F₁(2, (1/2)*(-3 + m), (1/2)*(-1 + m), -((c*x^2)/b)))/(b^2*(3 - m)*x^3))]
    @test_int [(c*x)^m/(b*x^2 + c*x^4)^3, x, 3, -(((c*x)^m*HypergeometricFunctions._₂F₁(3, (1/2)*(-5 + m), (1/2)*(-3 + m), -((c*x^2)/b)))/(b^3*(5 - m)*x^5))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p*with*b^2-4*a*c=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a^2+2*a*b*x^2+b^2*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (a^2*x^4)/4 + (1/3)*a*b*x^6 + (b^2*x^8)/8]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (a^2*x^3)/3 + (2/5)*a*b*x^5 + (b^2*x^7)/7]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (a^2*x^2)/2 + (1/2)*a*b*x^4 + (b^2*x^6)/6]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 1, a^2*x + (2/3)*a*b*x^3 + (b^2*x^5)/5]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^1, x, 2, a*b*x^2 + (b^2*x^4)/4 + a^2*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^2, x, 2, -(a^2/x) + 2*a*b*x + (b^2*x^3)/3]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3, x, 2, -(a^2/(2*x^2)) + (b^2*x^2)/2 + 2*a*b*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^4, x, 2, -(a^2/(3*x^3)) - (2*a*b)/x + b^2*x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5, x, 2, -(a^2/(4*x^4)) - (a*b)/x^2 + b^2*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^6, x, 2, -(a^2/(5*x^5)) - (2*a*b)/(3*x^3) - b^2/x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7, x, 2, -(a^2/(6*x^6)) - (a*b)/(2*x^4) - b^2/(2*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8, x, 2, -(a^2/(7*x^7)) - (2*a*b)/(5*x^5) - b^2/(3*x^3)]

    @test_int [x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (a^4*x^7)/7 + (4/9)*a^3*b*x^9 + (6/11)*a^2*b^2*x^11 + (4/13)*a*b^3*x^13 + (b^4*x^15)/15]
    @test_int [x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, (a^2*(a + b*x^2)^5)/(10*b^3) - (a*(a + b*x^2)^6)/(6*b^3) + (a + b*x^2)^7/(14*b^3)]
    @test_int [x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (a^4*x^5)/5 + (4/7)*a^3*b*x^7 + (2/3)*a^2*b^2*x^9 + (4/11)*a*b^3*x^11 + (b^4*x^13)/13]
    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, -((a*(a + b*x^2)^5)/(10*b^2)) + (a + b*x^2)^6/(12*b^2)]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (a^4*x^3)/3 + (4/5)*a^3*b*x^5 + (6/7)*a^2*b^2*x^7 + (4/9)*a*b^3*x^9 + (b^4*x^11)/11]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 2, (a + b*x^2)^5/(10*b)]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, a^4*x + (4/3)*a^3*b*x^3 + (6/5)*a^2*b^2*x^5 + (4/7)*a*b^3*x^7 + (b^4*x^9)/9]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^1, x, 4, 2*a^3*b*x^2 + (3/2)*a^2*b^2*x^4 + (2/3)*a*b^3*x^6 + (b^4*x^8)/8 + a^4*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^2, x, 3, -(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4/5)*a*b^3*x^5 + (b^4*x^7)/7]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^3, x, 4, -(a^4/(2*x^2)) + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^4, x, 3, -(a^4/(3*x^3)) - (4*a^3*b)/x + 6*a^2*b^2*x + (4/3)*a*b^3*x^3 + (b^4*x^5)/5]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^5, x, 4, -(a^4/(4*x^4)) - (2*a^3*b)/x^2 + 2*a*b^3*x^2 + (b^4*x^4)/4 + 6*a^2*b^2*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^6, x, 3, -(a^4/(5*x^5)) - (4*a^3*b)/(3*x^3) - (6*a^2*b^2)/x + 4*a*b^3*x + (b^4*x^3)/3]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^7, x, 4, -(a^4/(6*x^6)) - (a^3*b)/x^4 - (3*a^2*b^2)/x^2 + (b^4*x^2)/2 + 4*a*b^3*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^8, x, 3, -(a^4/(7*x^7)) - (4*a^3*b)/(5*x^5) - (2*a^2*b^2)/x^3 - (4*a*b^3)/x + b^4*x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^9, x, 4, -(a^4/(8*x^8)) - (2*a^3*b)/(3*x^6) - (3*a^2*b^2)/(2*x^4) - (2*a*b^3)/x^2 + b^4*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^10, x, 3, -(a^4/(9*x^9)) - (4*a^3*b)/(7*x^7) - (6*a^2*b^2)/(5*x^5) - (4*a*b^3)/(3*x^3) - b^4/x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^11, x, 2, -((a + b*x^2)^5/(10*a*x^10))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^12, x, 3, -(a^4/(11*x^11)) - (4*a^3*b)/(9*x^9) - (6*a^2*b^2)/(7*x^7) - (4*a*b^3)/(5*x^5) - b^4/(3*x^3)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^13, x, 4, -((a + b*x^2)^5/(12*a*x^12)) + (b*(a + b*x^2)^5)/(60*a^2*x^10)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^14, x, 3, -(a^4/(13*x^13)) - (4*a^3*b)/(11*x^11) - (2*a^2*b^2)/(3*x^9) - (4*a*b^3)/(7*x^7) - b^4/(5*x^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^15, x, 4, -(a^4/(14*x^14)) - (a^3*b)/(3*x^12) - (3*a^2*b^2)/(5*x^10) - (a*b^3)/(2*x^8) - b^4/(6*x^6)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16, x, 3, -(a^4/(15*x^15)) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9) - b^4/(7*x^7)]

    @test_int [x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (a^6*x^9)/9 + (6/11)*a^5*b*x^11 + (15/13)*a^4*b^2*x^13 + (4/3)*a^3*b^3*x^15 + (15/17)*a^2*b^4*x^17 + (6/19)*a*b^5*x^19 + (b^6*x^21)/21]
    @test_int [x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, -((a^3*(a + b*x^2)^7)/(14*b^4)) + (3*a^2*(a + b*x^2)^8)/(16*b^4) - (a*(a + b*x^2)^9)/(6*b^4) + (a + b*x^2)^10/(20*b^4)]
    @test_int [x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (a^6*x^7)/7 + (2/3)*a^5*b*x^9 + (15/11)*a^4*b^2*x^11 + (20/13)*a^3*b^3*x^13 + a^2*b^4*x^15 + (6/17)*a*b^5*x^17 + (b^6*x^19)/19]
    @test_int [x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, (a^2*(a + b*x^2)^7)/(14*b^3) - (a*(a + b*x^2)^8)/(8*b^3) + (a + b*x^2)^9/(18*b^3)]
    @test_int [x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (a^6*x^5)/5 + (6/7)*a^5*b*x^7 + (5/3)*a^4*b^2*x^9 + (20/11)*a^3*b^3*x^11 + (15/13)*a^2*b^4*x^13 + (2/5)*a*b^5*x^15 + (b^6*x^17)/17]
    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, -((a*(a + b*x^2)^7)/(14*b^2)) + (a + b*x^2)^8/(16*b^2)]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (a^6*x^3)/3 + (6/5)*a^5*b*x^5 + (15/7)*a^4*b^2*x^7 + (20/9)*a^3*b^3*x^9 + (15/11)*a^2*b^4*x^11 + (6/13)*a*b^5*x^13 + (b^6*x^15)/15]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 2, (a + b*x^2)^7/(14*b)]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, a^6*x + 2*a^5*b*x^3 + 3*a^4*b^2*x^5 + (20/7)*a^3*b^3*x^7 + (5/3)*a^2*b^4*x^9 + (6/11)*a*b^5*x^11 + (b^6*x^13)/13]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^1, x, 4, 3*a^5*b*x^2 + (15/4)*a^4*b^2*x^4 + (10/3)*a^3*b^3*x^6 + (15/8)*a^2*b^4*x^8 + (3/5)*a*b^5*x^10 + (b^6*x^12)/12 + a^6*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^2, x, 3, -(a^6/x) + 6*a^5*b*x + 5*a^4*b^2*x^3 + 4*a^3*b^3*x^5 + (15/7)*a^2*b^4*x^7 + (2/3)*a*b^5*x^9 + (b^6*x^11)/11]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^3, x, 4, -(a^6/(2*x^2)) + (15/2)*a^4*b^2*x^2 + 5*a^3*b^3*x^4 + (5/2)*a^2*b^4*x^6 + (3/4)*a*b^5*x^8 + (b^6*x^10)/10 + 6*a^5*b*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^4, x, 3, -(a^6/(3*x^3)) - (6*a^5*b)/x + 15*a^4*b^2*x + (20/3)*a^3*b^3*x^3 + 3*a^2*b^4*x^5 + (6/7)*a*b^5*x^7 + (b^6*x^9)/9]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^5, x, 4, -(a^6/(4*x^4)) - (3*a^5*b)/x^2 + 10*a^3*b^3*x^2 + (15/4)*a^2*b^4*x^4 + a*b^5*x^6 + (b^6*x^8)/8 + 15*a^4*b^2*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^6, x, 3, -(a^6/(5*x^5)) - (2*a^5*b)/x^3 - (15*a^4*b^2)/x + 20*a^3*b^3*x + 5*a^2*b^4*x^3 + (6/5)*a*b^5*x^5 + (b^6*x^7)/7]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7, x, 4, -(a^6/(6*x^6)) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15/2)*a^2*b^4*x^2 + (3/2)*a*b^5*x^4 + (b^6*x^6)/6 + 20*a^3*b^3*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8, x, 3, -(a^6/(7*x^7)) - (6*a^5*b)/(5*x^5) - (5*a^4*b^2)/x^3 - (20*a^3*b^3)/x + 15*a^2*b^4*x + 2*a*b^5*x^3 + (b^6*x^5)/5]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^9, x, 4, -(a^6/(8*x^8)) - (a^5*b)/x^6 - (15*a^4*b^2)/(4*x^4) - (10*a^3*b^3)/x^2 + 3*a*b^5*x^2 + (b^6*x^4)/4 + 15*a^2*b^4*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10, x, 3, -(a^6/(9*x^9)) - (6*a^5*b)/(7*x^7) - (3*a^4*b^2)/x^5 - (20*a^3*b^3)/(3*x^3) - (15*a^2*b^4)/x + 6*a*b^5*x + (b^6*x^3)/3]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11, x, 4, -(a^6/(10*x^10)) - (3*a^5*b)/(4*x^8) - (5*a^4*b^2)/(2*x^6) - (5*a^3*b^3)/x^4 - (15*a^2*b^4)/(2*x^2) + (b^6*x^2)/2 + 6*a*b^5*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^12, x, 3, -(a^6/(11*x^11)) - (2*a^5*b)/(3*x^9) - (15*a^4*b^2)/(7*x^7) - (4*a^3*b^3)/x^5 - (5*a^2*b^4)/x^3 - (6*a*b^5)/x + b^6*x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13, x, 4, -(a^6/(12*x^12)) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6) - (15*a^2*b^4)/(4*x^4) - (3*a*b^5)/x^2 + b^6*log(x)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^14, x, 3, -(a^6/(13*x^13)) - (6*a^5*b)/(11*x^11) - (5*a^4*b^2)/(3*x^9) - (20*a^3*b^3)/(7*x^7) - (3*a^2*b^4)/x^5 - (2*a*b^5)/x^3 - b^6/x]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^15, x, 2, -((a + b*x^2)^7/(14*a*x^14))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^16, x, 3, -(a^6/(15*x^15)) - (6*a^5*b)/(13*x^13) - (15*a^4*b^2)/(11*x^11) - (20*a^3*b^3)/(9*x^9) - (15*a^2*b^4)/(7*x^7) - (6*a*b^5)/(5*x^5) - b^6/(3*x^3)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^17, x, 4, -((a + b*x^2)^7/(16*a*x^16)) + (b*(a + b*x^2)^7)/(112*a^2*x^14)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^18, x, 3, -(a^6/(17*x^17)) - (2*a^5*b)/(5*x^15) - (15*a^4*b^2)/(13*x^13) - (20*a^3*b^3)/(11*x^11) - (5*a^2*b^4)/(3*x^9) - (6*a*b^5)/(7*x^7) - b^6/(5*x^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19, x, 5, -((a + b*x^2)^7/(18*a*x^18)) + (b*(a + b*x^2)^7)/(72*a^2*x^16) - (b^2*(a + b*x^2)^7)/(504*a^3*x^14)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^20, x, 3, -(a^6/(19*x^19)) - (6*a^5*b)/(17*x^17) - (a^4*b^2)/x^15 - (20*a^3*b^3)/(13*x^13) - (15*a^2*b^4)/(11*x^11) - (2*a*b^5)/(3*x^9) - b^6/(7*x^7)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^21, x, 6, -((a + b*x^2)^7/(20*a*x^20)) + (b*(a + b*x^2)^7)/(60*a^2*x^18) - (b^2*(a + b*x^2)^7)/(240*a^3*x^16) + (b^3*(a + b*x^2)^7)/(1680*a^4*x^14)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^22, x, 3, -(a^6/(21*x^21)) - (6*a^5*b)/(19*x^19) - (15*a^4*b^2)/(17*x^17) - (4*a^3*b^3)/(3*x^15) - (15*a^2*b^4)/(13*x^13) - (6*a*b^5)/(11*x^11) - b^6/(9*x^9)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^11/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, -((2*a^3*x^2)/b^5) + (3*a^2*x^4)/(4*b^4) - (a*x^6)/(3*b^3) + x^8/(8*b^2) + a^5/(2*b^6*(a + b*x^2)) + (5*a^4*log(a + b*x^2))/(2*b^6)]
    @test_int [x^9/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, (3*a^2*x^2)/(2*b^4) - (a*x^4)/(2*b^3) + x^6/(6*b^2) - a^4/(2*b^5*(a + b*x^2)) - (2*a^3*log(a + b*x^2))/b^5]
    @test_int [x^7/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, -((a*x^2)/b^3) + x^4/(4*b^2) + a^3/(2*b^4*(a + b*x^2)) + (3*a^2*log(a + b*x^2))/(2*b^4)]
    @test_int [x^5/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, x^2/(2*b^2) - a^2/(2*b^3*(a + b*x^2)) - (a*log(a + b*x^2))/b^3]
    @test_int [x^3/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, a/(2*b^2*(a + b*x^2)) + log(a + b*x^2)/(2*b^2)]
    @test_int [x^1/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, -(1/(2*b*(a + b*x^2)))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, 1/(2*a*(a + b*x^2)) + log(x)/a^2 - log(a + b*x^2)/(2*a^2)]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, -(1/(2*a^2*x^2)) - b/(2*a^2*(a + b*x^2)) - (2*b*log(x))/a^3 + (b*log(a + b*x^2))/a^3]
    @test_int [1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, -(1/(4*a^2*x^4)) + b/(a^3*x^2) + b^2/(2*a^3*(a + b*x^2)) + (3*b^2*log(x))/a^4 - (3*b^2*log(a + b*x^2))/(2*a^4)]

    @test_int [x^10/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 5, -((9*a^3*x)/(2*b^5)) + (3*a^2*x^3)/(2*b^4) - (9*a*x^5)/(10*b^3) + (9*x^7)/(14*b^2) - x^9/(2*b*(a + b*x^2)) + (9*a^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(11/2))]
    @test_int [x^8/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 5, (7*a^2*x)/(2*b^4) - (7*a*x^3)/(6*b^3) + (7*x^5)/(10*b^2) - x^7/(2*b*(a + b*x^2)) - (7*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(9/2))]
    @test_int [x^6/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 5, -((5*a*x)/(2*b^3)) + (5*x^3)/(6*b^2) - x^5/(2*b*(a + b*x^2)) + (5*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(7/2))]
    @test_int [x^4/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, (3*x)/(2*b^2) - x^3/(2*b*(a + b*x^2)) - (3*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(5/2))]
    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, -(x/(2*b*(a + b*x^2))) + atan((sqrt(b)*x)/sqrt(a))/(2*sqrt(a)*b^(3/2))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, x/(2*a*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(2*a^(3/2)*sqrt(b))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, -(3/(2*a^2*x)) + 1/(2*a*x*(a + b*x^2)) - (3*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 5, -(5/(6*a^2*x^3)) + (5*b)/(2*a^3*x) + 1/(2*a*x^3*(a + b*x^2)) + (5*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2))]
    @test_int [1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 6, -(7/(10*a^2*x^5)) + (7*b)/(6*a^3*x^3) - (7*b^2)/(2*a^4*x) + 1/(2*a*x^5*(a + b*x^2)) - (7*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(9/2))]

    @test_int [x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, -((2*a*x^2)/b^5) + x^4/(4*b^4) + a^5/(6*b^6*(a + b*x^2)^3) - (5*a^4)/(4*b^6*(a + b*x^2)^2) + (5*a^3)/(b^6*(a + b*x^2)) + (5*a^2*log(a + b*x^2))/b^6]
    @test_int [x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, x^2/(2*b^4) - a^4/(6*b^5*(a + b*x^2)^3) + a^3/(b^5*(a + b*x^2)^2) - (3*a^2)/(b^5*(a + b*x^2)) - (2*a*log(a + b*x^2))/b^5]
    @test_int [x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, a^3/(6*b^4*(a + b*x^2)^3) - (3*a^2)/(4*b^4*(a + b*x^2)^2) + (3*a)/(2*b^4*(a + b*x^2)) + log(a + b*x^2)/(2*b^4)]
    @test_int [x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 2, x^6/(6*a*(a + b*x^2)^3)]
    @test_int [x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 4, a/(6*b^2*(a + b*x^2)^3) - 1/(4*b^2*(a + b*x^2)^2)]
    @test_int [x^1/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 2, -(1/(6*b*(a + b*x^2)^3))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 4, 1/(6*a*(a + b*x^2)^3) + 1/(4*a^2*(a + b*x^2)^2) + 1/(2*a^3*(a + b*x^2)) + log(x)/a^4 - log(a + b*x^2)/(2*a^4)]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 4, -(1/(2*a^4*x^2)) - b/(6*a^2*(a + b*x^2)^3) - b/(2*a^3*(a + b*x^2)^2) - (3*b)/(2*a^4*(a + b*x^2)) - (4*b*log(x))/a^5 + (2*b*log(a + b*x^2))/a^5]
    @test_int [1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 4, -(1/(4*a^4*x^4)) + (2*b)/(a^5*x^2) + b^2/(6*a^3*(a + b*x^2)^3) + (3*b^2)/(4*a^4*(a + b*x^2)^2) + (3*b^2)/(a^5*(a + b*x^2)) + (10*b^2*log(x))/a^6 - (5*b^2*log(a + b*x^2))/a^6]

    @test_int [x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 7, (231*a^2*x)/(16*b^6) - (77*a*x^3)/(16*b^5) + (231*x^5)/(80*b^4) - x^11/(6*b*(a + b*x^2)^3) - (11*x^9)/(24*b^2*(a + b*x^2)^2) - (33*x^7)/(16*b^3*(a + b*x^2)) - (231*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(16*b^(13/2))]
    @test_int [x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 7, -((105*a*x)/(16*b^5)) + (35*x^3)/(16*b^4) - x^9/(6*b*(a + b*x^2)^3) - (3*x^7)/(8*b^2*(a + b*x^2)^2) - (21*x^5)/(16*b^3*(a + b*x^2)) + (105*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(16*b^(11/2))]
    @test_int [x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 6, (35*x)/(16*b^4) - x^7/(6*b*(a + b*x^2)^3) - (7*x^5)/(24*b^2*(a + b*x^2)^2) - (35*x^3)/(48*b^3*(a + b*x^2)) - (35*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(16*b^(9/2))]
    @test_int [x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 5, -(x^5/(6*b*(a + b*x^2)^3)) - (5*x^3)/(24*b^2*(a + b*x^2)^2) - (5*x)/(16*b^3*(a + b*x^2)) + (5*atan((sqrt(b)*x)/sqrt(a)))/(16*sqrt(a)*b^(7/2))]
    @test_int [x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 5, -(x^3/(6*b*(a + b*x^2)^3)) - x/(8*b^2*(a + b*x^2)^2) + x/(16*a*b^2*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(16*a^(3/2)*b^(5/2))]
    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 5, -(x/(6*b*(a + b*x^2)^3)) + x/(24*a*b*(a + b*x^2)^2) + x/(16*a^2*b*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(16*a^(5/2)*b^(3/2))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 5, x/(6*a*(a + b*x^2)^3) + (5*x)/(24*a^2*(a + b*x^2)^2) + (5*x)/(16*a^3*(a + b*x^2)) + (5*atan((sqrt(b)*x)/sqrt(a)))/(16*a^(7/2)*sqrt(b))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 6, -(35/(16*a^4*x)) + 1/(6*a*x*(a + b*x^2)^3) + 7/(24*a^2*x*(a + b*x^2)^2) + 35/(48*a^3*x*(a + b*x^2)) - (35*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(16*a^(9/2))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 7, -(35/(16*a^4*x^3)) + (105*b)/(16*a^5*x) + 1/(6*a*x^3*(a + b*x^2)^3) + 3/(8*a^2*x^3*(a + b*x^2)^2) + 21/(16*a^3*x^3*(a + b*x^2)) + (105*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(16*a^(11/2))]
    @test_int [1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 8, -(231/(80*a^4*x^5)) + (77*b)/(16*a^5*x^3) - (231*b^2)/(16*a^6*x) + 1/(6*a*x^5*(a + b*x^2)^3) + 11/(24*a^2*x^5*(a + b*x^2)^2) + 33/(16*a^3*x^5*(a + b*x^2)) - (231*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(16*a^(13/2))]

    @test_int [x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, -((3*a*x^2)/b^7) + x^4/(4*b^6) + a^7/(10*b^8*(a + b*x^2)^5) - (7*a^6)/(8*b^8*(a + b*x^2)^4) + (7*a^5)/(2*b^8*(a + b*x^2)^3) - (35*a^4)/(4*b^8*(a + b*x^2)^2) + (35*a^3)/(2*b^8*(a + b*x^2)) + (21*a^2*log(a + b*x^2))/(2*b^8)]
    @test_int [x^13/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, x^2/(2*b^6) - a^6/(10*b^7*(a + b*x^2)^5) + (3*a^5)/(4*b^7*(a + b*x^2)^4) - (5*a^4)/(2*b^7*(a + b*x^2)^3) + (5*a^3)/(b^7*(a + b*x^2)^2) - (15*a^2)/(2*b^7*(a + b*x^2)) - (3*a*log(a + b*x^2))/b^7]
    @test_int [x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, a^5/(10*b^6*(a + b*x^2)^5) - (5*a^4)/(8*b^6*(a + b*x^2)^4) + (5*a^3)/(3*b^6*(a + b*x^2)^3) - (5*a^2)/(2*b^6*(a + b*x^2)^2) + (5*a)/(2*b^6*(a + b*x^2)) + log(a + b*x^2)/(2*b^6)]
    @test_int [x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 2, x^10/(10*a*(a + b*x^2)^5)]
    @test_int [x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, x^8/(10*a*(a + b*x^2)^5) + x^8/(40*a^2*(a + b*x^2)^4)]
    @test_int [x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, -(a^2/(10*b^3*(a + b*x^2)^5)) + a/(4*b^3*(a + b*x^2)^4) - 1/(6*b^3*(a + b*x^2)^3)]
    @test_int [x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 4, a/(10*b^2*(a + b*x^2)^5) - 1/(8*b^2*(a + b*x^2)^4)]
    @test_int [x^1/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 2, -(1/(10*b*(a + b*x^2)^5))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 4, 1/(10*a*(a + b*x^2)^5) + 1/(8*a^2*(a + b*x^2)^4) + 1/(6*a^3*(a + b*x^2)^3) + 1/(4*a^4*(a + b*x^2)^2) + 1/(2*a^5*(a + b*x^2)) + log(x)/a^6 - log(a + b*x^2)/(2*a^6)]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 4, -(1/(2*a^6*x^2)) - b/(10*a^2*(a + b*x^2)^5) - b/(4*a^3*(a + b*x^2)^4) - b/(2*a^4*(a + b*x^2)^3) - b/(a^5*(a + b*x^2)^2) - (5*b)/(2*a^6*(a + b*x^2)) - (6*b*log(x))/a^7 + (3*b*log(a + b*x^2))/a^7]
    @test_int [1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 4, -(1/(4*a^6*x^4)) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(a + b*x^2)^4) + b^2/(a^5*(a + b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^2)/(2*a^7*(a + b*x^2)) + (21*b^2*log(x))/a^8 - (21*b^2*log(a + b*x^2))/(2*a^8)]

    @test_int [x^16/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 9, (9009*a^2*x)/(256*b^8) - (3003*a*x^3)/(256*b^7) + (9009*x^5)/(1280*b^6) - x^15/(10*b*(a + b*x^2)^5) - (3*x^13)/(16*b^2*(a + b*x^2)^4) - (13*x^11)/(32*b^3*(a + b*x^2)^3) - (143*x^9)/(128*b^4*(a + b*x^2)^2) - (1287*x^7)/(256*b^5*(a + b*x^2)) - (9009*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(256*b^(17/2))]
    @test_int [x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 9, -((3003*a*x)/(256*b^7)) + (1001*x^3)/(256*b^6) - x^13/(10*b*(a + b*x^2)^5) - (13*x^11)/(80*b^2*(a + b*x^2)^4) - (143*x^9)/(480*b^3*(a + b*x^2)^3) - (429*x^7)/(640*b^4*(a + b*x^2)^2) - (3003*x^5)/(1280*b^5*(a + b*x^2)) + (3003*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(256*b^(15/2))]
    @test_int [x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 8, (693*x)/(256*b^6) - x^11/(10*b*(a + b*x^2)^5) - (11*x^9)/(80*b^2*(a + b*x^2)^4) - (33*x^7)/(160*b^3*(a + b*x^2)^3) - (231*x^5)/(640*b^4*(a + b*x^2)^2) - (231*x^3)/(256*b^5*(a + b*x^2)) - (693*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(256*b^(13/2))]
    @test_int [x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, -(x^9/(10*b*(a + b*x^2)^5)) - (9*x^7)/(80*b^2*(a + b*x^2)^4) - (21*x^5)/(160*b^3*(a + b*x^2)^3) - (21*x^3)/(128*b^4*(a + b*x^2)^2) - (63*x)/(256*b^5*(a + b*x^2)) + (63*atan((sqrt(b)*x)/sqrt(a)))/(256*sqrt(a)*b^(11/2))]
    @test_int [x^8/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, -(x^7/(10*b*(a + b*x^2)^5)) - (7*x^5)/(80*b^2*(a + b*x^2)^4) - (7*x^3)/(96*b^3*(a + b*x^2)^3) - (7*x)/(128*b^4*(a + b*x^2)^2) + (7*x)/(256*a*b^4*(a + b*x^2)) + (7*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(3/2)*b^(9/2))]
    @test_int [x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, -(x^5/(10*b*(a + b*x^2)^5)) - x^3/(16*b^2*(a + b*x^2)^4) - x/(32*b^3*(a + b*x^2)^3) + x/(128*a*b^3*(a + b*x^2)^2) + (3*x)/(256*a^2*b^3*(a + b*x^2)) + (3*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(5/2)*b^(7/2))]
    @test_int [x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, -(x^3/(10*b*(a + b*x^2)^5)) - (3*x)/(80*b^2*(a + b*x^2)^4) + x/(160*a*b^2*(a + b*x^2)^3) + x/(128*a^2*b^2*(a + b*x^2)^2) + (3*x)/(256*a^3*b^2*(a + b*x^2)) + (3*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(7/2)*b^(5/2))]
    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, -(x/(10*b*(a + b*x^2)^5)) + x/(80*a*b*(a + b*x^2)^4) + (7*x)/(480*a^2*b*(a + b*x^2)^3) + (7*x)/(384*a^3*b*(a + b*x^2)^2) + (7*x)/(256*a^4*b*(a + b*x^2)) + (7*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(9/2)*b^(3/2))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 7, x/(10*a*(a + b*x^2)^5) + (9*x)/(80*a^2*(a + b*x^2)^4) + (21*x)/(160*a^3*(a + b*x^2)^3) + (21*x)/(128*a^4*(a + b*x^2)^2) + (63*x)/(256*a^5*(a + b*x^2)) + (63*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(11/2)*sqrt(b))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 8, -(693/(256*a^6*x)) + 1/(10*a*x*(a + b*x^2)^5) + 11/(80*a^2*x*(a + b*x^2)^4) + 33/(160*a^3*x*(a + b*x^2)^3) + 231/(640*a^4*x*(a + b*x^2)^2) + 231/(256*a^5*x*(a + b*x^2)) - (693*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(13/2))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 9, -(1001/(256*a^6*x^3)) + (3003*b)/(256*a^7*x) + 1/(10*a*x^3*(a + b*x^2)^5) + 13/(80*a^2*x^3*(a + b*x^2)^4) + 143/(480*a^3*x^3*(a + b*x^2)^3) + 429/(640*a^4*x^3*(a + b*x^2)^2) + 3003/(1280*a^5*x^3*(a + b*x^2)) + (3003*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(15/2))]
    @test_int [1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 10, -(9009/(1280*a^6*x^5)) + (3003*b)/(256*a^7*x^3) - (9009*b^2)/(256*a^8*x) + 1/(10*a*x^5*(a + b*x^2)^5) + 3/(16*a^2*x^5*(a + b*x^2)^4) + 13/(32*a^3*x^5*(a + b*x^2)^3) + 143/(128*a^4*x^5*(a + b*x^2)^2) + 1287/(256*a^5*x^5*(a + b*x^2)) - (9009*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(256*a^(17/2))]

    @test_int [x^0/(1 + 2*x^2 + x^4), x, 3, x/(2*(1 + x^2)) + atan(x)/2]
    @test_int [x^1/(1 + 2*x^2 + x^4), x, 2, -(1/(2*(1 + x^2)))]
    @test_int [x^2/(1 + 2*x^2 + x^4), x, 3, -(x/(2*(1 + x^2))) + atan(x)/2]
    @test_int [x^3/(1 + 2*x^2 + x^4), x, 4, 1/(2*(1 + x^2)) + (1/2)*log(1 + x^2)]

    @test_int [x/(81 - 18*x^2 + x^4), x, 2, 1/(2*(9 - x^2))]
    @test_int [x^3/(16 - 8*x^2 + x^4), x, 4, 2/(4 - x^2) + (1/2)*log(4 - x^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a^2+2*a*b*x^2+b^2*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, (a*x^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*(a + b*x^2)) + (b*x^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*(a + b*x^2))]
    @test_int [x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, -((a*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*b^2)) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(6*b^2)]
    @test_int [x^1*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, ((a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*b)]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^1, x, 3, (b*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^3, x, 3, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^5, x, 3, -(((a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*a*x^4))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7, x, 1, -((a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*a*x^6) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(12*a^2*x^6)]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^9, x, 4, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^6*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^11, x, 4, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2))]

    @test_int [x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (a*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2)) + (b*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2))]
    @test_int [x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (a*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (b*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2))]
    @test_int [x^0*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (a*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^2, x, 3, -((a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))) + (b*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2)]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^4, x, 3, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^6, x, 3, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8, x, 3, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/x^10, x, 3, -(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2))]

    @test_int [x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, (a^3*x^10*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*(a + b*x^2)) + (a^2*b*x^12*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (3*a*b^2*x^14*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*(a + b*x^2)) + (b^3*x^16*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*(a + b*x^2))]
    @test_int [x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, (a^3*x^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*(a + b*x^2)) + (3*a^2*b*x^10*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*(a + b*x^2)) + (a*b^2*x^12*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (b^3*x^14*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*(a + b*x^2))]
    @test_int [x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^2*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*b^3) - (a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(5*b^3) + ((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*b^3), (a^2*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*b^3) - (a*(a + b*x^2)^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*b^3) + ((a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*b^3)]
    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, -((a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*b^2)) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(10*b^2)]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 2, ((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*b)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^1, x, 4, (3*a^2*b*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (3*a*b^2*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (b^3*x^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*(a + b*x^2)) + (a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^3, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (3*a*b^2*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (b^3*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^5, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^4*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (b^3*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^7, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^6*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^4*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^9, x, 3, -((a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*a*x^8)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^11, x, 1, -((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*a*x^10) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(40*a^2*x^10)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^13, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*x^12*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^6*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^15, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*x^14*(a + b*x^2)) - (a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^12*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^17, x, 4, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*x^16*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*x^14*(a + b*x^2)) - (a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^12*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2))]

    @test_int [x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (3*a^2*b*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (3*a*b^2*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (b^3*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*(a + b*x^2))]
    @test_int [x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (a^2*b*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (3*a*b^2*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (b^3*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2))]
    @test_int [x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2)) + (3*a^2*b*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (a*b^2*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (b^3*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2))]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (3*a^2*b*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2)) + (3*a*b^2*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (b^3*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2))]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(a + b*x^2)^3 + (a^2*b*x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(a + b*x^2)^3 + (3*a*b^2*x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(5*(a + b*x^2)^3) + (b^3*x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(7*(a + b*x^2)^3)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^2, x, 3, -((a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))) + (3*a^2*b*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (a*b^2*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b^3*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^4, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (3*a*b^2*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b^3*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^6, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2)) - (a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^3*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (b^3*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^8, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2)) - (a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^3*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^10, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^9*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^14, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^9*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^16, x, 3, -(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*x^15*(a + b*x^2)) - (3*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (3*a*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2))]

    @test_int [x^13*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, (a^5*x^14*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*(a + b*x^2)) + (5*a^4*b*x^16*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*(a + b*x^2)) + (5*a^3*b^2*x^18*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (a^2*b^3*x^20*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (5*a*b^4*x^22*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(22*(a + b*x^2)) + (b^5*x^24*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(24*(a + b*x^2))]
    @test_int [x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, (a^5*x^12*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*(a + b*x^2)) + (5*a^4*b*x^14*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*(a + b*x^2)) + (5*a^3*b^2*x^16*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*(a + b*x^2)) + (5*a^2*b^3*x^18*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (a*b^4*x^20*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (b^5*x^22*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(22*(a + b*x^2))]
    @test_int [x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^4*(a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*b^5) - (2*a^3*(a + b*x^2)^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*b^5) + (3*a^2*(a + b*x^2)^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*b^5) - (2*a*(a + b*x^2)^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*b^5) + ((a + b*x^2)^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(20*b^5)]
    @test_int [x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, -(a^3*(a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*b^4) + (3*a^2*(a + b*x^2)^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*b^4) - (3*a*(a + b*x^2)^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*b^4) + ((a + b*x^2)^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(18*b^4)]
    @test_int [x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, (a^2*(a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*b^3) - (a*(a + b*x^2)^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*b^3) + ((a + b*x^2)^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*b^3)]
    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, -((a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*b^2)) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(14*b^2)]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 2, ((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*b)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^1, x, 4, (5*a^4*b*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (5*a^3*b^2*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (5*a^2*b^3*x^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (5*a*b^4*x^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*(a + b*x^2)) + (b^5*x^10*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*(a + b*x^2)) + (a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^3, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (5*a^3*b^2*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (5*a^2*b^3*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (5*a*b^4*x^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*(a + b*x^2)) + (b^5*x^8*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*(a + b*x^2)) + (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^5, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^4*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (5*a^2*b^3*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (5*a*b^4*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (b^5*x^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*(a + b*x^2)) + (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^7, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^6*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^4*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^2*(a + b*x^2)) + (5*a*b^4*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (b^5*x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*(a + b*x^2)) + (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^9, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^6*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^4*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^2*(a + b*x^2)) + (b^5*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*(a + b*x^2)) + (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^11, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^6*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^4*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^2*(a + b*x^2)) + (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*log(x))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^13, x, 3, -((a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*a*x^12)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15, x, 1, -((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*a*x^14) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(84*a^2*x^14)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^17, x, 5, -((a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*a*x^16) + (b*(a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(56*a^2*x^14) - (b^2*(a + b*x^2)^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(336*a^3*x^12)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(18*x^18*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*x^16*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^14*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*x^12*(a + b*x^2)) - (a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^10*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^8*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^21, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(20*x^20*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(18*x^18*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^16*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^14*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*x^12*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(10*x^10*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^23, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(22*x^22*(a + b*x^2)) - (a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*x^20*(a + b*x^2)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^18*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*x^16*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*x^14*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(12*x^12*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^25, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(24*x^24*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(22*x^22*(a + b*x^2)) - (a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*x^20*(a + b*x^2)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^18*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*x^16*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(14*x^14*(a + b*x^2))]

    @test_int [x^12*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (a^4*b*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (10*a^3*b^2*x^17*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*(a + b*x^2)) + (10*a^2*b^3*x^19*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*(a + b*x^2)) + (5*a*b^4*x^21*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*(a + b*x^2)) + (b^5*x^23*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(23*(a + b*x^2))]
    @test_int [x^10*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (5*a^4*b*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (2*a^3*b^2*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (10*a^2*b^3*x^17*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*(a + b*x^2)) + (5*a*b^4*x^19*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*(a + b*x^2)) + (b^5*x^21*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*(a + b*x^2))]
    @test_int [x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (5*a^4*b*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (10*a^3*b^2*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (2*a^2*b^3*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (5*a*b^4*x^17*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*(a + b*x^2)) + (b^5*x^19*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*(a + b*x^2))]
    @test_int [x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (5*a^4*b*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (10*a^3*b^2*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (10*a^2*b^3*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (a*b^4*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (b^5*x^17*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*(a + b*x^2))]
    @test_int [x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2)) + (5*a^4*b*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (10*a^3*b^2*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (10*a^2*b^3*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (5*a*b^4*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2)) + (b^5*x^15*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*(a + b*x^2))]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (a^4*b*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (10*a^3*b^2*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (10*a^2*b^3*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2)) + (5*a*b^4*x^11*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*(a + b*x^2)) + (b^5*x^13*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*(a + b*x^2))]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(a + b*x^2)^5 + (5*a^4*b*x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(3*(a + b*x^2)^5) + (2*a^3*b^2*x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(a + b*x^2)^5 + (10*a^2*b^3*x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(7*(a + b*x^2)^5) + (5*a*b^4*x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(9*(a + b*x^2)^5) + (b^5*x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(11*(a + b*x^2)^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^2, x, 3, -((a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))) + (5*a^4*b*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (10*a^3*b^2*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (2*a^2*b^3*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (5*a*b^4*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2)) + (b^5*x^9*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^4, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (10*a^3*b^2*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (10*a^2*b^3*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (a*b^4*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b^5*x^7*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (10*a^2*b^3*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (5*a*b^4*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2)) + (b^5*x^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^8, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^5*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (5*a*b^4*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b^5*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^10, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (2*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^5*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2)) + (b^5*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^12, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (2*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^5*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^14, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(x^5*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^3*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^16, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*x^15*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*x^5*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^18, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*x^17*(a + b*x^2)) - (a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^15*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*x^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^20, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*x^19*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*x^17*(a + b*x^2)) - (2*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^15*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*x^9*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^22, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*x^21*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*x^19*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*x^17*(a + b*x^2)) - (2*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^15*(a + b*x^2)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*x^11*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^24, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(23*x^23*(a + b*x^2)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*x^21*(a + b*x^2)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*x^19*(a + b*x^2)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*x^17*(a + b*x^2)) - (a*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*x^15*(a + b*x^2)) - (b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*x^13*(a + b*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, -((a*x^2*(a + b*x^2))/(2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + (x^4*(a + b*x^2))/(4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^2*(a + b*x^2)*log(a + b*x^2))/(2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^3/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/(2*b^2) - (a*(a + b*x^2)*log(a + b*x^2))/(2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^1/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, ((a + b*x^2)*log(a + b*x^2))/(2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^1*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 5, ((a + b*x^2)*log(x))/(a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - ((a + b*x^2)*log(a + b*x^2))/(2*a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, -(a + b*x^2)/(2*a*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b*(a + b*x^2)*log(x))/(a^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b*(a + b*x^2)*log(a + b*x^2))/(2*a^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [x^4/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 4, -((a*x*(a + b*x^2))/(b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + (x^3*(a + b*x^2))/(3*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(3/2)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(b^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^2/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (x*(a + b*x^2))/(b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (sqrt(a)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(b^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^0/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, ((a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(sqrt(a)*sqrt(b)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 3, -((a + b*x^2)/(a*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) - (sqrt(b)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 4, -(a + b*x^2)/(3*a*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b*(a + b*x^2))/(a^2*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(3/2)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(a^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, -((3*a^2)/(2*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + a^3/(4*b^4*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (x^2*(a + b*x^2))/(2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*a*(a + b*x^2)*log(a + b*x^2))/(2*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, a/(b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - a^2/(4*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*log(a + b*x^2))/(2*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, x^4/(4*a*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), -(1/(2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + a/(4*b^2*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 2, -(1/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 4, 1/(2*a^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*log(x))/(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - ((a + b*x^2)*log(a + b*x^2))/(2*a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 4, -(b/(a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) - b/(4*a^2*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a + b*x^2)/(2*a^3*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*b*(a + b*x^2)*log(x))/(a^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*b*(a + b*x^2)*log(a + b*x^2))/(2*a^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, (-3*x)/(8*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x^3/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*sqrt(a)*b^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, x/(8*a*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(3/2)*b^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 4, (x*(a + b*x^2))/(4*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*x*(a + b*x^2)^2)/(8*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*(a + b*x^2)^3*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*sqrt(b)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 5, 5/(8*a^2*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*x*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (15*(a + b*x^2))/(8*a^3*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (15*sqrt(b)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 6, 7/(8*a^2*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*x^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (35*(a + b*x^2))/(24*a^3*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*b*(a + b*x^2))/(8*a^4*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*b^(3/2)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, -((5*a^2)/(b^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + a^5/(8*b^6*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*a^4)/(6*b^6*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*a^3)/(2*b^6*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (x^2*(a + b*x^2))/(2*b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*a*(a + b*x^2)*log(a + b*x^2))/(2*b^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 4, (2*a)/(b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - a^4/(8*b^5*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (2*a^3)/(3*b^5*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*a^2)/(2*b^5*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*log(a + b*x^2))/(2*b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^7/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, x^8/(8*a*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 1, x^6/(24*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + x^6/(8*a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))]
    @test_int [x^3/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, a/(8*b^2*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) - 1/(6*b^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))]
    @test_int [x^1/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 2, -(1/(8*b*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 4, 1/(2*a^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(6*a^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*log(x))/(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - ((a + b*x^2)*log(a + b*x^2))/(2*a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 4, (-2*b)/(a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - b/(8*a^2*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - b/(3*a^3*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*b)/(4*a^4*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a + b*x^2)/(2*a^5*x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*b*(a + b*x^2)*log(x))/(a^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*b*(a + b*x^2)*log(a + b*x^2))/(2*a^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 6, (5*x)/(128*a*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x^5/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*x^3)/(48*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*x)/(64*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(3/2)*b^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^4/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 6, (3*x)/(128*a^2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x^3/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x/(16*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + x/(64*a*b^2*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(5/2)*b^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 6, (5*x)/(128*a^3*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - x/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + x/(48*a*b*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*x)/(192*a^2*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(7/2)*b^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 6, (x*(a + b*x^2))/(8*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (7*x*(a + b*x^2)^2)/(48*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^3)/(192*a^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^4)/(128*a^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*(a + b*x^2)^5*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(9/2)*sqrt(b)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 7, 105/(128*a^4*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*x*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 3/(16*a^2*x*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 21/(64*a^3*x*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (315*(a + b*x^2))/(128*a^5*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (315*sqrt(b)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 8, 231/(128*a^4*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*x^3*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 11/(48*a^2*x^3*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 33/(64*a^3*x^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (385*(a + b*x^2))/(128*a^5*x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*b*(a + b*x^2))/(128*a^6*x*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*b^(3/2)*(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a)))/(128*a^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a^2+2*a*b*x^2+b^2*x^4)^(p/3)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3), x, 4, (3*x*(a + b*x^2))/(5*b*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)) + (3*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(1 + (b*x^2)/a)^(2/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(5*b^2*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3), x, 3, -((3^(3/4)*sqrt(2 - sqrt(3))*a*(1 + (b*x^2)/a)^(2/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(b*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2))))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)), x, 4, -((a + b*x^2)/(a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3))) + (sqrt(2 - sqrt(3))*(1 + (b*x^2)/a)^(2/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)))]

    @test_int [x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3), x, 6, -((3*x*(a + b*x^2))/(2*b*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3))) - (9*a*x*(1 + (b*x^2)/a)^(4/3))/(2*b*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))) + (9*3^(1/4)*sqrt(2 + sqrt(3))*a^2*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(4*b^2*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2))) - (3*3^(3/4)*a^2*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*b^2*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)))]
    @test_int [x^0/(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3), x, 6, (3*x*(a + b*x^2))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) + (3*x*(1 + (b*x^2)/a)^(4/3))/(2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))) - (3*3^(1/4)*sqrt(2 + sqrt(3))*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(4*b*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2))) + (3^(3/4)*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*b*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)), x, 7, (3*(a + b*x^2))/(2*a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) - (5*(a + b*x^2)^2)/(2*a^2*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) - (5*b*x*(1 + (b*x^2)/a)^(4/3))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))) + (5*3^(1/4)*sqrt(2 + sqrt(3))*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(4*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2))) - (5*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*sqrt((1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*3^(1/4)*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*sqrt(-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - sqrt(3) - (1 + (b*x^2)/a)^(1/3))^2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^(m/2)*(a^2+2*a*b*x^2+b^2*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (2*a^2*(d*x)^(7/2))/(7*d) + (4*a*b*(d*x)^(11/2))/(11*d^3) + (2*b^2*(d*x)^(15/2))/(15*d^5)]
    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(13*d^5)]
    @test_int [sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (2*a^2*(d*x)^(3/2))/(3*d) + (4*a*b*(d*x)^(7/2))/(7*d^3) + (2*b^2*(d*x)^(11/2))/(11*d^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/sqrt(d*x), x, 2, (2*a^2*sqrt(d*x))/d + (4*a*b*(d*x)^(5/2))/(5*d^3) + (2*b^2*(d*x)^(9/2))/(9*d^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(3/2), x, 2, (-2*a^2)/(d*sqrt(d*x)) + (4*a*b*(d*x)^(3/2))/(3*d^3) + (2*b^2*(d*x)^(7/2))/(7*d^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(5/2), x, 2, (-2*a^2)/(3*d*(d*x)^(3/2)) + (4*a*b*sqrt(d*x))/d^3 + (2*b^2*(d*x)^(5/2))/(5*d^5)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(7/2), x, 2, (-2*a^2)/(5*d*(d*x)^(5/2)) - (4*a*b)/(d^3*sqrt(d*x)) + (2*b^2*(d*x)^(3/2))/(3*d^5)]

    @test_int [(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (2*a^4*(d*x)^(7/2))/(7*d) + (8*a^3*b*(d*x)^(11/2))/(11*d^3) + (4*a^2*b^2*(d*x)^(15/2))/(5*d^5) + (8*a*b^3*(d*x)^(19/2))/(19*d^7) + (2*b^4*(d*x)^(23/2))/(23*d^9)]
    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (2*a^4*(d*x)^(5/2))/(5*d) + (8*a^3*b*(d*x)^(9/2))/(9*d^3) + (12*a^2*b^2*(d*x)^(13/2))/(13*d^5) + (8*a*b^3*(d*x)^(17/2))/(17*d^7) + (2*b^4*(d*x)^(21/2))/(21*d^9)]
    @test_int [sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (2*a^4*(d*x)^(3/2))/(3*d) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (12*a^2*b^2*(d*x)^(11/2))/(11*d^5) + (8*a*b^3*(d*x)^(15/2))/(15*d^7) + (2*b^4*(d*x)^(19/2))/(19*d^9)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/sqrt(d*x), x, 3, (2*a^4*sqrt(d*x))/d + (8*a^3*b*(d*x)^(5/2))/(5*d^3) + (4*a^2*b^2*(d*x)^(9/2))/(3*d^5) + (8*a*b^3*(d*x)^(13/2))/(13*d^7) + (2*b^4*(d*x)^(17/2))/(17*d^9)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(3/2), x, 3, (-2*a^4)/(d*sqrt(d*x)) + (8*a^3*b*(d*x)^(3/2))/(3*d^3) + (12*a^2*b^2*(d*x)^(7/2))/(7*d^5) + (8*a*b^3*(d*x)^(11/2))/(11*d^7) + (2*b^4*(d*x)^(15/2))/(15*d^9)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(5/2), x, 3, (-2*a^4)/(3*d*(d*x)^(3/2)) + (8*a^3*b*sqrt(d*x))/d^3 + (12*a^2*b^2*(d*x)^(5/2))/(5*d^5) + (8*a*b^3*(d*x)^(9/2))/(9*d^7) + (2*b^4*(d*x)^(13/2))/(13*d^9)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2), x, 3, (-2*a^4)/(5*d*(d*x)^(5/2)) - (8*a^3*b)/(d^3*sqrt(d*x)) + (4*a^2*b^2*(d*x)^(3/2))/d^5 + (8*a*b^3*(d*x)^(7/2))/(7*d^7) + (2*b^4*(d*x)^(11/2))/(11*d^9)]

    @test_int [(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (2*a^6*(d*x)^(7/2))/(7*d) + (12*a^5*b*(d*x)^(11/2))/(11*d^3) + (2*a^4*b^2*(d*x)^(15/2))/d^5 + (40*a^3*b^3*(d*x)^(19/2))/(19*d^7) + (30*a^2*b^4*(d*x)^(23/2))/(23*d^9) + (4*a*b^5*(d*x)^(27/2))/(9*d^11) + (2*b^6*(d*x)^(31/2))/(31*d^13)]
    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (2*a^6*(d*x)^(5/2))/(5*d) + (4*a^5*b*(d*x)^(9/2))/(3*d^3) + (30*a^4*b^2*(d*x)^(13/2))/(13*d^5) + (40*a^3*b^3*(d*x)^(17/2))/(17*d^7) + (10*a^2*b^4*(d*x)^(21/2))/(7*d^9) + (12*a*b^5*(d*x)^(25/2))/(25*d^11) + (2*b^6*(d*x)^(29/2))/(29*d^13)]
    @test_int [sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (2*a^6*(d*x)^(3/2))/(3*d) + (12*a^5*b*(d*x)^(7/2))/(7*d^3) + (30*a^4*b^2*(d*x)^(11/2))/(11*d^5) + (8*a^3*b^3*(d*x)^(15/2))/(3*d^7) + (30*a^2*b^4*(d*x)^(19/2))/(19*d^9) + (12*a*b^5*(d*x)^(23/2))/(23*d^11) + (2*b^6*(d*x)^(27/2))/(27*d^13)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/sqrt(d*x), x, 3, (2*a^6*sqrt(d*x))/d + (12*a^5*b*(d*x)^(5/2))/(5*d^3) + (10*a^4*b^2*(d*x)^(9/2))/(3*d^5) + (40*a^3*b^3*(d*x)^(13/2))/(13*d^7) + (30*a^2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a*b^5*(d*x)^(21/2))/(7*d^11) + (2*b^6*(d*x)^(25/2))/(25*d^13)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(3/2), x, 3, (-2*a^6)/(d*sqrt(d*x)) + (4*a^5*b*(d*x)^(3/2))/d^3 + (30*a^4*b^2*(d*x)^(7/2))/(7*d^5) + (40*a^3*b^3*(d*x)^(11/2))/(11*d^7) + (2*a^2*b^4*(d*x)^(15/2))/d^9 + (12*a*b^5*(d*x)^(19/2))/(19*d^11) + (2*b^6*(d*x)^(23/2))/(23*d^13)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2), x, 3, (-2*a^6)/(3*d*(d*x)^(3/2)) + (12*a^5*b*sqrt(d*x))/d^3 + (6*a^4*b^2*(d*x)^(5/2))/d^5 + (40*a^3*b^3*(d*x)^(9/2))/(9*d^7) + (30*a^2*b^4*(d*x)^(13/2))/(13*d^9) + (12*a*b^5*(d*x)^(17/2))/(17*d^11) + (2*b^6*(d*x)^(21/2))/(21*d^13)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(7/2), x, 3, (-2*a^6)/(5*d*(d*x)^(5/2)) - (12*a^5*b)/(d^3*sqrt(d*x)) + (10*a^4*b^2*(d*x)^(3/2))/d^5 + (40*a^3*b^3*(d*x)^(7/2))/(7*d^7) + (30*a^2*b^4*(d*x)^(11/2))/(11*d^9) + (4*a*b^5*(d*x)^(15/2))/(5*d^11) + (2*b^6*(d*x)^(19/2))/(19*d^13)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 14, (-9*a*d^5*sqrt(d*x))/(2*b^3) + (9*d^3*(d*x)^(5/2))/(10*b^2) - (d*(d*x)^(9/2))/(2*b*(a + b*x^2)) - (9*a^(5/4)*d^(11/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(13/4)) + (9*a^(5/4)*d^(11/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(13/4)) - (9*a^(5/4)*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(13/4)) + (9*a^(5/4)*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(13/4))]
    @test_int [(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 13, (7*d^3*(d*x)^(3/2))/(6*b^2) - (d*(d*x)^(7/2))/(2*b*(a + b*x^2)) + (7*a^(3/4)*d^(9/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(11/4)) - (7*a^(3/4)*d^(9/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(11/4)) - (7*a^(3/4)*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(11/4)) + (7*a^(3/4)*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(11/4))]
    @test_int [(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 13, (5*d^3*sqrt(d*x))/(2*b^2) - (d*(d*x)^(5/2))/(2*b*(a + b*x^2)) + (5*a^(1/4)*d^(7/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(9/4)) - (5*a^(1/4)*d^(7/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*b^(9/4)) + (5*a^(1/4)*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(9/4)) - (5*a^(1/4)*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*b^(9/4))]
    @test_int [(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 12, -(d*(d*x)^(3/2))/(2*b*(a + b*x^2)) - (3*d^(5/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(1/4)*b^(7/4)) + (3*d^(5/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(1/4)*b^(7/4)) + (3*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(1/4)*b^(7/4)) - (3*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(1/4)*b^(7/4))]
    @test_int [(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 12, -(d*sqrt(d*x))/(2*b*(a + b*x^2)) - (d^(3/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(3/4)*b^(5/4)) + (d^(3/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(3/4)*b^(5/4)) - (d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(3/4)*b^(5/4)) + (d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(3/4)*b^(5/4))]
    @test_int [sqrt(d*x)/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 12, (d*x)^(3/2)/(2*a*d*(a + b*x^2)) - (sqrt(d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(5/4)*b^(3/4)) + (sqrt(d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(5/4)*b^(3/4)) + (sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(5/4)*b^(3/4)) - (sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(5/4)*b^(3/4))]
    @test_int [1/(sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 12, sqrt(d*x)/(2*a*d*(a + b*x^2)) - (3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(7/4)*b^(1/4)*sqrt(d)) + (3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(7/4)*b^(1/4)*sqrt(d)) - (3*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(7/4)*b^(1/4)*sqrt(d)) + (3*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(7/4)*b^(1/4)*sqrt(d))]
    @test_int [1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 13, -5/(2*a^2*d*sqrt(d*x)) + 1/(2*a*d*sqrt(d*x)*(a + b*x^2)) + (5*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(9/4)*d^(3/2)) - (5*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(9/4)*d^(3/2)) - (5*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(9/4)*d^(3/2)) + (5*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(9/4)*d^(3/2))]
    @test_int [1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 13, -7/(6*a^2*d*(d*x)^(3/2)) + 1/(2*a*d*(d*x)^(3/2)*(a + b*x^2)) + (7*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(11/4)*d^(5/2)) - (7*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(11/4)*d^(5/2)) + (7*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(11/4)*d^(5/2)) - (7*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(11/4)*d^(5/2))]
    @test_int [1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 14, -9/(10*a^2*d*(d*x)^(5/2)) + (9*b)/(2*a^3*d^3*sqrt(d*x)) + 1/(2*a*d*(d*x)^(5/2)*(a + b*x^2)) - (9*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(13/4)*d^(7/2)) + (9*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(4*sqrt(2)*a^(13/4)*d^(7/2)) + (9*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(13/4)*d^(7/2)) - (9*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(8*sqrt(2)*a^(13/4)*d^(7/2))]

    @test_int [(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 16, (-663*a*d^9*sqrt(d*x))/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17/2))/(6*b*(a + b*x^2)^3) - (17*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(21/4)) + (663*a^(5/4)*d^(19/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(21/4)) - (663*a^(5/4)*d^(19/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(21/4)) + (663*a^(5/4)*d^(19/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(21/4))]
    @test_int [(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 15, (385*d^7*(d*x)^(3/2))/(192*b^4) - (d*(d*x)^(15/2))/(6*b*(a + b*x^2)^3) - (5*d^3*(d*x)^(11/2))/(16*b^2*(a + b*x^2)^2) - (55*d^5*(d*x)^(7/2))/(64*b^3*(a + b*x^2)) + (385*a^(3/4)*d^(17/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(19/4)) - (385*a^(3/4)*d^(17/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(19/4)) - (385*a^(3/4)*d^(17/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(19/4)) + (385*a^(3/4)*d^(17/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(19/4))]
    @test_int [(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 15, (195*d^7*sqrt(d*x))/(64*b^4) - (d*(d*x)^(13/2))/(6*b*(a + b*x^2)^3) - (13*d^3*(d*x)^(9/2))/(48*b^2*(a + b*x^2)^2) - (39*d^5*(d*x)^(5/2))/(64*b^3*(a + b*x^2)) + (195*a^(1/4)*d^(15/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(17/4)) - (195*a^(1/4)*d^(15/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*b^(17/4)) + (195*a^(1/4)*d^(15/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(17/4)) - (195*a^(1/4)*d^(15/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*b^(17/4))]
    @test_int [(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*(d*x)^(11/2))/(6*b*(a + b*x^2)^3) - (11*d^3*(d*x)^(7/2))/(48*b^2*(a + b*x^2)^2) - (77*d^5*(d*x)^(3/2))/(192*b^3*(a + b*x^2)) - (77*d^(13/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(1/4)*b^(15/4)) + (77*d^(13/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(1/4)*b^(15/4)) + (77*d^(13/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(1/4)*b^(15/4)) - (77*d^(13/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(1/4)*b^(15/4))]
    @test_int [(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*(d*x)^(9/2))/(6*b*(a + b*x^2)^3) - (3*d^3*(d*x)^(5/2))/(16*b^2*(a + b*x^2)^2) - (15*d^5*sqrt(d*x))/(64*b^3*(a + b*x^2)) - (15*d^(11/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(3/4)*b^(13/4)) + (15*d^(11/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(3/4)*b^(13/4)) - (15*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(3/4)*b^(13/4)) + (15*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(3/4)*b^(13/4))]
    @test_int [(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*(d*x)^(7/2))/(6*b*(a + b*x^2)^3) - (7*d^3*(d*x)^(3/2))/(48*b^2*(a + b*x^2)^2) + (7*d^3*(d*x)^(3/2))/(64*a*b^2*(a + b*x^2)) - (7*d^(9/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(5/4)*b^(11/4)) + (7*d^(9/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(5/4)*b^(11/4)) + (7*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(5/4)*b^(11/4)) - (7*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(5/4)*b^(11/4))]
    @test_int [(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*(d*x)^(5/2))/(6*b*(a + b*x^2)^3) - (5*d^3*sqrt(d*x))/(48*b^2*(a + b*x^2)^2) + (5*d^3*sqrt(d*x))/(192*a*b^2*(a + b*x^2)) - (5*d^(7/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(7/4)*b^(9/4)) + (5*d^(7/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(7/4)*b^(9/4)) - (5*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(7/4)*b^(9/4)) + (5*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(7/4)*b^(9/4))]
    @test_int [(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*(d*x)^(3/2))/(6*b*(a + b*x^2)^3) + (d*(d*x)^(3/2))/(16*a*b*(a + b*x^2)^2) + (5*d*(d*x)^(3/2))/(64*a^2*b*(a + b*x^2)) - (5*d^(5/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(9/4)*b^(7/4)) + (5*d^(5/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(9/4)*b^(7/4)) + (5*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(9/4)*b^(7/4)) - (5*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(9/4)*b^(7/4))]
    @test_int [(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, -(d*sqrt(d*x))/(6*b*(a + b*x^2)^3) + (d*sqrt(d*x))/(48*a*b*(a + b*x^2)^2) + (7*d*sqrt(d*x))/(192*a^2*b*(a + b*x^2)) - (7*d^(3/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(11/4)*b^(5/4)) + (7*d^(3/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(11/4)*b^(5/4)) - (7*d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(11/4)*b^(5/4)) + (7*d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(11/4)*b^(5/4))]
    @test_int [sqrt(d*x)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 14, (d*x)^(3/2)/(6*a*d*(a + b*x^2)^3) + (3*(d*x)^(3/2))/(16*a^2*d*(a + b*x^2)^2) + (15*(d*x)^(3/2))/(64*a^3*d*(a + b*x^2)) - (15*sqrt(d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(13/4)*b^(3/4)) + (15*sqrt(d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(13/4)*b^(3/4)) + (15*sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(13/4)*b^(3/4)) - (15*sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(13/4)*b^(3/4))]
    @test_int [1/(sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 14, sqrt(d*x)/(6*a*d*(a + b*x^2)^3) + (11*sqrt(d*x))/(48*a^2*d*(a + b*x^2)^2) + (77*sqrt(d*x))/(192*a^3*d*(a + b*x^2)) - (77*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(15/4)*b^(1/4)*sqrt(d)) + (77*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(15/4)*b^(1/4)*sqrt(d)) - (77*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(15/4)*b^(1/4)*sqrt(d)) + (77*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(15/4)*b^(1/4)*sqrt(d))]
    @test_int [1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 15, -195/(64*a^4*d*sqrt(d*x)) + 1/(6*a*d*sqrt(d*x)*(a + b*x^2)^3) + 13/(48*a^2*d*sqrt(d*x)*(a + b*x^2)^2) + 39/(64*a^3*d*sqrt(d*x)*(a + b*x^2)) + (195*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(17/4)*d^(3/2)) - (195*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(17/4)*d^(3/2)) - (195*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(17/4)*d^(3/2)) + (195*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(17/4)*d^(3/2))]
    @test_int [1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 15, -385/(192*a^4*d*(d*x)^(3/2)) + 1/(6*a*d*(d*x)^(3/2)*(a + b*x^2)^3) + 5/(16*a^2*d*(d*x)^(3/2)*(a + b*x^2)^2) + 55/(64*a^3*d*(d*x)^(3/2)*(a + b*x^2)) + (385*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(19/4)*d^(5/2)) - (385*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(19/4)*d^(5/2)) + (385*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(19/4)*d^(5/2)) - (385*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(19/4)*d^(5/2))]
    @test_int [1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2), x, 16, -663/(320*a^4*d*(d*x)^(5/2)) + (663*b)/(64*a^5*d^3*sqrt(d*x)) + 1/(6*a*d*(d*x)^(5/2)*(a + b*x^2)^3) + 17/(48*a^2*d*(d*x)^(5/2)*(a + b*x^2)^2) + 221/(192*a^3*d*(d*x)^(5/2)*(a + b*x^2)) - (663*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(21/4)*d^(7/2)) + (663*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(128*sqrt(2)*a^(21/4)*d^(7/2)) + (663*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(21/4)*d^(7/2)) - (663*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(256*sqrt(2)*a^(21/4)*d^(7/2))]

    @test_int [(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 18, (-69615*a*d^13*sqrt(d*x))/(4096*b^7) + (13923*d^11*(d*x)^(5/2))/(4096*b^6) - (d*(d*x)^(25/2))/(10*b*(a + b*x^2)^5) - (5*d^3*(d*x)^(21/2))/(32*b^2*(a + b*x^2)^4) - (35*d^5*(d*x)^(17/2))/(128*b^3*(a + b*x^2)^3) - (595*d^7*(d*x)^(13/2))/(1024*b^4*(a + b*x^2)^2) - (7735*d^9*(d*x)^(9/2))/(4096*b^5*(a + b*x^2)) - (69615*a^(5/4)*d^(27/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(29/4)) + (69615*a^(5/4)*d^(27/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(29/4)) - (69615*a^(5/4)*d^(27/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(29/4)) + (69615*a^(5/4)*d^(27/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(29/4))]
    @test_int [(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 17, (33649*d^11*(d*x)^(3/2))/(12288*b^6) - (d*(d*x)^(23/2))/(10*b*(a + b*x^2)^5) - (23*d^3*(d*x)^(19/2))/(160*b^2*(a + b*x^2)^4) - (437*d^5*(d*x)^(15/2))/(1920*b^3*(a + b*x^2)^3) - (437*d^7*(d*x)^(11/2))/(1024*b^4*(a + b*x^2)^2) - (4807*d^9*(d*x)^(7/2))/(4096*b^5*(a + b*x^2)) + (33649*a^(3/4)*d^(25/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(27/4)) + (33649*a^(3/4)*d^(25/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(27/4))]
    @test_int [(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 17, (13923*d^11*sqrt(d*x))/(4096*b^6) - (d*(d*x)^(21/2))/(10*b*(a + b*x^2)^5) - (21*d^3*(d*x)^(17/2))/(160*b^2*(a + b*x^2)^4) - (119*d^5*(d*x)^(13/2))/(640*b^3*(a + b*x^2)^3) - (1547*d^7*(d*x)^(9/2))/(5120*b^4*(a + b*x^2)^2) - (13923*d^9*(d*x)^(5/2))/(20480*b^5*(a + b*x^2)) + (13923*a^(1/4)*d^(23/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*b^(25/4)) + (13923*a^(1/4)*d^(23/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*b^(25/4))]
    @test_int [(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x^2)^4) - (19*d^5*(d*x)^(11/2))/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(a + b*x^2)) - (4389*d^(21/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(1/4)*b^(23/4)) - (4389*d^(21/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(1/4)*b^(23/4))]
    @test_int [(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(17/2))/(10*b*(a + b*x^2)^5) - (17*d^3*(d*x)^(13/2))/(160*b^2*(a + b*x^2)^4) - (221*d^5*(d*x)^(9/2))/(1920*b^3*(a + b*x^2)^3) - (663*d^7*(d*x)^(5/2))/(5120*b^4*(a + b*x^2)^2) - (663*d^9*sqrt(d*x))/(4096*b^5*(a + b*x^2)) - (663*d^(19/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(3/4)*b^(21/4)) + (663*d^(19/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(3/4)*b^(21/4)) - (663*d^(19/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(3/4)*b^(21/4)) + (663*d^(19/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(3/4)*b^(21/4))]
    @test_int [(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(15/2))/(10*b*(a + b*x^2)^5) - (3*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2)^4) - (11*d^5*(d*x)^(7/2))/(128*b^3*(a + b*x^2)^3) - (77*d^7*(d*x)^(3/2))/(1024*b^4*(a + b*x^2)^2) + (231*d^7*(d*x)^(3/2))/(4096*a*b^4*(a + b*x^2)) - (231*d^(17/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(5/4)*b^(19/4)) + (231*d^(17/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(5/4)*b^(19/4)) + (231*d^(17/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(5/4)*b^(19/4)) - (231*d^(17/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(5/4)*b^(19/4))]
    @test_int [(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(13/2))/(10*b*(a + b*x^2)^5) - (13*d^3*(d*x)^(9/2))/(160*b^2*(a + b*x^2)^4) - (39*d^5*(d*x)^(5/2))/(640*b^3*(a + b*x^2)^3) - (39*d^7*sqrt(d*x))/(1024*b^4*(a + b*x^2)^2) + (39*d^7*sqrt(d*x))/(4096*a*b^4*(a + b*x^2)) - (117*d^(15/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(7/4)*b^(17/4)) + (117*d^(15/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(7/4)*b^(17/4)) - (117*d^(15/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(7/4)*b^(17/4)) + (117*d^(15/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(7/4)*b^(17/4))]
    @test_int [(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(11/2))/(10*b*(a + b*x^2)^5) - (11*d^3*(d*x)^(7/2))/(160*b^2*(a + b*x^2)^4) - (77*d^5*(d*x)^(3/2))/(1920*b^3*(a + b*x^2)^3) + (77*d^5*(d*x)^(3/2))/(5120*a*b^3*(a + b*x^2)^2) + (77*d^5*(d*x)^(3/2))/(4096*a^2*b^3*(a + b*x^2)) - (77*d^(13/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(9/4)*b^(15/4)) + (77*d^(13/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(9/4)*b^(15/4)) + (77*d^(13/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(9/4)*b^(15/4)) - (77*d^(13/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(9/4)*b^(15/4))]
    @test_int [(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(9/2))/(10*b*(a + b*x^2)^5) - (9*d^3*(d*x)^(5/2))/(160*b^2*(a + b*x^2)^4) - (3*d^5*sqrt(d*x))/(128*b^3*(a + b*x^2)^3) + (3*d^5*sqrt(d*x))/(1024*a*b^3*(a + b*x^2)^2) + (21*d^5*sqrt(d*x))/(4096*a^2*b^3*(a + b*x^2)) - (63*d^(11/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(11/4)*b^(13/4)) + (63*d^(11/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(11/4)*b^(13/4)) - (63*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(11/4)*b^(13/4)) + (63*d^(11/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(11/4)*b^(13/4))]
    @test_int [(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(7/2))/(10*b*(a + b*x^2)^5) - (7*d^3*(d*x)^(3/2))/(160*b^2*(a + b*x^2)^4) + (7*d^3*(d*x)^(3/2))/(640*a*b^2*(a + b*x^2)^3) + (63*d^3*(d*x)^(3/2))/(5120*a^2*b^2*(a + b*x^2)^2) + (63*d^3*(d*x)^(3/2))/(4096*a^3*b^2*(a + b*x^2)) - (63*d^(9/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(13/4)*b^(11/4)) + (63*d^(9/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(13/4)*b^(11/4)) + (63*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(13/4)*b^(11/4)) - (63*d^(9/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(13/4)*b^(11/4))]
    @test_int [(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(5/2))/(10*b*(a + b*x^2)^5) - (d^3*sqrt(d*x))/(32*b^2*(a + b*x^2)^4) + (d^3*sqrt(d*x))/(384*a*b^2*(a + b*x^2)^3) + (11*d^3*sqrt(d*x))/(3072*a^2*b^2*(a + b*x^2)^2) + (77*d^3*sqrt(d*x))/(12288*a^3*b^2*(a + b*x^2)) - (77*d^(7/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(15/4)*b^(9/4)) + (77*d^(7/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(15/4)*b^(9/4)) - (77*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(15/4)*b^(9/4)) + (77*d^(7/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(15/4)*b^(9/4))]
    @test_int [(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*(d*x)^(3/2))/(10*b*(a + b*x^2)^5) + (3*d*(d*x)^(3/2))/(160*a*b*(a + b*x^2)^4) + (13*d*(d*x)^(3/2))/(640*a^2*b*(a + b*x^2)^3) + (117*d*(d*x)^(3/2))/(5120*a^3*b*(a + b*x^2)^2) + (117*d*(d*x)^(3/2))/(4096*a^4*b*(a + b*x^2)) - (117*d^(5/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(17/4)*b^(7/4)) + (117*d^(5/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(17/4)*b^(7/4)) + (117*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(17/4)*b^(7/4)) - (117*d^(5/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(17/4)*b^(7/4))]
    @test_int [(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, -(d*sqrt(d*x))/(10*b*(a + b*x^2)^5) + (d*sqrt(d*x))/(160*a*b*(a + b*x^2)^4) + (d*sqrt(d*x))/(128*a^2*b*(a + b*x^2)^3) + (11*d*sqrt(d*x))/(1024*a^3*b*(a + b*x^2)^2) + (77*d*sqrt(d*x))/(4096*a^4*b*(a + b*x^2)) - (231*d^(3/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(19/4)*b^(5/4)) + (231*d^(3/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(19/4)*b^(5/4)) - (231*d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(19/4)*b^(5/4)) + (231*d^(3/2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(19/4)*b^(5/4))]
    @test_int [sqrt(d*x)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 16, (d*x)^(3/2)/(10*a*d*(a + b*x^2)^5) + (17*(d*x)^(3/2))/(160*a^2*d*(a + b*x^2)^4) + (221*(d*x)^(3/2))/(1920*a^3*d*(a + b*x^2)^3) + (663*(d*x)^(3/2))/(5120*a^4*d*(a + b*x^2)^2) + (663*(d*x)^(3/2))/(4096*a^5*d*(a + b*x^2)) - (663*sqrt(d)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(21/4)*b^(3/4)) + (663*sqrt(d)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(21/4)*b^(3/4)) + (663*sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(21/4)*b^(3/4)) - (663*sqrt(d)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(21/4)*b^(3/4))]
    @test_int [1/(sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 16, sqrt(d*x)/(10*a*d*(a + b*x^2)^5) + (19*sqrt(d*x))/(160*a^2*d*(a + b*x^2)^4) + (19*sqrt(d*x))/(128*a^3*d*(a + b*x^2)^3) + (209*sqrt(d*x))/(1024*a^4*d*(a + b*x^2)^2) + (1463*sqrt(d*x))/(4096*a^5*d*(a + b*x^2)) - (4389*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(23/4)*b^(1/4)*sqrt(d)) + (4389*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(23/4)*b^(1/4)*sqrt(d)) - (4389*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(23/4)*b^(1/4)*sqrt(d)) + (4389*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(23/4)*b^(1/4)*sqrt(d))]
    @test_int [1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 17, -13923/(4096*a^6*d*sqrt(d*x)) + 1/(10*a*d*sqrt(d*x)*(a + b*x^2)^5) + 21/(160*a^2*d*sqrt(d*x)*(a + b*x^2)^4) + 119/(640*a^3*d*sqrt(d*x)*(a + b*x^2)^3) + 1547/(5120*a^4*d*sqrt(d*x)*(a + b*x^2)^2) + 13923/(20480*a^5*d*sqrt(d*x)*(a + b*x^2)) + (13923*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(25/4)*d^(3/2)) + (13923*b^(1/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(25/4)*d^(3/2))]
    @test_int [1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 17, -33649/(12288*a^6*d*(d*x)^(3/2)) + 1/(10*a*d*(d*x)^(3/2)*(a + b*x^2)^5) + 23/(160*a^2*d*(d*x)^(3/2)*(a + b*x^2)^4) + 437/(1920*a^3*d*(d*x)^(3/2)*(a + b*x^2)^3) + 437/(1024*a^4*d*(d*x)^(3/2)*(a + b*x^2)^2) + 4807/(4096*a^5*d*(d*x)^(3/2)*(a + b*x^2)) + (33649*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(27/4)*d^(5/2)) - (33649*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(27/4)*d^(5/2)) + (33649*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(27/4)*d^(5/2)) - (33649*b^(3/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(27/4)*d^(5/2))]
    @test_int [1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3), x, 18, -13923/(4096*a^6*d*(d*x)^(5/2)) + (69615*b)/(4096*a^7*d^3*sqrt(d*x)) + 1/(10*a*d*(d*x)^(5/2)*(a + b*x^2)^5) + 5/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^4) + 35/(128*a^3*d*(d*x)^(5/2)*(a + b*x^2)^3) + 595/(1024*a^4*d*(d*x)^(5/2)*(a + b*x^2)^2) + 7735/(4096*a^5*d*(d*x)^(5/2)*(a + b*x^2)) - (69615*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(29/4)*d^(7/2)) + (69615*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(8192*sqrt(2)*a^(29/4)*d^(7/2)) + (69615*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(29/4)*d^(7/2)) - (69615*b^(5/4)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(16384*sqrt(2)*a^(29/4)*d^(7/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^(m/2)*(a^2+2*a*b*x^2+b^2*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (2*a*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d*(a + b*x^2)) + (2*b*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^3*(a + b*x^2))]
    @test_int [(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (2*a*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(a + b*x^2)) + (2*b*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*d^3*(a + b*x^2))]
    @test_int [sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 3, (2*a*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(a + b*x^2)) + (2*b*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/sqrt(d*x), x, 3, (2*a*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(a + b*x^2)) + (2*b*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d^3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(3/2), x, 3, (-2*a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*sqrt(d*x)*(a + b*x^2)) + (2*b*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(5/2), x, 3, (-2*a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (2*b*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(a + b*x^2))]
    @test_int [sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(7/2), x, 3, (-2*a*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(d*x)^(5/2)*(a + b*x^2)) - (2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*sqrt(d*x)*(a + b*x^2))]

    @test_int [(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (2*a^3*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d*(a + b*x^2)) + (6*a^2*b*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^3*(a + b*x^2)) + (2*a*b^2*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(19/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*d^7*(a + b*x^2))]
    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (2*a^3*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(a + b*x^2)) + (2*a^2*b*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(17/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*d^7*(a + b*x^2))]
    @test_int [sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (2*a^3*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(a + b*x^2)) + (6*a^2*b*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*d^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/sqrt(d*x), x, 3, (2*a^3*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(a + b*x^2)) + (6*a^2*b*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d^3*(a + b*x^2)) + (2*a*b^2*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*d^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(3/2), x, 3, (-2*a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*sqrt(d*x)*(a + b*x^2)) + (2*a^2*b*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(5/2), x, 3, (-2*a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (6*a^2*b*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(a + b*x^2)) + (6*a*b^2*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d^5*(a + b*x^2)) + (2*b^3*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*d^7*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(d*x)^(7/2), x, 3, (-2*a^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(d*x)^(5/2)*(a + b*x^2)) - (6*a^2*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*sqrt(d*x)*(a + b*x^2)) + (2*a*b^2*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^5*(a + b*x^2)) + (2*b^3*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^7*(a + b*x^2))]

    @test_int [(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (2*a^5*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^3*(a + b*x^2)) + (4*a^3*b^2*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(19/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(23/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(23*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(27/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(27*d^11*(a + b*x^2))]
    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (2*a^5*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(17/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(21/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(25/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(25*d^11*(a + b*x^2))]
    @test_int [sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (2*a^5*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(a + b*x^2)) + (10*a^4*b*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^5*(a + b*x^2)) + (4*a^2*b^3*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(19/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(23/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(23*d^11*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/sqrt(d*x), x, 3, (2*a^5*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(a + b*x^2)) + (2*a^4*b*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(17/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(21/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(21*d^11*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(3/2), x, 3, (-2*a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*sqrt(d*x)*(a + b*x^2)) + (10*a^4*b*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^3*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^7*(a + b*x^2)) + (2*a*b^4*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(19/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(19*d^11*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(5/2), x, 3, (-2*a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(d*x)^(3/2)*(a + b*x^2)) + (10*a^4*b*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(a + b*x^2)) + (4*a^3*b^2*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(9/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(9*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(13/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(13*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(17/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(17*d^11*(a + b*x^2))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(d*x)^(7/2), x, 3, (-2*a^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d*(d*x)^(5/2)*(a + b*x^2)) - (10*a^4*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*sqrt(d*x)*(a + b*x^2)) + (20*a^3*b^2*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^5*(a + b*x^2)) + (20*a^2*b^3*(d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(7*d^7*(a + b*x^2)) + (10*a*b^4*(d*x)^(11/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(11*d^9*(a + b*x^2)) + (2*b^5*(d*x)^(15/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(15*d^11*(a + b*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d*x)^(7/2)/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 13, (-2*a*d^3*sqrt(d*x)*(a + b*x^2))/(b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (2*d*(d*x)^(5/2)*(a + b*x^2))/(5*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(5/4)*d^(7/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(5/4)*d^(7/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(5/4)*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(5/4)*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(5/2)/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 12, (2*d*(d*x)^(3/2)*(a + b*x^2))/(3*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(3/4)*d^(5/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(3/4)*d^(5/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(3/4)*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(3/4)*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(3/2)/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 12, (2*d*sqrt(d*x)*(a + b*x^2))/(b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(1/4)*d^(3/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(1/4)*d^(3/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (a^(1/4)*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (a^(1/4)*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [sqrt(d*x)/sqrt(a^2 + 2*a*b*x^2 + b^2*x^4), x, 11, -((sqrt(d)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(1/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + (sqrt(d)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(1/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(1/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(1/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 11, -(((a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(3/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))) + ((a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(3/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - ((a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(3/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(3/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 12, (-2*(a + b*x^2))/(a*d*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(1/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(5/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(1/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(5/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(5/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(5/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 12, (-2*(a + b*x^2))/(3*a*d*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(3/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(7/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(3/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(7/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(7/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(7/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)), x, 13, (-2*(a + b*x^2))/(5*a*d*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (2*b*(a + b*x^2))/(a^2*d^3*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(5/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(9/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(5/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(sqrt(2)*a^(9/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(9/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(2*sqrt(2)*a^(9/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 15, (-13*d^3*(d*x)^(9/2))/(16*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(13/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*a*d^7*sqrt(d*x)*(a + b*x^2))/(16*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*d^5*(d*x)^(5/2)*(a + b*x^2))/(80*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*a^(5/4)*d^(15/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*a^(5/4)*d^(15/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 14, (-11*d^3*(d*x)^(7/2))/(16*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(11/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*d^5*(d*x)^(3/2)*(a + b*x^2))/(48*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*a^(3/4)*d^(13/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*a^(3/4)*d^(13/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*a^(3/4)*d^(13/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*a^(3/4)*d^(13/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 14, (-9*d^3*(d*x)^(5/2))/(16*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(9/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*d^5*sqrt(d*x)*(a + b*x^2))/(16*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*a^(1/4)*d^(11/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*a^(1/4)*d^(11/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 13, (-7*d^3*(d*x)^(3/2))/(16*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(7/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (21*d^(9/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(1/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (21*d^(9/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(1/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (21*d^(9/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(1/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (21*d^(9/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(1/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 13, (-5*d^3*sqrt(d*x))/(16*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(5/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*d^(7/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(3/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*d^(7/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(3/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(3/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(3/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 13, (3*d*(d*x)^(3/2))/(16*a*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(3/2))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*d^(5/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(5/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*d^(5/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(5/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(5/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(5/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 13, (d*sqrt(d*x))/(16*a*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*sqrt(d*x))/(4*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*d^(3/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(7/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*d^(3/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(7/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(7/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(7/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [sqrt(d*x)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 13, (5*(d*x)^(3/2))/(16*a^2*d*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (d*x)^(3/2)/(4*a*d*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*sqrt(d)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(9/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*sqrt(d)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(9/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(9/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(9/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 13, (7*sqrt(d*x))/(16*a^2*d*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + sqrt(d*x)/(4*a*d*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (21*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(11/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (21*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(11/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (21*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(11/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (21*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(11/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 14, 9/(16*a^2*d*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*d*sqrt(d*x)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*(a + b*x^2))/(16*a^3*d*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*b^(1/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(13/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*b^(1/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(13/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(13/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(13/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 14, 11/(16*a^2*d*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*d*(d*x)^(3/2)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*(a + b*x^2))/(48*a^3*d*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*b^(3/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(15/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*b^(3/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(15/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(15/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(15/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)), x, 15, 13/(16*a^2*d*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(4*a*d*(d*x)^(5/2)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*(a + b*x^2))/(80*a^3*d*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*b*(a + b*x^2))/(16*a^4*d^3*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*b^(5/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(17/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*b^(5/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(32*sqrt(2)*a^(17/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (117*b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(17/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (117*b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(64*sqrt(2)*a^(17/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    @test_int [(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 17, (-1547*d^7*(d*x)^(9/2))/(1024*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(21/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7*d^3*(d*x)^(17/2))/(32*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (119*d^5*(d*x)^(13/2))/(256*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*a*d^11*sqrt(d*x)*(a + b*x^2))/(1024*b^6*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*d^9*(d*x)^(5/2)*(a + b*x^2))/(5120*b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(25/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(25/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(25/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(25/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 16, (-1045*d^7*(d*x)^(7/2))/(1024*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(19/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (19*d^3*(d*x)^(15/2))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (95*d^5*(d*x)^(11/2))/(256*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7315*d^9*(d*x)^(3/2)*(a + b*x^2))/(3072*b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7315*a^(3/4)*d^(21/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(23/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7315*a^(3/4)*d^(21/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(23/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7315*a^(3/4)*d^(21/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(23/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7315*a^(3/4)*d^(21/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(23/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 16, (-663*d^7*(d*x)^(5/2))/(1024*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(17/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (17*d^3*(d*x)^(13/2))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (221*d^5*(d*x)^(9/2))/(768*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3315*d^9*sqrt(d*x)*(a + b*x^2))/(1024*b^5*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(21/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*b^(21/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(21/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*b^(21/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (-385*d^7*(d*x)^(3/2))/(1024*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(15/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (55*d^5*(d*x)^(7/2))/(256*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (1155*d^(17/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(1/4)*b^(19/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*d^(17/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(1/4)*b^(19/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*d^(17/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(1/4)*b^(19/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (1155*d^(17/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(1/4)*b^(19/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (-195*d^7*sqrt(d*x))/(1024*b^4*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(13/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13*d^3*(d*x)^(9/2))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (39*d^5*(d*x)^(5/2))/(256*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (195*d^(15/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(3/4)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (195*d^(15/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(3/4)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (195*d^(15/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(3/4)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (195*d^(15/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(3/4)*b^(17/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (77*d^5*(d*x)^(3/2))/(1024*a*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(11/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (11*d^3*(d*x)^(7/2))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*d^5*(d*x)^(3/2))/(768*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*d^(13/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(5/4)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*d^(13/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(5/4)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*d^(13/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(5/4)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*d^(13/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(5/4)*b^(15/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (15*d^5*sqrt(d*x))/(1024*a*b^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(9/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3*d^3*(d*x)^(5/2))/(32*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (15*d^5*sqrt(d*x))/(256*b^3*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*d^(11/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(7/4)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*d^(11/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(7/4)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*d^(11/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(7/4)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*d^(11/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(7/4)*b^(13/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (35*d^3*(d*x)^(3/2))/(1024*a^2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(7/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7*d^3*(d*x)^(3/2))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7*d^3*(d*x)^(3/2))/(256*a*b^2*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (35*d^(9/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(9/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*d^(9/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(9/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*d^(9/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(9/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (35*d^(9/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(9/4)*b^(11/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (35*d^3*sqrt(d*x))/(3072*a^2*b^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(5/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (5*d^3*sqrt(d*x))/(96*b^2*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*d^3*sqrt(d*x))/(768*a*b^2*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (35*d^(7/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(11/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*d^(7/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(11/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (35*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(11/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (35*d^(7/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(11/4)*b^(9/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (45*d*(d*x)^(3/2))/(1024*a^3*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*(d*x)^(3/2))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (d*(d*x)^(3/2))/(32*a*b*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (9*d*(d*x)^(3/2))/(256*a^2*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*d^(5/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(13/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*d^(5/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(13/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (45*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(13/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (45*d^(5/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(13/4)*b^(7/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (77*d*sqrt(d*x))/(3072*a^3*b*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (d*sqrt(d*x))/(8*b*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (d*sqrt(d*x))/(96*a*b*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (11*d*sqrt(d*x))/(768*a^2*b*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*d^(3/2)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(15/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*d^(3/2)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(15/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (77*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(15/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (77*d^(3/2)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(15/4)*b^(5/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [sqrt(d*x)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 15, (195*(d*x)^(3/2))/(1024*a^4*d*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (d*x)^(3/2)/(8*a*d*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13*(d*x)^(3/2))/(96*a^2*d*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (39*(d*x)^(3/2))/(256*a^3*d*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (195*sqrt(d)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(17/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (195*sqrt(d)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(17/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (195*sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(17/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (195*sqrt(d)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(17/4)*b^(3/4)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/(sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 15, (385*sqrt(d*x))/(1024*a^4*d*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + sqrt(d*x)/(8*a*d*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (5*sqrt(d*x))/(32*a^2*d*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (55*sqrt(d*x))/(256*a^3*d*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (1155*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(19/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(19/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (1155*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(19/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (1155*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(19/4)*b^(1/4)*sqrt(d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 16, 663/(1024*a^4*d*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*d*sqrt(d*x)*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 17/(96*a^2*d*sqrt(d*x)*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 221/(768*a^3*d*sqrt(d*x)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3315*(a + b*x^2))/(1024*a^5*d*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3315*b^(1/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(21/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3315*b^(1/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(21/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (3315*b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(21/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (3315*b^(1/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(21/4)*d^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 16, 1045/(1024*a^4*d*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*d*(d*x)^(3/2)*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 19/(96*a^2*d*(d*x)^(3/2)*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 95/(256*a^3*d*(d*x)^(3/2)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7315*(a + b*x^2))/(3072*a^5*d*(d*x)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7315*b^(3/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(23/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7315*b^(3/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(23/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (7315*b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(23/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (7315*b^(3/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(23/4)*d^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)), x, 17, 1547/(1024*a^4*d*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 1/(8*a*d*(d*x)^(5/2)*(a + b*x^2)^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 7/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + 119/(256*a^3*d*(d*x)^(5/2)*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*(a + b*x^2))/(5120*a^5*d*(d*x)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*b*(a + b*x^2))/(1024*a^6*d^3*sqrt(d*x)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*b^(5/4)*(a + b*x^2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(25/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*b^(5/4)*(a + b*x^2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(d*x))/(a^(1/4)*sqrt(d))))/(2048*sqrt(2)*a^(25/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + (13923*b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(25/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) - (13923*b^(5/4)*(a + b*x^2)*log(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(d)*x + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(d*x)))/(4096*sqrt(2)*a^(25/4)*d^(7/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^p*with*m*symbolic=#

    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 3, (a^6*(d*x)^(1 + m))/(d*(1 + m)) + (6*a^5*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (15*a^4*b^2*(d*x)^(5 + m))/(d^5*(5 + m)) + (20*a^3*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (15*a^2*b^4*(d*x)^(9 + m))/(d^9*(9 + m)) + (6*a*b^5*(d*x)^(11 + m))/(d^11*(11 + m)) + (b^6*(d*x)^(13 + m))/(d^13*(13 + m))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 3, (a^4*(d*x)^(1 + m))/(d*(1 + m)) + (4*a^3*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (6*a^2*b^2*(d*x)^(5 + m))/(d^5*(5 + m)) + (4*a*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (b^4*(d*x)^(9 + m))/(d^9*(9 + m))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, (a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (b^2*(d*x)^(5 + m))/(d^5*(5 + m))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4), x, 2, ((d*x)^(1 + m)*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^2*d*(1 + m))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^2, x, 2, ((d*x)^(1 + m)*HypergeometricFunctions._₂F₁(4, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^4*d*(1 + m))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x, 2, ((d*x)^(1 + m)*HypergeometricFunctions._₂F₁(6, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^6*d*(1 + m))]

    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(1 + m)*(a + b*x^2)) + (5*a^4*b*(d*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(3 + m)*(a + b*x^2)) + (10*a^3*b^2*(d*x)^(5 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^5*(5 + m)*(a + b*x^2)) + (10*a^2*b^3*(d*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^7*(7 + m)*(a + b*x^2)) + (5*a*b^4*(d*x)^(9 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^9*(9 + m)*(a + b*x^2)) + (b^5*(d*x)^(11 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^11*(11 + m)*(a + b*x^2))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(1 + m)*(a + b*x^2)) + (3*a^2*b*(d*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(3 + m)*(a + b*x^2)) + (3*a*b^2*(d*x)^(5 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^5*(5 + m)*(a + b*x^2)) + (b^3*(d*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^7*(7 + m)*(a + b*x^2))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/2), x, 3, (a*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d*(1 + m)*(a + b*x^2)) + (b*(d*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(d^3*(3 + m)*(a + b*x^2))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/2), x, 2, ((d*x)^(1 + m)*(a + b*x^2)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*d*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 2, ((d*x)^(1 + m)*(a + b*x^2)*HypergeometricFunctions._₂F₁(3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^3*d*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 2, ((d*x)^(1 + m)*(a + b*x^2)*HypergeometricFunctions._₂F₁(5, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^5*d*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^p*with*p*symbolic=#

    @test_int [(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, ((d*x)^(1 + m)*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(1, (1/2)*(3 + m + 4*p), (3 + m)/2, -((b*x^2)/a)))/(a*d*(1 + m)), ((d*x)^(1 + m)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -2*p, (3 + m)/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(d*(1 + m)))]

    @test_int [x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 4, -((a^3*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^4*(1 + 2*p))) + (3*a^2*(a + b*x^2)^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(4*b^4*(1 + p)) - (3*a*(a + b*x^2)^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^4*(3 + 2*p)) + ((a + b*x^2)^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(4*b^4*(2 + p))]
    @test_int [x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 4, (a^2*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^3*(1 + 2*p)) - (a*(a + b*x^2)^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^3*(1 + p)) + ((a + b*x^2)^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^3*(3 + 2*p))]
    @test_int [x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 4, -((a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b^2*(1 + 2*p))) + ((a + b*x^2)^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(4*b^2*(1 + p))]
    @test_int [x^1*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, ((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p)/(2*b*(1 + 2*p))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^1, x, 3, -(((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(1, 1 + 2*p, 2*(1 + p), 1 + (b*x^2)/a))/(2*a*(1 + 2*p)))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^3, x, 3, (b*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(2, 1 + 2*p, 2*(1 + p), 1 + (b*x^2)/a))/(2*a^2*(1 + 2*p))]

    @test_int [x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, ((1/5)*x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(5/2, -2*p, 7/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^(2*p)]
    @test_int [x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, ((1/3)*x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(3/2, -2*p, 5/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^(2*p)]
    @test_int [x^0*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, (x*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(1/2, -2*p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^(2*p)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^2, x, 2, -(((a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(-(1/2), -2*p, 1/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*x))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x^4, x, 2, -(((a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(-(3/2), -2*p, -(1/2), -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(3*x^3)))]

    @test_int [(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, (2*(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(5/4, -2*p, 9/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(5*d))]
    @test_int [(d*x)^(1/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p, x, 2, (2*(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(3/4, -2*p, 7/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(3*d))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(1/2), x, 2, (2*sqrt(d*x)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(1/4, -2*p, 5/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*d)]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2), x, 2, -((2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(-(1/4), -2*p, 3/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(d*sqrt(d*x))))]
    @test_int [(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(5/2), x, 2, -((2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*HypergeometricFunctions._₂F₁(-(3/4), -2*p, 1/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^(2*p)*(3*d*(d*x)^(3/2))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(a + b*x^2 + c*x^4), x, 2, (a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7]
    @test_int [x^1*(a + b*x^2 + c*x^4), x, 2, (a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6]
    @test_int [x^0*(a + b*x^2 + c*x^4), x, 1, a*x + (b*x^3)/3 + (c*x^5)/5]
    @test_int [(a + b*x^2 + c*x^4)/x^1, x, 2, (b*x^2)/2 + (c*x^4)/4 + a*log(x)]
    @test_int [(a + b*x^2 + c*x^4)/x^2, x, 2, -(a/x) + b*x + (c*x^3)/3]
    @test_int [(a + b*x^2 + c*x^4)/x^3, x, 2, -(a/(2*x^2)) + (c*x^2)/2 + b*log(x)]
    @test_int [(a + b*x^2 + c*x^4)/x^4, x, 2, -(a/(3*x^3)) - b/x + c*x]
    @test_int [(a + b*x^2 + c*x^4)/x^5, x, 2, -(a/(4*x^4)) - b/(2*x^2) + c*log(x)]
    @test_int [(a + b*x^2 + c*x^4)/x^6, x, 2, -(a/(5*x^5)) - b/(3*x^3) - c/x]
    @test_int [(a + b*x^2 + c*x^4)/x^7, x, 2, -(a/(6*x^6)) - b/(4*x^4) - c/(2*x^2)]
    @test_int [(a + b*x^2 + c*x^4)/x^8, x, 2, -(a/(7*x^7)) - b/(5*x^5) - c/(3*x^3)]

    @test_int [x^2*(a + b*x^2 + c*x^4)^2, x, 2, (a^2*x^3)/3 + (2/5)*a*b*x^5 + (1/7)*(b^2 + 2*a*c)*x^7 + (2/9)*b*c*x^9 + (c^2*x^11)/11]
    @test_int [x^1*(a + b*x^2 + c*x^4)^2, x, 3, (a^2*x^2)/2 + (1/2)*a*b*x^4 + (1/6)*(b^2 + 2*a*c)*x^6 + (1/4)*b*c*x^8 + (c^2*x^10)/10]
    @test_int [x^0*(a + b*x^2 + c*x^4)^2, x, 2, a^2*x + (2/3)*a*b*x^3 + (1/5)*(b^2 + 2*a*c)*x^5 + (2/7)*b*c*x^7 + (c^2*x^9)/9]
    @test_int [(a + b*x^2 + c*x^4)^2/x^1, x, 3, a*b*x^2 + (1/4)*(b^2 + 2*a*c)*x^4 + (1/3)*b*c*x^6 + (c^2*x^8)/8 + a^2*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^2, x, 2, -(a^2/x) + 2*a*b*x + (1/3)*(b^2 + 2*a*c)*x^3 + (2/5)*b*c*x^5 + (c^2*x^7)/7]
    @test_int [(a + b*x^2 + c*x^4)^2/x^3, x, 3, -(a^2/(2*x^2)) + (1/2)*(b^2 + 2*a*c)*x^2 + (1/2)*b*c*x^4 + (c^2*x^6)/6 + 2*a*b*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^4, x, 2, -(a^2/(3*x^3)) - (2*a*b)/x + (b^2 + 2*a*c)*x + (2/3)*b*c*x^3 + (c^2*x^5)/5]
    @test_int [(a + b*x^2 + c*x^4)^2/x^5, x, 3, -(a^2/(4*x^4)) - (a*b)/x^2 + b*c*x^2 + (c^2*x^4)/4 + (b^2 + 2*a*c)*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^6, x, 2, -(a^2/(5*x^5)) - (2*a*b)/(3*x^3) - (b^2 + 2*a*c)/x + 2*b*c*x + (c^2*x^3)/3]
    @test_int [(a + b*x^2 + c*x^4)^2/x^7, x, 3, -(a^2/(6*x^6)) - (a*b)/(2*x^4) - (b^2 + 2*a*c)/(2*x^2) + (c^2*x^2)/2 + 2*b*c*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^8, x, 2, -(a^2/(7*x^7)) - (2*a*b)/(5*x^5) - (b^2 + 2*a*c)/(3*x^3) - (2*b*c)/x + c^2*x]
    @test_int [(a + b*x^2 + c*x^4)^2/x^9, x, 3, -(a^2/(8*x^8)) - (a*b)/(3*x^6) - (b^2 + 2*a*c)/(4*x^4) - (b*c)/x^2 + c^2*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^10, x, 2, -(a^2/(9*x^9)) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - c^2/x]
    @test_int [(a + b*x^2 + c*x^4)^2/x^11, x, 3, -(a^2/(10*x^10)) - (a*b)/(4*x^8) - (b^2 + 2*a*c)/(6*x^6) - (b*c)/(2*x^4) - c^2/(2*x^2)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^12, x, 2, -(a^2/(11*x^11)) - (2*a*b)/(9*x^9) - (b^2 + 2*a*c)/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^13, x, 3, -(a^2/(12*x^12)) - (a*b)/(5*x^10) - (b^2 + 2*a*c)/(8*x^8) - (b*c)/(3*x^6) - c^2/(4*x^4)]

    @test_int [x^2*(a + b*x^2 + c*x^4)^3, x, 2, (a^3*x^3)/3 + (3/5)*a^2*b*x^5 + (3/7)*a*(b^2 + a*c)*x^7 + (1/9)*b*(b^2 + 6*a*c)*x^9 + (3/11)*c*(b^2 + a*c)*x^11 + (3/13)*b*c^2*x^13 + (c^3*x^15)/15]
    @test_int [x^1*(a + b*x^2 + c*x^4)^3, x, 3, (a^3*x^2)/2 + (3/4)*a^2*b*x^4 + (1/2)*a*(b^2 + a*c)*x^6 + (1/8)*b*(b^2 + 6*a*c)*x^8 + (3/10)*c*(b^2 + a*c)*x^10 + (1/4)*b*c^2*x^12 + (c^3*x^14)/14]
    @test_int [x^0*(a + b*x^2 + c*x^4)^3, x, 2, a^3*x + a^2*b*x^3 + (3/5)*a*(b^2 + a*c)*x^5 + (1/7)*b*(b^2 + 6*a*c)*x^7 + (1/3)*c*(b^2 + a*c)*x^9 + (3/11)*b*c^2*x^11 + (c^3*x^13)/13]
    @test_int [(a + b*x^2 + c*x^4)^3/x^1, x, 3, (3/2)*a^2*b*x^2 + (3/4)*a*(b^2 + a*c)*x^4 + (1/6)*b*(b^2 + 6*a*c)*x^6 + (3/8)*c*(b^2 + a*c)*x^8 + (3/10)*b*c^2*x^10 + (c^3*x^12)/12 + a^3*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^2, x, 2, -(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (1/5)*b*(b^2 + 6*a*c)*x^5 + (3/7)*c*(b^2 + a*c)*x^7 + (1/3)*b*c^2*x^9 + (c^3*x^11)/11]
    @test_int [(a + b*x^2 + c*x^4)^3/x^3, x, 3, -(a^3/(2*x^2)) + (3/2)*a*(b^2 + a*c)*x^2 + (1/4)*b*(b^2 + 6*a*c)*x^4 + (1/2)*c*(b^2 + a*c)*x^6 + (3/8)*b*c^2*x^8 + (c^3*x^10)/10 + 3*a^2*b*log(x)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^4, x, 2, -(a^3/(3*x^3)) - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (1/3)*b*(b^2 + 6*a*c)*x^3 + (3/5)*c*(b^2 + a*c)*x^5 + (3/7)*b*c^2*x^7 + (c^3*x^9)/9]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^7/(a + b*x^2 + c*x^4), x, 7, -((b*x^2)/(2*c^2)) + x^4/(4*c) + (b*(b^2 - 3*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^3*sqrt(b^2 - 4*a*c)) + ((b^2 - a*c)*log(a + b*x^2 + c*x^4))/(4*c^3)]
    @test_int [x^5/(a + b*x^2 + c*x^4), x, 6, x^2/(2*c) - ((b^2 - 2*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^2*sqrt(b^2 - 4*a*c)) - (b*log(a + b*x^2 + c*x^4))/(4*c^2)]
    @test_int [x^3/(a + b*x^2 + c*x^4), x, 5, (b*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c*sqrt(b^2 - 4*a*c)) + log(a + b*x^2 + c*x^4)/(4*c)]
    @test_int [x^1/(a + b*x^2 + c*x^4), x, 3, -(atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c))/sqrt(b^2 - 4*a*c))]
    @test_int [1/(x^1*(a + b*x^2 + c*x^4)), x, 7, (b*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a*sqrt(b^2 - 4*a*c)) + log(x)/a - log(a + b*x^2 + c*x^4)/(4*a)]
    @test_int [1/(x^3*(a + b*x^2 + c*x^4)), x, 8, -(1/(2*a*x^2)) - ((b^2 - 2*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*sqrt(b^2 - 4*a*c)) - (b*log(x))/a^2 + (b*log(a + b*x^2 + c*x^4))/(4*a^2)]
    @test_int [1/(x^5*(a + b*x^2 + c*x^4)), x, 8, -(1/(4*a*x^4)) + b/(2*a^2*x^2) + (b*(b^2 - 3*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*sqrt(b^2 - 4*a*c)) + ((b^2 - a*c)*log(x))/a^3 - ((b^2 - a*c)*log(a + b*x^2 + c*x^4))/(4*a^3)]

    @test_int [x^6/(a + b*x^2 + c*x^4), x, 5, -((b*x)/c^2) + x^3/(3*c) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^4/(a + b*x^2 + c*x^4), x, 4, x/c - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2/(a + b*x^2 + c*x^4), x, 3, -((sqrt(b - sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c))) + (sqrt(b + sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c))]
    @test_int [x^0/(a + b*x^2 + c*x^4), x, 3, (sqrt(2)*sqrt(c)*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(2)*sqrt(c)*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [1/(x^2*(a + b*x^2 + c*x^4)), x, 4, -(1/(a*x)) - (sqrt(c)*(1 + b/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(1 - b/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [1/(x^4*(a + b*x^2 + c*x^4)), x, 5, -(1/(3*a*x^3)) + b/(a^2*x) + (sqrt(c)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^2*sqrt(b + sqrt(b^2 - 4*a*c)))]

    @test_int [x^7/(a + b*x^2 + c*x^4)^2, x, 7, -((b*x^2)/(2*c*(b^2 - 4*a*c))) + (x^4*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^2*(b^2 - 4*a*c)^(3/2)) + log(a + b*x^2 + c*x^4)/(4*c^2)]
    @test_int [x^5/(a + b*x^2 + c*x^4)^2, x, 4, (x^2*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*a*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [x^3/(a + b*x^2 + c*x^4)^2, x, 4, (2*a + b*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [x^1/(a + b*x^2 + c*x^4)^2, x, 4, -((b + 2*c*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (2*c*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [1/(x^1*(a + b*x^2 + c*x^4)^2), x, 8, (b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*(b^2 - 4*a*c)^(3/2)) + log(x)/a^2 - log(a + b*x^2 + c*x^4)/(4*a^2)]
    @test_int [1/(x^3*(a + b*x^2 + c*x^4)^2), x, 8, -((b^2 - 3*a*c)/(a^2*(b^2 - 4*a*c)*x^2)) + (b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) - (2*b*log(x))/a^3 + (b*log(a + b*x^2 + c*x^4))/(2*a^3)]

    @test_int [x^8/(a + b*x^2 + c*x^4)^2, x, 6, ((3*b^2 - 10*a*c)*x)/(2*c^2*(b^2 - 4*a*c)) - (b*x^3)/(2*c*(b^2 - 4*a*c)) + (x^5*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((3*b^3 - 13*a*b*c - (3*b^4 - 19*a*b^2*c + 20*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - ((3*b^3 - 13*a*b*c + (3*b^4 - 19*a*b^2*c + 20*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^6/(a + b*x^2 + c*x^4)^2, x, 5, -((b*x)/(2*c*(b^2 - 4*a*c))) + (x^3*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b^2 - 6*a*c - (b*(b^2 - 8*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^2 - 6*a*c + (b*(b^2 - 8*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^4/(a + b*x^2 + c*x^4)^2, x, 4, (x*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b - (b^2 + 4*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^2 + 4*a*c + b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2/(a + b*x^2 + c*x^4)^2, x, 4, -((x*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (sqrt(c)*(2*b - sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(2*b + sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^0/(a + b*x^2 + c*x^4)^2, x, 4, (x*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (sqrt(c)*(b^2 - 12*a*c + b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(b^2 - 12*a*c - b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [1/(x^2*(a + b*x^2 + c*x^4)^2), x, 5, -((3*b^2 - 10*a*c)/(2*a^2*(b^2 - 4*a*c)*x)) + (b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) - (sqrt(c)*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]

    @test_int [x^11/(a + b*x^2 + c*x^4)^3, x, 8, -((b*(b^2 - 7*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2)) + (x^8*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^4*(a*(b^2 - 16*a*c) + b*(b^2 - 10*a*c)*x^2))/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^3*(b^2 - 4*a*c)^(5/2)) + log(a + b*x^2 + c*x^4)/(4*c^3)]
    @test_int [x^9/(a + b*x^2 + c*x^4)^3, x, 5, (x^6*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*a*x^2*(2*a + b*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (6*a^2*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^7/(a + b*x^2 + c*x^4)^3, x, 5, -((x^6*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*b*x^2*(2*a + b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*a*b*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^5/(a + b*x^2 + c*x^4)^3, x, 5, (x^2*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*a*b + (b^2 + 2*a*c)*x^2)/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((b^2 + 2*a*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^3/(a + b*x^2 + c*x^4)^3, x, 5, (2*a + b*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*b*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*b*c*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^1/(a + b*x^2 + c*x^4)^3, x, 5, -((b + 2*c*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*c*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (6*c^2*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/(x^1*(a + b*x^2 + c*x^4)^3), x, 9, (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*(b^2 - 4*a*c)^(5/2)) + log(x)/a^3 - log(a + b*x^2 + c*x^4)/(4*a^3)]
    @test_int [1/(x^3*(a + b*x^2 + c*x^4)^3), x, 9, -((3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*x^2)) + (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x^2 + c*x^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x^2 + c*x^4))/(4*a^4)]

    @test_int [x^10/(a + b*x^2 + c*x^4)^3, x, 7, -((3*b*(b^2 - 8*a*c)*x)/(8*c^2*(b^2 - 4*a*c)^2)) + ((b^2 - 28*a*c)*x^3)/(8*c*(b^2 - 4*a*c)^2) + (x^7*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^5*(12*a*b - (b^2 - 28*a*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^2 - (b^5 - 11*a*b^3*c + 44*a^2*b*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^5 - 11*a*b^3*c + 44*a^2*b*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^8/(a + b*x^2 + c*x^4)^3, x, 6, -(((b^2 + 20*a*c)*x)/(8*c*(b^2 - 4*a*c)^2)) + (x^5*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^3*(12*a*b + (b^2 + 20*a*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((b^3 - 16*a*b*c - (b^4 - 18*a*b^2*c - 40*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^3 - 16*a*b*c + (b^4 - 18*a*b^2*c - 40*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^6/(a + b*x^2 + c*x^4)^3, x, 5, (x^3*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x*(4*a*b + (b^2 + 4*a*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*(b^2 + 4*a*c - (b*(b^2 + 12*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (3*(b^2 + 4*a*c + (b*(b^2 + 12*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^4/(a + b*x^2 + c*x^4)^3, x, 5, (x*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x*(7*b^2 - 4*a*c + 12*b*c*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*sqrt(c)*(3*b^2 + 4*a*c - 2*b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (3*sqrt(c)*(3*b^2 + 4*a*c + 2*b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2/(a + b*x^2 + c*x^4)^3, x, 5, -((x*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (x*(b*(b^2 + 8*a*c) + c*(b^2 + 20*a*c)*x^2))/(8*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (sqrt(c)*(b^2 + 20*a*c + (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(b^2 + 20*a*c - (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^0/(a + b*x^2 + c*x^4)^3, x, 5, (x*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*sqrt(c)*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b*(b^2 - 8*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (3*sqrt(c)*(b^3 - 8*a*b*c - (b^4 - 10*a*b^2*c + 56*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [1/(x^2*(a + b*x^2 + c*x^4)^3), x, 6, -((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(8*a^3*(b^2 - 4*a*c)^2*x)) + (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*x^2)/(8*a^2*(b^2 - 4*a*c)^2*x*(a + b*x^2 + c*x^4)) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) - (5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]

    @test_int [x^5/(a - b*x^2 + c*x^4), x, 6, x^2/(2*c) + ((b^2 - 2*a*c)*atanh((b - 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^2*sqrt(b^2 - 4*a*c)) + (b*log(a - b*x^2 + c*x^4))/(4*c^2)]
    @test_int [x^3/(a - b*x^2 + c*x^4), x, 5, (b*atanh((b - 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c*sqrt(b^2 - 4*a*c)) + log(a - b*x^2 + c*x^4)/(4*c)]
    @test_int [x^1/(a - b*x^2 + c*x^4), x, 3, atanh((b - 2*c*x^2)/sqrt(b^2 - 4*a*c))/sqrt(b^2 - 4*a*c)]
    @test_int [1/(x^1*(a - b*x^2 + c*x^4)), x, 7, (b*atanh((b - 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a*sqrt(b^2 - 4*a*c)) + log(x)/a - log(a - b*x^2 + c*x^4)/(4*a)]
    @test_int [1/(x^3*(a - b*x^2 + c*x^4)), x, 8, -(1/(2*a*x^2)) + ((b^2 - 2*a*c)*atanh((b - 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*sqrt(b^2 - 4*a*c)) + (b*log(x))/a^2 - (b*log(a - b*x^2 + c*x^4))/(4*a^2)]

    @test_int [x^4/(a - b*x^2 + c*x^4), x, 4, x/c - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*c^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2/(a - b*x^2 + c*x^4), x, 3, (sqrt(b - sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c)) - (sqrt(b + sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c))]
    @test_int [x^0/(a - b*x^2 + c*x^4), x, 3, (sqrt(2)*sqrt(c)*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(2)*sqrt(c)*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [1/(x^2*(a - b*x^2 + c*x^4)), x, 4, -(1/(a*x)) + (sqrt(c)*(1 + b/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(1 - b/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a*sqrt(b + sqrt(b^2 - 4*a*c)))]

    @test_int [x^5/(a - b + 2*a*x^2 + a*x^4), x, 6, x^2/(2*a) - ((a + b)*atanh((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*a^(3/2)*sqrt(b)) - log(a - b + 2*a*x^2 + a*x^4)/(2*a)]
    @test_int [x^3/(a - b + 2*a*x^2 + a*x^4), x, 5, atanh((sqrt(a)*(1 + x^2))/sqrt(b))/(2*sqrt(a)*sqrt(b)) + log(a - b + 2*a*x^2 + a*x^4)/(4*a)]
    @test_int [x^1/(a - b + 2*a*x^2 + a*x^4), x, 3, -(atanh((sqrt(a)*(1 + x^2))/sqrt(b))/(2*sqrt(a)*sqrt(b)))]
    @test_int [1/(x^1*(a - b + 2*a*x^2 + a*x^4)), x, 7, (sqrt(a)*atanh((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*(a - b)*sqrt(b)) + log(x)/(a - b) - log(a - b + 2*a*x^2 + a*x^4)/(4*(a - b))]
    @test_int [1/(x^3*(a - b + 2*a*x^2 + a*x^4)), x, 8, -(1/(2*(a - b)*x^2)) - (sqrt(a)*(a + b)*atanh((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*(a - b)^2*sqrt(b)) - (2*a*log(x))/(a - b)^2 + (a*log(a - b + 2*a*x^2 + a*x^4))/(2*(a - b)^2)]

    @test_int [x^4/(a - b + 2*a*x^2 + a*x^4), x, 4, x/a + ((sqrt(a) - sqrt(b))^(3/2)*atan((a^(1/4)*x)/sqrt(sqrt(a) - sqrt(b))))/(2*a^(5/4)*sqrt(b)) - ((sqrt(a) + sqrt(b))^(3/2)*atan((a^(1/4)*x)/sqrt(sqrt(a) + sqrt(b))))/(2*a^(5/4)*sqrt(b))]
    @test_int [x^2/(a - b + 2*a*x^2 + a*x^4), x, 3, -((sqrt(sqrt(a) - sqrt(b))*atan((a^(1/4)*x)/sqrt(sqrt(a) - sqrt(b))))/(2*a^(3/4)*sqrt(b))) + (sqrt(sqrt(a) + sqrt(b))*atan((a^(1/4)*x)/sqrt(sqrt(a) + sqrt(b))))/(2*a^(3/4)*sqrt(b))]
    @test_int [x^0/(a - b + 2*a*x^2 + a*x^4), x, 3, atan((a^(1/4)*x)/sqrt(sqrt(a) - sqrt(b)))/(2*a^(1/4)*sqrt(sqrt(a) - sqrt(b))*sqrt(b)) - atan((a^(1/4)*x)/sqrt(sqrt(a) + sqrt(b)))/(2*a^(1/4)*sqrt(sqrt(a) + sqrt(b))*sqrt(b))]
    @test_int [1/(x^2*(a - b + 2*a*x^2 + a*x^4)), x, 4, -(1/((a - b)*x)) - (a^(1/4)*atan((a^(1/4)*x)/sqrt(sqrt(a) - sqrt(b))))/(2*(sqrt(a) - sqrt(b))^(3/2)*sqrt(b)) + (a^(1/4)*atan((a^(1/4)*x)/sqrt(sqrt(a) + sqrt(b))))/(2*(sqrt(a) + sqrt(b))^(3/2)*sqrt(b))]

    @test_int [x^5/(a + b + 2*a*x^2 + a*x^4), x, 6, x^2/(2*a) + ((a - b)*atan((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*a^(3/2)*sqrt(b)) - log(a + b + 2*a*x^2 + a*x^4)/(2*a)]
    @test_int [x^3/(a + b + 2*a*x^2 + a*x^4), x, 5, -(atan((sqrt(a)*(1 + x^2))/sqrt(b))/(2*sqrt(a)*sqrt(b))) + log(a + b + 2*a*x^2 + a*x^4)/(4*a)]
    @test_int [x^1/(a + b + 2*a*x^2 + a*x^4), x, 3, atan((sqrt(a)*(1 + x^2))/sqrt(b))/(2*sqrt(a)*sqrt(b))]
    @test_int [1/(x^1*(a + b + 2*a*x^2 + a*x^4)), x, 7, -((sqrt(a)*atan((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*sqrt(b)*(a + b))) + log(x)/(a + b) - log(a + b + 2*a*x^2 + a*x^4)/(4*(a + b))]
    @test_int [1/(x^3*(a + b + 2*a*x^2 + a*x^4)), x, 8, -(1/(2*(a + b)*x^2)) + (sqrt(a)*(a - b)*atan((sqrt(a)*(1 + x^2))/sqrt(b)))/(2*sqrt(b)*(a + b)^2) - (2*a*log(x))/(a + b)^2 + (a*log(a + b + 2*a*x^2 + a*x^4))/(2*(a + b)^2)]

    @test_int [x^4/(a + b + 2*a*x^2 + a*x^4), x, 10, x/a + ((a + b + 2*sqrt(a)*sqrt(a + b))*atan((sqrt(-sqrt(a) + sqrt(a + b)) - sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b))))/(2*sqrt(2)*a^(5/4)*sqrt(a + b)*sqrt(sqrt(a) + sqrt(a + b))) - ((a + b + 2*sqrt(a)*sqrt(a + b))*atan((sqrt(-sqrt(a) + sqrt(a + b)) + sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b))))/(2*sqrt(2)*a^(5/4)*sqrt(a + b)*sqrt(sqrt(a) + sqrt(a + b))) + ((a + b - 2*sqrt(a)*sqrt(a + b))*log(sqrt(a + b) - sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2))/(4*sqrt(2)*a^(5/4)*sqrt(a + b)*sqrt(-sqrt(a) + sqrt(a + b))) - ((a + b - 2*sqrt(a)*sqrt(a + b))*log(sqrt(a + b) + sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2))/(4*sqrt(2)*a^(5/4)*sqrt(a + b)*sqrt(-sqrt(a) + sqrt(a + b)))]
    @test_int [x^2/(a + b + 2*a*x^2 + a*x^4), x, 9, -(atan((sqrt(-sqrt(a) + sqrt(a + b)) - sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b)))/(2*sqrt(2)*a^(3/4)*sqrt(sqrt(a) + sqrt(a + b)))) + atan((sqrt(-sqrt(a) + sqrt(a + b)) + sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b)))/(2*sqrt(2)*a^(3/4)*sqrt(sqrt(a) + sqrt(a + b))) + log(sqrt(a + b) - sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2)/(4*sqrt(2)*a^(3/4)*sqrt(-sqrt(a) + sqrt(a + b))) - log(sqrt(a + b) + sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2)/(4*sqrt(2)*a^(3/4)*sqrt(-sqrt(a) + sqrt(a + b)))]
    @test_int [x^0/(a + b + 2*a*x^2 + a*x^4), x, 9, -(atan((sqrt(-sqrt(a) + sqrt(a + b)) - sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b)))/(2*sqrt(2)*a^(1/4)*sqrt(a + b)*sqrt(sqrt(a) + sqrt(a + b)))) + atan((sqrt(-sqrt(a) + sqrt(a + b)) + sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b)))/(2*sqrt(2)*a^(1/4)*sqrt(a + b)*sqrt(sqrt(a) + sqrt(a + b))) - log(sqrt(a + b) - sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2)/(4*sqrt(2)*a^(1/4)*sqrt(a + b)*sqrt(-sqrt(a) + sqrt(a + b))) + log(sqrt(a + b) + sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2)/(4*sqrt(2)*a^(1/4)*sqrt(a + b)*sqrt(-sqrt(a) + sqrt(a + b)))]
    @test_int [1/(x^2*(a + b + 2*a*x^2 + a*x^4)), x, 10, -(1/((a + b)*x)) + (a^(1/4)*(2*sqrt(a) + sqrt(a + b))*atan((sqrt(-sqrt(a) + sqrt(a + b)) - sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b))))/(2*sqrt(2)*(a + b)^(3/2)*sqrt(sqrt(a) + sqrt(a + b))) - (a^(1/4)*(2*sqrt(a) + sqrt(a + b))*atan((sqrt(-sqrt(a) + sqrt(a + b)) + sqrt(2)*a^(1/4)*x)/sqrt(sqrt(a) + sqrt(a + b))))/(2*sqrt(2)*(a + b)^(3/2)*sqrt(sqrt(a) + sqrt(a + b))) + (a^(1/4)*(2*sqrt(a) - sqrt(a + b))*log(sqrt(a + b) - sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2))/(4*sqrt(2)*(a + b)^(3/2)*sqrt(-sqrt(a) + sqrt(a + b))) - (a^(1/4)*(2*sqrt(a) - sqrt(a + b))*log(sqrt(a + b) + sqrt(2)*a^(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*x + sqrt(a)*x^2))/(4*sqrt(2)*(a + b)^(3/2)*sqrt(-sqrt(a) + sqrt(a + b)))]

    @test_int [x/(1 + x^2 + x^4), x, 3, atan((1 + 2*x^2)/sqrt(3))/sqrt(3)]
    @test_int [x/(10 + 2*x^2 + x^4), x, 3, (1/6)*atan((1/3)*(1 + x^2))]

    @test_int [x^2/(20 + 9*x^2 + x^4), x, 3, -2*atan(x/2) + sqrt(5)*atan(x/sqrt(5))]
    @test_int [x^2/(1 - x^2 + x^4), x, 9, (-(1/2))*atan(sqrt(3) - 2*x) + (1/2)*atan(sqrt(3) + 2*x) + log(1 - sqrt(3)*x + x^2)/(4*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(4*sqrt(3))]
    @test_int [x^2/(2 - 2*x^2 + x^4), x, 9, (-(1/2))*sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*x)/sqrt(2*(-1 + sqrt(2)))) + (1/2)*sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*x)/sqrt(2*(-1 + sqrt(2)))) + log(sqrt(2) - sqrt(2*(1 + sqrt(2)))*x + x^2)/(4*sqrt(2*(1 + sqrt(2)))) - log(sqrt(2) + sqrt(2*(1 + sqrt(2)))*x + x^2)/(4*sqrt(2*(1 + sqrt(2))))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^7*sqrt(a + b*x^2 + c*x^4), x, 6, -((b*(7*b^2 - 12*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(256*c^4)) + (x^4*(a + b*x^2 + c*x^4)^(3/2))/(10*c) + ((35*b^2 - 32*a*c - 42*b*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(480*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(512*c^(9/2))]
    @test_int [x^5*sqrt(a + b*x^2 + c*x^4), x, 6, ((5*b^2 - 4*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(128*c^3) - (5*b*(a + b*x^2 + c*x^4)^(3/2))/(48*c^2) + (x^2*(a + b*x^2 + c*x^4)^(3/2))/(8*c) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(256*c^(7/2))]
    @test_int [x^3*sqrt(a + b*x^2 + c*x^4), x, 5, -((b*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c^2)) + (a + b*x^2 + c*x^4)^(3/2)/(6*c) + (b*(b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(5/2))]
    @test_int [x^1*sqrt(a + b*x^2 + c*x^4), x, 4, ((b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*c) - ((b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*c^(3/2))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^1, x, 7, (1/2)*sqrt(a + b*x^2 + c*x^4) - (1/2)*sqrt(a)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))) + (b*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(c))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^3, x, 7, -(sqrt(a + b*x^2 + c*x^4)/(2*x^2)) - (b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(a)) + (1/2)*sqrt(c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^5, x, 4, -(((2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*a*x^4)) + ((b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(16*a^(3/2))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^7, x, 5, (b*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*a^2*x^4) - (a + b*x^2 + c*x^4)^(3/2)/(6*a*x^6) - (b*(b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(32*a^(5/2))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^9, x, 6, -(((5*b^2 - 4*a*c)*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(128*a^3*x^4)) - (a + b*x^2 + c*x^4)^(3/2)/(8*a*x^8) + (5*b*(a + b*x^2 + c*x^4)^(3/2))/(48*a^2*x^6) + ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(256*a^(7/2))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^11, x, 7, (b*(7*b^2 - 12*a*c)*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(256*a^4*x^4) - (a + b*x^2 + c*x^4)^(3/2)/(10*a*x^10) + (7*b*(a + b*x^2 + c*x^4)^(3/2))/(80*a^2*x^8) - ((35*b^2 - 32*a*c)*(a + b*x^2 + c*x^4)^(3/2))/(480*a^3*x^6) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(512*a^(9/2))]

    @test_int [x^4*sqrt(a + b*x^2 + c*x^4), x, 5, -((2*(2*b^2 - 5*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(105*c^2)) + (b*(8*b^2 - 29*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(105*c^(5/2)*(sqrt(a) + sqrt(c)*x^2)) + (x^3*(b + 5*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(35*c) - (a^(1/4)*b*(8*b^2 - 29*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(105*c^(11/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(8*b^3 - 29*a*b*c + 2*sqrt(a)*sqrt(c)*(2*b^2 - 5*a*c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(210*c^(11/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^2*sqrt(a + b*x^2 + c*x^4), x, 4, -((2*(b^2 - 3*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(15*c^(3/2)*(sqrt(a) + sqrt(c)*x^2))) + (x*(b + 3*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(15*c) + (2*a^(1/4)*(b^2 - 3*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(15*c^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(2*b^2 + sqrt(a)*b*sqrt(c) - 6*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(30*c^(7/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^0*sqrt(a + b*x^2 + c*x^4), x, 4, (1/3)*x*sqrt(a + b*x^2 + c*x^4) + (b*x*sqrt(a + b*x^2 + c*x^4))/(3*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*b*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*c^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(b + 2*sqrt(a)*sqrt(c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*c^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^2, x, 4, -(sqrt(a + b*x^2 + c*x^4)/x) + (2*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(sqrt(a) + sqrt(c)*x^2) - (2*a^(1/4)*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/sqrt(a + b*x^2 + c*x^4) + ((b + 2*sqrt(a)*sqrt(c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(1/4)*c^(1/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^4, x, 5, -(sqrt(a + b*x^2 + c*x^4)/(3*x^3)) - (b*sqrt(a + b*x^2 + c*x^4))/(3*a*x) + (b*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(3*a*(sqrt(a) + sqrt(c)*x^2)) - (b*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*a^(3/4)*sqrt(a + b*x^2 + c*x^4)) + ((b + 2*sqrt(a)*sqrt(c))*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*a^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [sqrt(a + b*x^2 + c*x^4)/x^6, x, 6, -(sqrt(a + b*x^2 + c*x^4)/(5*x^5)) - (b*sqrt(a + b*x^2 + c*x^4))/(15*a*x^3) + (2*(b^2 - 3*a*c)*sqrt(a + b*x^2 + c*x^4))/(15*a^2*x) - (2*sqrt(c)*(b^2 - 3*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(15*a^2*(sqrt(a) + sqrt(c)*x^2)) + (2*c^(1/4)*(b^2 - 3*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(15*a^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (c^(1/4)*(2*b^2 + sqrt(a)*b*sqrt(c) - 6*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(30*a^(7/4)*sqrt(a + b*x^2 + c*x^4))]

    @test_int [x^7*(a + b*x^2 + c*x^4)^(3/2), x, 7, (3*b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(2048*c^5) - (b*(3*b^2 - 4*a*c)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(256*c^4) + (x^4*(a + b*x^2 + c*x^4)^(5/2))/(14*c) + ((21*b^2 - 16*a*c - 30*b*c*x^2)*(a + b*x^2 + c*x^4)^(5/2))/(560*c^3) - (3*b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4096*c^(11/2))]
    @test_int [x^5*(a + b*x^2 + c*x^4)^(3/2), x, 7, -(((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(1024*c^4)) + ((7*b^2 - 4*a*c)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*c^3) - (7*b*(a + b*x^2 + c*x^4)^(5/2))/(120*c^2) + (x^2*(a + b*x^2 + c*x^4)^(5/2))/(12*c) + ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(2048*c^(9/2))]
    @test_int [x^3*(a + b*x^2 + c*x^4)^(3/2), x, 6, (3*b*(b^2 - 4*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(256*c^3) - (b*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*c^2) + (a + b*x^2 + c*x^4)^(5/2)/(10*c) - (3*b*(b^2 - 4*a*c)^2*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(512*c^(7/2))]
    @test_int [x^1*(a + b*x^2 + c*x^4)^(3/2), x, 5, -((3*(b^2 - 4*a*c)*(b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(128*c^2)) + ((b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(16*c) + (3*(b^2 - 4*a*c)^2*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(256*c^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^1, x, 8, ((b^2 + 8*a*c + 2*b*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c) + (1/6)*(a + b*x^2 + c*x^4)^(3/2) - (1/2)*a^(3/2)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))) - (b*(b^2 - 12*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(3/2))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^3, x, 8, (3/8)*(3*b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4) - (a + b*x^2 + c*x^4)^(3/2)/(2*x^2) - (3/4)*sqrt(a)*b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))) + (3*(b^2 + 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*sqrt(c))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^5, x, 8, -((3*(b - 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*x^2)) - (a + b*x^2 + c*x^4)^(3/2)/(4*x^4) - (3*(b^2 + 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(16*sqrt(a)) + (3/4)*b*sqrt(c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^7, x, 8, -(((2*a*b + (b^2 + 8*a*c)*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*a*x^4)) - (a + b*x^2 + c*x^4)^(3/2)/(6*x^6) + (b*(b^2 - 12*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(32*a^(3/2)) + (1/2)*c^(3/2)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^9, x, 5, (3*(b^2 - 4*a*c)*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(128*a^2*x^4) - ((2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(16*a*x^8) - (3*(b^2 - 4*a*c)^2*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(256*a^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^11, x, 6, -((3*b*(b^2 - 4*a*c)*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(256*a^3*x^4)) + (b*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*a^2*x^8) - (a + b*x^2 + c*x^4)^(5/2)/(10*a*x^10) + (3*b*(b^2 - 4*a*c)^2*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(512*a^(7/2))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^13, x, 7, ((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^2)*sqrt(a + b*x^2 + c*x^4))/(1024*a^4*x^4) - ((7*b^2 - 4*a*c)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*a^3*x^8) - (a + b*x^2 + c*x^4)^(5/2)/(12*a*x^12) + (7*b*(a + b*x^2 + c*x^4)^(5/2))/(120*a^2*x^10) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2048*a^(9/2))]

    @test_int [x^4*(a + b*x^2 + c*x^4)^(3/2), x, 6, ((8*b^4 - 51*a*b^2*c + 60*a^2*c^2)*x*sqrt(a + b*x^2 + c*x^4))/(1155*c^3) - (8*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(1155*c^(7/2)*(sqrt(a) + sqrt(c)*x^2)) - (x^3*(b*(2*b^2 + a*c) + 10*c*(b^2 - 3*a*c)*x^2)*sqrt(a + b*x^2 + c*x^4))/(385*c^2) + (x^3*(b + 3*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(33*c) + (8*a^(1/4)*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(1155*c^(15/4)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(8*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c) + sqrt(a)*sqrt(c)*(8*b^4 - 51*a*b^2*c + 60*a^2*c^2))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2310*c^(15/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^2*(a + b*x^2 + c*x^4)^(3/2), x, 5, ((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x*sqrt(a + b*x^2 + c*x^4))/(315*c^(5/2)*(sqrt(a) + sqrt(c)*x^2)) - (x*(b*(4*b^2 - 9*a*c) + 6*c*(2*b^2 - 7*a*c)*x^2)*sqrt(a + b*x^2 + c*x^4))/(315*c^2) + (x*(3*b + 7*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) - (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(315*c^(11/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2 + 4*sqrt(a)*b*sqrt(c)*(b^2 - 6*a*c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(630*c^(11/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^0*(a + b*x^2 + c*x^4)^(3/2), x, 5, -((2*b*(b^2 - 8*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(35*c^(3/2)*(sqrt(a) + sqrt(c)*x^2))) + (x*(b^2 + 10*a*c + 3*b*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(35*c) + (1/7)*x*(a + b*x^2 + c*x^4)^(3/2) + (2*a^(1/4)*b*(b^2 - 8*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(35*c^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(sqrt(a)*sqrt(c)*(b^2 - 20*a*c) + 2*b*(b^2 - 8*a*c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(70*c^(7/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^2, x, 5, ((b^2 + 12*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(5*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (1/5)*x*(7*b + 6*c*x^2)*sqrt(a + b*x^2 + c*x^4) - (a + b*x^2 + c*x^4)^(3/2)/x - (a^(1/4)*(b^2 + 12*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(5*c^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(b^2 + 8*sqrt(a)*b*sqrt(c) + 12*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(10*c^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^4, x, 5, (8*b*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(3*(sqrt(a) + sqrt(c)*x^2)) - ((3*b - 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(3*x) - (a + b*x^2 + c*x^4)^(3/2)/(3*x^3) - (8*a^(1/4)*b*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*sqrt(a + b*x^2 + c*x^4)) + ((3*b^2 + 8*sqrt(a)*b*sqrt(c) + 4*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*a^(1/4)*c^(1/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^6, x, 6, -(((b^2 + 12*a*c)*sqrt(a + b*x^2 + c*x^4))/(5*a*x)) + (sqrt(c)*(b^2 + 12*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(5*a*(sqrt(a) + sqrt(c)*x^2)) - ((b - 6*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(5*x^3) - (a + b*x^2 + c*x^4)^(3/2)/(5*x^5) - (c^(1/4)*(b^2 + 12*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(5*a^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (c^(1/4)*(b^2 + 8*sqrt(a)*b*sqrt(c) + 12*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(10*a^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/x^8, x, 7, -(((b^2 - 20*a*c)*sqrt(a + b*x^2 + c*x^4))/(35*a*x^3)) + (2*b*(b^2 - 8*a*c)*sqrt(a + b*x^2 + c*x^4))/(35*a^2*x) - (2*b*sqrt(c)*(b^2 - 8*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(35*a^2*(sqrt(a) + sqrt(c)*x^2)) - (3*(b + 10*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(35*x^5) - (a + b*x^2 + c*x^4)^(3/2)/(7*x^7) + (2*b*c^(1/4)*(b^2 - 8*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(35*a^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (c^(1/4)*(sqrt(a)*sqrt(c)*(b^2 - 20*a*c) + 2*b*(b^2 - 8*a*c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(70*a^(7/4)*sqrt(a + b*x^2 + c*x^4))]

    @test_int [sqrt(3 - 2*x^2 - x^4), x, 5, (1/3)*x*sqrt(3 - 2*x^2 - x^4) - (2*Elliptic.E(asin(x), -(1/3)))/sqrt(3) + (4*Elliptic.F(asin(x), -(1/3)))/sqrt(3)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^7/sqrt(a + b*x^2 + c*x^4), x, 5, (x^4*sqrt(a + b*x^2 + c*x^4))/(6*c) + ((15*b^2 - 16*a*c - 10*b*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(48*c^3) - (b*(5*b^2 - 12*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(7/2))]
    @test_int [x^5/sqrt(a + b*x^2 + c*x^4), x, 5, -((3*b*sqrt(a + b*x^2 + c*x^4))/(8*c^2)) + (x^2*sqrt(a + b*x^2 + c*x^4))/(4*c) + ((3*b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*c^(5/2))]
    @test_int [x^3/sqrt(a + b*x^2 + c*x^4), x, 4, sqrt(a + b*x^2 + c*x^4)/(2*c) - (b*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*c^(3/2))]
    @test_int [x^1/sqrt(a + b*x^2 + c*x^4), x, 3, atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))/(2*sqrt(c))]
    @test_int [1/(x^1*sqrt(a + b*x^2 + c*x^4)), x, 3, -(atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4)))/(2*sqrt(a)))]
    @test_int [1/(x^3*sqrt(a + b*x^2 + c*x^4)), x, 4, -(sqrt(a + b*x^2 + c*x^4)/(2*a*x^2)) + (b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*a^(3/2))]
    @test_int [1/(x^5*sqrt(a + b*x^2 + c*x^4)), x, 5, -(sqrt(a + b*x^2 + c*x^4)/(4*a*x^4)) + (3*b*sqrt(a + b*x^2 + c*x^4))/(8*a^2*x^2) - ((3*b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(16*a^(5/2))]
    @test_int [1/(x^7*sqrt(a + b*x^2 + c*x^4)), x, 6, -(sqrt(a + b*x^2 + c*x^4)/(6*a*x^6)) + (5*b*sqrt(a + b*x^2 + c*x^4))/(24*a^2*x^4) - ((15*b^2 - 16*a*c)*sqrt(a + b*x^2 + c*x^4))/(48*a^3*x^2) + (b*(5*b^2 - 12*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(32*a^(7/2))]

    @test_int [x^4/sqrt(a + b*x^2 + c*x^4), x, 4, (x*sqrt(a + b*x^2 + c*x^4))/(3*c) - (2*b*x*sqrt(a + b*x^2 + c*x^4))/(3*c^(3/2)*(sqrt(a) + sqrt(c)*x^2)) + (2*a^(1/4)*b*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*c^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(2*b + sqrt(a)*sqrt(c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*c^(7/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^2/sqrt(a + b*x^2 + c*x^4), x, 3, (x*sqrt(a + b*x^2 + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(c^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*c^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^0/sqrt(a + b*x^2 + c*x^4), x, 1, ((sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(1/4)*c^(1/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(x^2*sqrt(a + b*x^2 + c*x^4)), x, 5, -(sqrt(a + b*x^2 + c*x^4)/(a*x)) + (sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(a*(sqrt(a) + sqrt(c)*x^2)) - (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(a^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(x^4*sqrt(a + b*x^2 + c*x^4)), x, 5, -(sqrt(a + b*x^2 + c*x^4)/(3*a*x^3)) + (2*b*sqrt(a + b*x^2 + c*x^4))/(3*a^2*x) - (2*b*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(3*a^2*(sqrt(a) + sqrt(c)*x^2)) + (2*b*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*a^(7/4)*sqrt(a + b*x^2 + c*x^4)) - ((2*b + sqrt(a)*sqrt(c))*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*a^(7/4)*sqrt(a + b*x^2 + c*x^4))]

    @test_int [x^7/sqrt(a + b*x^2 - c*x^4), x, 5, -((x^4*sqrt(a + b*x^2 - c*x^4))/(6*c)) - ((15*b^2 + 16*a*c + 10*b*c*x^2)*sqrt(a + b*x^2 - c*x^4))/(48*c^3) - (b*(5*b^2 + 12*a*c)*atan((b - 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 - c*x^4))))/(32*c^(7/2))]
    @test_int [x^5/sqrt(a + b*x^2 - c*x^4), x, 5, -((3*b*sqrt(a + b*x^2 - c*x^4))/(8*c^2)) - (x^2*sqrt(a + b*x^2 - c*x^4))/(4*c) - ((3*b^2 + 4*a*c)*atan((b - 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 - c*x^4))))/(16*c^(5/2))]
    @test_int [x^3/sqrt(a + b*x^2 - c*x^4), x, 4, -(sqrt(a + b*x^2 - c*x^4)/(2*c)) - (b*atan((b - 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 - c*x^4))))/(4*c^(3/2))]
    @test_int [x^1/sqrt(a + b*x^2 - c*x^4), x, 3, -(atan((b - 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 - c*x^4)))/(2*sqrt(c)))]
    @test_int [1/(x^1*sqrt(-a + b*x^2 + c*x^4)), x, 3, -(atan((2*a - b*x^2)/(2*sqrt(a)*sqrt(-a + b*x^2 + c*x^4)))/(2*sqrt(a)))]
    @test_int [1/(x^3*sqrt(-a + b*x^2 + c*x^4)), x, 4, sqrt(-a + b*x^2 + c*x^4)/(2*a*x^2) - (b*atan((2*a - b*x^2)/(2*sqrt(a)*sqrt(-a + b*x^2 + c*x^4))))/(4*a^(3/2))]
    @test_int [1/(x^5*sqrt(-a + b*x^2 + c*x^4)), x, 5, sqrt(-a + b*x^2 + c*x^4)/(4*a*x^4) + (3*b*sqrt(-a + b*x^2 + c*x^4))/(8*a^2*x^2) - ((3*b^2 + 4*a*c)*atan((2*a - b*x^2)/(2*sqrt(a)*sqrt(-a + b*x^2 + c*x^4))))/(16*a^(5/2))]
    @test_int [1/(x^7*sqrt(-a + b*x^2 + c*x^4)), x, 6, sqrt(-a + b*x^2 + c*x^4)/(6*a*x^6) + (5*b*sqrt(-a + b*x^2 + c*x^4))/(24*a^2*x^4) + ((15*b^2 + 16*a*c)*sqrt(-a + b*x^2 + c*x^4))/(48*a^3*x^2) - (b*(5*b^2 + 12*a*c)*atan((2*a - b*x^2)/(2*sqrt(a)*sqrt(-a + b*x^2 + c*x^4))))/(32*a^(7/2))]

    @test_int [x^4/sqrt(a + b*x^2 - c*x^4), x, 5, -((x*sqrt(a + b*x^2 - c*x^4))/(3*c)) - (b*(b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.E(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(3*sqrt(2)*c^(5/2)*sqrt(a + b*x^2 - c*x^4)) + (sqrt(b + sqrt(b^2 + 4*a*c))*(b^2 + a*c - b*sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.F(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(3*sqrt(2)*c^(5/2)*sqrt(a + b*x^2 - c*x^4))]
    @test_int [x^2/sqrt(a + b*x^2 - c*x^4), x, 4, -(((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.E(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(2*sqrt(2)*c^(3/2)*sqrt(a + b*x^2 - c*x^4))) + ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.F(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(2*sqrt(2)*c^(3/2)*sqrt(a + b*x^2 - c*x^4))]
    @test_int [x^0/sqrt(a + b*x^2 - c*x^4), x, 2, (sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.F(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(a + b*x^2 - c*x^4))]
    @test_int [1/(x^2*sqrt(a + b*x^2 - c*x^4)), x, 6, -(sqrt(a + b*x^2 - c*x^4)/(a*x)) + ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.E(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(2*sqrt(2)*a*sqrt(c)*sqrt(a + b*x^2 - c*x^4)) - ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.F(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(2*sqrt(2)*a*sqrt(c)*sqrt(a + b*x^2 - c*x^4))]
    @test_int [1/(x^4*sqrt(a + b*x^2 - c*x^4)), x, 6, -(sqrt(a + b*x^2 - c*x^4)/(3*a*x^3)) + (2*b*sqrt(a + b*x^2 - c*x^4))/(3*a^2*x) - (b*(b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.E(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(3*sqrt(2)*a^2*sqrt(c)*sqrt(a + b*x^2 - c*x^4)) + (sqrt(b + sqrt(b^2 + 4*a*c))*(b^2 + a*c - b*sqrt(b^2 + 4*a*c))*sqrt(1 - (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*sqrt(1 - (2*c*x^2)/(b + sqrt(b^2 + 4*a*c)))*Elliptic.F(asin((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), (b + sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c))))/(3*sqrt(2)*a^2*sqrt(c)*sqrt(a + b*x^2 - c*x^4))]

    @test_int [x^9/(a + b*x^2 + c*x^4)^(3/2), x, 6, (x^6*(2*a + b*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (b*x^4*sqrt(a + b*x^2 + c*x^4))/(c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 12*a*c)*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*c^3*(b^2 - 4*a*c)) + (3*(5*b^2 - 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*c^(7/2))]
    @test_int [x^7/(a + b*x^2 + c*x^4)^(3/2), x, 5, (x^4*(2*a + b*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) + ((3*b^2 - 8*a*c - 2*b*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(2*c^2*(b^2 - 4*a*c)) - (3*b*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*c^(5/2))]
    @test_int [x^5/(a + b*x^2 + c*x^4)^(3/2), x, 5, (x^2*(2*a + b*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (b*sqrt(a + b*x^2 + c*x^4))/(c*(b^2 - 4*a*c)) + atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))/(2*c^(3/2))]
    @test_int [x^3/(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*a + b*x^2)/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^1/(a + b*x^2 + c*x^4)^(3/2), x, 2, -((b + 2*c*x^2)/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)))]
    @test_int [1/(x^1*(a + b*x^2 + c*x^4)^(3/2)), x, 5, (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4)))/(2*a^(3/2))]
    @test_int [1/(x^3*(a + b*x^2 + c*x^4)^(3/2)), x, 5, (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*x^2*sqrt(a + b*x^2 + c*x^4)) - ((3*b^2 - 8*a*c)*sqrt(a + b*x^2 + c*x^4))/(2*a^2*(b^2 - 4*a*c)*x^2) + (3*b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*a^(5/2))]
    @test_int [1/(x^5*(a + b*x^2 + c*x^4)^(3/2)), x, 6, (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*x^4*sqrt(a + b*x^2 + c*x^4)) - ((5*b^2 - 12*a*c)*sqrt(a + b*x^2 + c*x^4))/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(15*b^2 - 52*a*c)*sqrt(a + b*x^2 + c*x^4))/(8*a^3*(b^2 - 4*a*c)*x^2) - (3*(5*b^2 - 4*a*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(16*a^(7/2))]

    @test_int [x^6/(a + b*x^2 + c*x^4)^(3/2), x, 5, (x^3*(2*a + b*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (b*x*sqrt(a + b*x^2 + c*x^4))/(c*(b^2 - 4*a*c)) + (2*(b^2 - 3*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(c^(3/2)*(b^2 - 4*a*c)*(sqrt(a) + sqrt(c)*x^2)) - (2*a^(1/4)*(b^2 - 3*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(c^(7/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(2*b^2 + sqrt(a)*b*sqrt(c) - 6*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*c^(7/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^4/(a + b*x^2 + c*x^4)^(3/2), x, 4, (x*(2*a + b*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (b*x*sqrt(a + b*x^2 + c*x^4))/(sqrt(c)*(b^2 - 4*a*c)*(sqrt(a) + sqrt(c)*x^2)) + (a^(1/4)*b*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(c^(3/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*(b - 2*sqrt(a)*sqrt(c))*c^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^2/(a + b*x^2 + c*x^4)^(3/2), x, 4, -((x*(b + 2*c*x^2))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4))) + (2*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/((b^2 - 4*a*c)*(sqrt(a) + sqrt(c)*x^2)) - (2*a^(1/4)*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/((b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) + ((sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(1/4)*(b - 2*sqrt(a)*sqrt(c))*c^(1/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^0/(a + b*x^2 + c*x^4)^(3/2), x, 4, (x*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (b*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(a*(b^2 - 4*a*c)*(sqrt(a) + sqrt(c)*x^2)) + (b*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(a^(3/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) - (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(3/4)*(b - 2*sqrt(a)*sqrt(c))*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(x^2*(a + b*x^2 + c*x^4)^(3/2)), x, 5, (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*x*sqrt(a + b*x^2 + c*x^4)) - (2*(b^2 - 3*a*c)*sqrt(a + b*x^2 + c*x^4))/(a^2*(b^2 - 4*a*c)*x) + (2*sqrt(c)*(b^2 - 3*a*c)*x*sqrt(a + b*x^2 + c*x^4))/(a^2*(b^2 - 4*a*c)*(sqrt(a) + sqrt(c)*x^2)) - (2*c^(1/4)*(b^2 - 3*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(a^(7/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4)) + (c^(1/4)*(2*b^2 + sqrt(a)*b*sqrt(c) - 6*a*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(7/4)*(b^2 - 4*a*c)*sqrt(a + b*x^2 + c*x^4))]

    #= Must*check*to*ensure*constant*term*is*not*zero! =#
    @test_int [x^4/sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4), x, 3, -((2*b*sqrt(b*x^2 + c*x^4))/(3*c^2*x)) + (x*sqrt(b*x^2 + c*x^4))/(3*c)]
    @test_int [x^3/sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4), x, 5, sqrt(b*x^2 + c*x^4)/(2*c) - (b*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*c^(3/2))]
    @test_int [x^2/sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4), x, 2, sqrt(b*x^2 + c*x^4)/(c*x)]
    @test_int [x^1/sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4), x, 4, atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))/sqrt(c)]
    @test_int [x^0/sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4), x, 3, -(atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))/sqrt(b))]
    @test_int [1/(x^1*sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4)), x, 2, -(sqrt(b*x^2 + c*x^4)/(b*x^2))]
    @test_int [1/(x^2*sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4)), x, 4, -(sqrt(b*x^2 + c*x^4)/(2*b*x^3)) + (c*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*b^(3/2))]
    @test_int [1/(x^3*sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4)), x, 3, -(sqrt(b*x^2 + c*x^4)/(3*b*x^4)) + (2*c*sqrt(b*x^2 + c*x^4))/(3*b^2*x^2)]
    @test_int [1/(x^4*sqrt(2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4)), x, 5, -(sqrt(b*x^2 + c*x^4)/(4*b*x^5)) + (3*c*sqrt(b*x^2 + c*x^4))/(8*b^2*x^3) - (3*c^2*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(5/2))]

    #= Must*check*to*ensure*coefficient*is*not*zero! =#
    @test_int [x^4/sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4), x, 3, (x*sqrt(a + c*x^4))/(3*c) - (a^(3/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(6*c^(5/4)*sqrt(a + c*x^4))]
    @test_int [x^3/sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4), x, 2, sqrt(a + c*x^4)/(2*c)]
    @test_int [x^2/sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4), x, 4, (x*sqrt(a + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(c^(3/4)*sqrt(a + c*x^4)) + (a^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [x^1/sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4), x, 4, atanh((sqrt(c)*x^2)/sqrt(a + c*x^4))/(2*sqrt(c))]
    @test_int [x^0/sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4), x, 2, ((sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [1/(x^1*sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4)), x, 4, -(atanh(sqrt(a + c*x^4)/sqrt(a))/(2*sqrt(a)))]
    @test_int [1/(x^2*sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4)), x, 5, -(sqrt(a + c*x^4)/(a*x)) + (sqrt(c)*x*sqrt(a + c*x^4))/(a*(sqrt(a) + sqrt(c)*x^2)) - (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(a^(3/4)*sqrt(a + c*x^4)) + (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(3/4)*sqrt(a + c*x^4))]
    @test_int [1/(x^3*sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4)), x, 2, -(sqrt(a + c*x^4)/(2*a*x^2))]
    @test_int [1/(x^4*sqrt(a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4)), x, 3, -(sqrt(a + c*x^4)/(3*a*x^3)) - (c^(3/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(6*a^(5/4)*sqrt(a + c*x^4))]

    #= Must*check*to*ensure*coefficient*is*not*zero! =#
    @test_int [x^4/sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4), x, 5, -((3*a*x*sqrt(a + b*x^2))/(8*b^2)) + (x^3*sqrt(a + b*x^2))/(4*b) + (3*a^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(5/2))]
    @test_int [x^3/sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4), x, 4, -((a*sqrt(a + b*x^2))/b^2) + (a + b*x^2)^(3/2)/(3*b^2)]
    @test_int [x^2/sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4), x, 4, (x*sqrt(a + b*x^2))/(2*b) - (a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(3/2))]
    @test_int [x^1/sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4), x, 2, sqrt(a + b*x^2)/b]
    @test_int [x^0/sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4), x, 3, atanh((sqrt(b)*x)/sqrt(a + b*x^2))/sqrt(b)]
    @test_int [1/(x^1*sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4)), x, 4, -(atanh(sqrt(a + b*x^2)/sqrt(a))/sqrt(a))]
    @test_int [1/(x^2*sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4)), x, 2, -(sqrt(a + b*x^2)/(a*x))]
    @test_int [1/(x^3*sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4)), x, 5, -(sqrt(a + b*x^2)/(2*a*x^2)) + (b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(3/2))]
    @test_int [1/(x^4*sqrt(a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4)), x, 3, -(sqrt(a + b*x^2)/(3*a*x^3)) + (2*b*sqrt(a + b*x^2))/(3*a^2*x)]

    #= Must*check*to*ensure*constant*term*is*not*zero! =#
    @test_int [x^4/sqrt(2 + 2*a - 2*(1 + a) + c*x^4), x, 3, x^5/(3*sqrt(c*x^4))]
    @test_int [x^3/sqrt(2 + 2*a - 2*(1 + a) + c*x^4), x, 3, x^4/(2*sqrt(c*x^4))]
    @test_int [x^2/sqrt(2 + 2*a - 2*(1 + a) + c*x^4), x, 3, x^3/sqrt(c*x^4)]
    @test_int [x^1/sqrt(2 + 2*a - 2*(1 + a) + c*x^4), x, 3, x^2*log(x)/sqrt(c*x^4)]
    @test_int [x^0/sqrt(2 + 2*a - 2*(1 + a) + c*x^4), x, 3, -(x/sqrt(c*x^4))]
    @test_int [1/(x^1*sqrt(2 + 2*a - 2*(1 + a) + c*x^4)), x, 3, -(1/(2*sqrt(c*x^4)))]
    @test_int [1/(x^2*sqrt(2 + 2*a - 2*(1 + a) + c*x^4)), x, 3, -(1/(3*x*sqrt(c*x^4)))]
    @test_int [1/(x^3*sqrt(2 + 2*a - 2*(1 + a) + c*x^4)), x, 3, -(1/(4*x^2*sqrt(c*x^4)))]
    @test_int [1/(x^4*sqrt(2 + 2*a - 2*(1 + a) + c*x^4)), x, 3, -(1/(5*x^3*sqrt(c*x^4)))]

    #= Must*check*to*ensure*coefficient*is*not*zero! =#
    @test_int [x^4/sqrt(a + (2 + 2*c - 2*(1 + c))*x^4), x, 3, x^5/(5*sqrt(a))]
    @test_int [x^3/sqrt(a + (2 + 2*c - 2*(1 + c))*x^4), x, 3, x^4/(4*sqrt(a))]
    @test_int [x^2/sqrt(a + (2 + 2*c - 2*(1 + c))*x^4), x, 3, x^3/(3*sqrt(a))]
    @test_int [x^1/sqrt(a + (2 + 2*c - 2*(1 + c))*x^4), x, 3, x^2/(2*sqrt(a))]
    @test_int [x^0/sqrt(a + (2 + 2*c - 2*(1 + c))*x^4), x, 2, x/sqrt(a)]
    @test_int [1/(x^1*sqrt(a + (2 + 2*c - 2*(1 + c))*x^4)), x, 3, log(x)/sqrt(a)]
    @test_int [1/(x^2*sqrt(a + (2 + 2*c - 2*(1 + c))*x^4)), x, 3, -(1/(sqrt(a)*x))]
    @test_int [1/(x^3*sqrt(a + (2 + 2*c - 2*(1 + c))*x^4)), x, 3, -(1/(2*sqrt(a)*x^2))]
    @test_int [1/(x^4*sqrt(a + (2 + 2*c - 2*(1 + c))*x^4)), x, 3, -(1/(3*sqrt(a)*x^3))]

    @test_int [1/sqrt(3 - 2*x^2 - x^4), x, 2, Elliptic.F(asin(x), -(1/3))/sqrt(3)]

    @test_int [1/sqrt(-1 + 5*x^2 - x^4), x, 2, -(Elliptic.F(acos(sqrt(2/(5 + sqrt(21)))*x), (1/42)*(21 + 5*sqrt(21)))/21^(1/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^(m/2)*(a+b*x^2+c*x^4)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^(5/2)*(a + b*x^2 + c*x^4), x, 2, (2/7)*a*x^(7/2) + (2/11)*b*x^(11/2) + (2/15)*c*x^(15/2)]
    @test_int [x^(3/2)*(a + b*x^2 + c*x^4), x, 2, (2/5)*a*x^(5/2) + (2/9)*b*x^(9/2) + (2/13)*c*x^(13/2)]
    @test_int [x^(1/2)*(a + b*x^2 + c*x^4), x, 2, (2/3)*a*x^(3/2) + (2/7)*b*x^(7/2) + (2/11)*c*x^(11/2)]
    @test_int [(a + b*x^2 + c*x^4)/x^(1/2), x, 2, 2*a*sqrt(x) + (2/5)*b*x^(5/2) + (2/9)*c*x^(9/2)]
    @test_int [(a + b*x^2 + c*x^4)/x^(3/2), x, 2, -((2*a)/sqrt(x)) + (2/3)*b*x^(3/2) + (2/7)*c*x^(7/2)]
    @test_int [(a + b*x^2 + c*x^4)/x^(5/2), x, 2, -((2*a)/(3*x^(3/2))) + 2*b*sqrt(x) + (2/5)*c*x^(5/2)]
    @test_int [(a + b*x^2 + c*x^4)/x^(7/2), x, 2, -((2*a)/(5*x^(5/2))) - (2*b)/sqrt(x) + (2/3)*c*x^(3/2)]

    @test_int [x^(5/2)*(a + b*x^2 + c*x^4)^2, x, 2, (2/7)*a^2*x^(7/2) + (4/11)*a*b*x^(11/2) + (2/15)*(b^2 + 2*a*c)*x^(15/2) + (4/19)*b*c*x^(19/2) + (2/23)*c^2*x^(23/2)]
    @test_int [x^(3/2)*(a + b*x^2 + c*x^4)^2, x, 2, (2/5)*a^2*x^(5/2) + (4/9)*a*b*x^(9/2) + (2/13)*(b^2 + 2*a*c)*x^(13/2) + (4/17)*b*c*x^(17/2) + (2/21)*c^2*x^(21/2)]
    @test_int [x^(1/2)*(a + b*x^2 + c*x^4)^2, x, 2, (2/3)*a^2*x^(3/2) + (4/7)*a*b*x^(7/2) + (2/11)*(b^2 + 2*a*c)*x^(11/2) + (4/15)*b*c*x^(15/2) + (2/19)*c^2*x^(19/2)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^(1/2), x, 2, 2*a^2*sqrt(x) + (4/5)*a*b*x^(5/2) + (2/9)*(b^2 + 2*a*c)*x^(9/2) + (4/13)*b*c*x^(13/2) + (2/17)*c^2*x^(17/2)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^(3/2), x, 2, -((2*a^2)/sqrt(x)) + (4/3)*a*b*x^(3/2) + (2/7)*(b^2 + 2*a*c)*x^(7/2) + (4/11)*b*c*x^(11/2) + (2/15)*c^2*x^(15/2)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^(5/2), x, 2, -((2*a^2)/(3*x^(3/2))) + 4*a*b*sqrt(x) + (2/5)*(b^2 + 2*a*c)*x^(5/2) + (4/9)*b*c*x^(9/2) + (2/13)*c^2*x^(13/2)]
    @test_int [(a + b*x^2 + c*x^4)^2/x^(7/2), x, 2, -((2*a^2)/(5*x^(5/2))) - (4*a*b)/sqrt(x) + (2/3)*(b^2 + 2*a*c)*x^(3/2) + (4/7)*b*c*x^(7/2) + (2/11)*c^2*x^(11/2)]

    @test_int [x^(5/2)*(a + b*x^2 + c*x^4)^3, x, 2, (2/7)*a^3*x^(7/2) + (6/11)*a^2*b*x^(11/2) + (2/5)*a*(b^2 + a*c)*x^(15/2) + (2/19)*b*(b^2 + 6*a*c)*x^(19/2) + (6/23)*c*(b^2 + a*c)*x^(23/2) + (2/9)*b*c^2*x^(27/2) + (2/31)*c^3*x^(31/2)]
    @test_int [x^(3/2)*(a + b*x^2 + c*x^4)^3, x, 2, (2/5)*a^3*x^(5/2) + (2/3)*a^2*b*x^(9/2) + (6/13)*a*(b^2 + a*c)*x^(13/2) + (2/17)*b*(b^2 + 6*a*c)*x^(17/2) + (2/7)*c*(b^2 + a*c)*x^(21/2) + (6/25)*b*c^2*x^(25/2) + (2/29)*c^3*x^(29/2)]
    @test_int [x^(1/2)*(a + b*x^2 + c*x^4)^3, x, 2, (2/3)*a^3*x^(3/2) + (6/7)*a^2*b*x^(7/2) + (6/11)*a*(b^2 + a*c)*x^(11/2) + (2/15)*b*(b^2 + 6*a*c)*x^(15/2) + (6/19)*c*(b^2 + a*c)*x^(19/2) + (6/23)*b*c^2*x^(23/2) + (2/27)*c^3*x^(27/2)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^(1/2), x, 2, 2*a^3*sqrt(x) + (6/5)*a^2*b*x^(5/2) + (2/3)*a*(b^2 + a*c)*x^(9/2) + (2/13)*b*(b^2 + 6*a*c)*x^(13/2) + (6/17)*c*(b^2 + a*c)*x^(17/2) + (2/7)*b*c^2*x^(21/2) + (2/25)*c^3*x^(25/2)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^(3/2), x, 2, -((2*a^3)/sqrt(x)) + 2*a^2*b*x^(3/2) + (6/7)*a*(b^2 + a*c)*x^(7/2) + (2/11)*b*(b^2 + 6*a*c)*x^(11/2) + (2/5)*c*(b^2 + a*c)*x^(15/2) + (6/19)*b*c^2*x^(19/2) + (2/23)*c^3*x^(23/2)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^(5/2), x, 2, -((2*a^3)/(3*x^(3/2))) + 6*a^2*b*sqrt(x) + (6/5)*a*(b^2 + a*c)*x^(5/2) + (2/9)*b*(b^2 + 6*a*c)*x^(9/2) + (6/13)*c*(b^2 + a*c)*x^(13/2) + (6/17)*b*c^2*x^(17/2) + (2/21)*c^3*x^(21/2)]
    @test_int [(a + b*x^2 + c*x^4)^3/x^(7/2), x, 2, -((2*a^3)/(5*x^(5/2))) - (6*a^2*b)/sqrt(x) + 2*a*(b^2 + a*c)*x^(3/2) + (2/7)*b*(b^2 + 6*a*c)*x^(7/2) + (6/11)*c*(b^2 + a*c)*x^(11/2) + (2/5)*b*c^2*x^(15/2) + (2/19)*c^3*x^(19/2)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^(9/2)/(a + b*x^2 + c*x^4), x, 9, (2*x^(3/2))/(3*c) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(7/4)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(7/4)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(7/4)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(7/4)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(7/2)/(a + b*x^2 + c*x^4), x, 9, (2*sqrt(x))/c + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(5/4)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(5/4)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(5/4)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(5/4)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(5/2)/(a + b*x^2 + c*x^4), x, 8, -(((-b - sqrt(b^2 - 4*a*c))^(3/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(3/4)*sqrt(b^2 - 4*a*c))) + ((-b + sqrt(b^2 - 4*a*c))^(3/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(3/4)*sqrt(b^2 - 4*a*c)) + ((-b - sqrt(b^2 - 4*a*c))^(3/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(3/4)*sqrt(b^2 - 4*a*c)) - ((-b + sqrt(b^2 - 4*a*c))^(3/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*c^(3/4)*sqrt(b^2 - 4*a*c))]
    @test_int [x^(3/2)/(a + b*x^2 + c*x^4), x, 8, ((-b - sqrt(b^2 - 4*a*c))^(1/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(1/4)*sqrt(b^2 - 4*a*c)) - ((-b + sqrt(b^2 - 4*a*c))^(1/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(1/4)*sqrt(b^2 - 4*a*c)) + ((-b - sqrt(b^2 - 4*a*c))^(1/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(1/4)*sqrt(b^2 - 4*a*c)) - ((-b + sqrt(b^2 - 4*a*c))^(1/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*c^(1/4)*sqrt(b^2 - 4*a*c))]
    @test_int [x^(1/2)/(a + b*x^2 + c*x^4), x, 8, -((2^(1/4)*c^(1/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(1/4))) + (2^(1/4)*c^(1/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + (2^(1/4)*c^(1/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (2^(1/4)*c^(1/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [1/(x^(1/2)*(a + b*x^2 + c*x^4)), x, 8, (2^(3/4)*c^(3/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (2^(3/4)*c^(3/4)*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (2^(3/4)*c^(3/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (2^(3/4)*c^(3/4)*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [1/(x^(3/2)*(a + b*x^2 + c*x^4)), x, 9, -(2/(a*sqrt(x))) - (c^(1/4)*(1 - b/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(1 + b/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(1 - b/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(1 + b/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [1/(x^(5/2)*(a + b*x^2 + c*x^4)), x, 9, -(2/(3*a*x^(3/2))) + (c^(3/4)*(1 - b/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*a*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(1 + b/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*a*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(1 - b/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*a*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(1 + b/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(1/4)*a*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [1/(x^(7/2)*(a + b*x^2 + c*x^4)), x, 10, -(2/(5*a*x^(5/2))) + (2*b)/(a^2*sqrt(x)) + (c^(1/4)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a^2*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a^2*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a^2*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2^(3/4)*a^2*(-b + sqrt(b^2 - 4*a*c))^(1/4))]

    @test_int [x^(13/2)/(a + b*x^2 + c*x^4)^2, x, 10, -((b*x^(3/2))/(2*c*(b^2 - 4*a*c))) + (x^(7/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - ((3*b^3 - 20*a*b*c - (3*b^2 - 14*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + ((3*b^3 - 20*a*b*c - (3*b^2 - 14*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(11/2)/(a + b*x^2 + c*x^4)^2, x, 10, -((b*sqrt(x))/(2*c*(b^2 - 4*a*c))) + (x^(5/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b^2 - 10*a*c + (b*(b^2 - 12*a*c))/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - ((b^2 - 10*a*c - (b*(b^2 - 12*a*c))/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) - ((b^2 - 10*a*c + (b*(b^2 - 12*a*c))/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - ((b^2 - 10*a*c - (b*(b^2 - 12*a*c))/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(9/2)/(a + b*x^2 + c*x^4)^2, x, 9, (x^(3/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b^2 + 12*a*c + b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + ((b - (b^2 + 12*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - ((b^2 + 12*a*c + b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - ((b - (b^2 + 12*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(7/2)/(a + b*x^2 + c*x^4)^2, x, 9, (sqrt(x)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((3*b^2 + 4*a*c + 3*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + ((3*b^2 + 4*a*c - 3*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) - ((3*b^2 + 4*a*c + 3*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + ((3*b^2 + 4*a*c - 3*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(5/2)/(a + b*x^2 + c*x^4)^2, x, 9, -((x^(3/2)*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) - (c^(1/4)*(4*b + sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(4*b - sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(4*b + sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(4*b - sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(3/4)*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(3/2)/(a + b*x^2 + c*x^4)^2, x, 9, -((sqrt(x)*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (c^(3/4)*(3 + (4*b)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(1/4)*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(3 - (4*b)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(1/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(3 + (4*b)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(1/4)*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(3 - (4*b)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(2*2^(1/4)*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(1/2)/(a + b*x^2 + c*x^4)^2, x, 9, (x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (c^(1/4)*(b - (b^2 - 20*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(b + (b^2 - 20*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(b - (b^2 - 20*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a*(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(b + (b^2 - 20*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a*(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [1/(x^(1/2)*(a + b*x^2 + c*x^4)^2), x, 9, (sqrt(x)*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (c^(3/4)*(3*b^2 - 28*a*c - 3*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*a*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (c^(3/4)*(3*b^2 - 28*a*c + 3*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*a*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(3*b^2 - 28*a*c - 3*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*a*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (c^(3/4)*(3*b^2 - 28*a*c + 3*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(1/4)*a*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [1/(x^(3/2)*(a + b*x^2 + c*x^4)^2), x, 10, -((5*b^2 - 18*a*c)/(2*a^2*(b^2 - 4*a*c)*sqrt(x))) + (b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*sqrt(x)*(a + b*x^2 + c*x^4)) + (c^(1/4)*(5*b^3 - 28*a*b*c - (5*b^2 - 18*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(5*b^3 - 28*a*b*c + (5*b^2 - 18*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(5*b^3 - 28*a*b*c - (5*b^2 - 18*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(5*b^3 - 28*a*b*c + (5*b^2 - 18*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(4*2^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]

    @test_int [x^(15/2)/(a + b*x^2 + c*x^4)^3, x, 11, -((3*(b^2 + 12*a*c)*sqrt(x))/(16*c*(b^2 - 4*a*c)^2)) + (x^(9/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x^(5/2)*(8*a*b + (b^2 + 12*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(b^3 - 28*a*b*c + (b^4 - 30*a*b^2*c - 24*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(3/4)) - (3*(b^3 - 28*a*b*c + (b^4 - 30*a*b^2*c - 24*a^2*c^2)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(13/2)/(a + b*x^2 + c*x^4)^3, x, 10, (x^(7/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^(3/2)*(24*a*b + (5*b^2 + 28*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((5*b^3 + 172*a*b*c + sqrt(b^2 - 4*a*c)*(5*b^2 + 28*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + ((5*b^2 + 28*a*c - (5*b^3 + 172*a*b*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - ((5*b^3 + 172*a*b*c + sqrt(b^2 - 4*a*c)*(5*b^2 + 28*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - ((5*b^2 + 28*a*c - (5*b^3 + 172*a*b*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(11/2)/(a + b*x^2 + c*x^4)^3, x, 10, (x^(5/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (sqrt(x)*(24*a*b + (7*b^2 + 20*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(7*b^3 + 36*a*b*c + sqrt(b^2 - 4*a*c)*(7*b^2 + 20*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(3/4)) - (3*(7*b^3 + 36*a*b*c + sqrt(b^2 - 4*a*c)*(7*b^2 + 20*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(9/2)/(a + b*x^2 + c*x^4)^3, x, 10, (x^(3/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*x^(3/2)*(5*b^2 - 4*a*c + 8*b*c*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*c^(1/4)*(11*b^2 + 20*a*c + 4*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (3*c^(1/4)*(11*b^2 + 20*a*c - 4*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + (3*c^(1/4)*(11*b^2 + 20*a*c + 4*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (3*c^(1/4)*(11*b^2 + 20*a*c - 4*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(7/2)/(a + b*x^2 + c*x^4)^3, x, 10, (sqrt(x)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (sqrt(x)*(13*b^2 - 4*a*c + 24*b*c*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (c^(3/4)*(41*b^2 + 28*a*c + 36*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (c^(3/4)*(41*b^2 + 28*a*c - 36*b*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (c^(3/4)*(41*b^2 + 28*a*c + 36*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (c^(3/4)*(41*b^2 + 28*a*c - 36*b*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(16*2^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(5/2)/(a + b*x^2 + c*x^4)^3, x, 10, -((x^(3/2)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*x^(3/2)*(b*(b^2 + 4*a*c) + c*(b^2 + 12*a*c)*x^2))/(16*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*c^(1/4)*(b^2 + 12*a*c - b^3/sqrt(b^2 - 4*a*c) + (68*a*b*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (3*c^(1/4)*(b^3 - 68*a*b*c + sqrt(b^2 - 4*a*c)*(b^2 + 12*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) - (3*c^(1/4)*(b^2 + 12*a*c - b^3/sqrt(b^2 - 4*a*c) + (68*a*b*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (3*c^(1/4)*(b^3 - 68*a*b*c + sqrt(b^2 - 4*a*c)*(b^2 + 12*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [x^(3/2)/(a + b*x^2 + c*x^4)^3, x, 10, -((sqrt(x)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (sqrt(x)*(b*(b^2 + 20*a*c) + c*(b^2 + 44*a*c)*x^2))/(16*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/sqrt(b^2 - 4*a*c) + (68*a*b*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*c + sqrt(b^2 - 4*a*c)*(b^2 + 44*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/sqrt(b^2 - 4*a*c) + (68*a*b*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a*(b^2 - 4*a*c)^2*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*c + sqrt(b^2 - 4*a*c)*(b^2 + 44*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]
    @test_int [x^(1/2)/(a + b*x^2 + c*x^4)^3, x, 10, (x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^(3/2)*(5*b^4 - 45*a*b^2*c + 52*a^2*c^2 + b*c*(5*b^2 - 44*a*c)*x^2))/(16*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (c^(1/4)*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 - b*(5*b^2 - 44*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 + b*(5*b^2 - 44*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4)) + (c^(1/4)*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 - b*(5*b^2 - 44*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(1/4)) - (c^(1/4)*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 + b*(5*b^2 - 44*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(1/4))]
    @test_int [1/(x^(1/2)*(a + b*x^2 + c*x^4)^3), x, 10, (sqrt(x)*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (sqrt(x)*(7*b^4 - 55*a*b^2*c + 60*a^2*c^2 + b*c*(7*b^2 - 52*a*c)*x^2))/(16*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*c^(3/4)*(7*b^4 - 66*a*b^2*c + 280*a^2*c^2 - b*(7*b^2 - 52*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*c^(3/4)*(7*b^4 - 66*a*b^2*c + 280*a^2*c^2 + b*(7*b^2 - 52*a*c)*sqrt(b^2 - 4*a*c))*atan((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4)) + (3*c^(3/4)*(7*b^4 - 66*a*b^2*c + 280*a^2*c^2 - b*(7*b^2 - 52*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - sqrt(b^2 - 4*a*c))^(3/4)) - (3*c^(3/4)*(7*b^4 - 66*a*b^2*c + 280*a^2*c^2 + b*(7*b^2 - 52*a*c)*sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*c^(1/4)*sqrt(x))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(32*2^(1/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + sqrt(b^2 - 4*a*c))^(3/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^(m/2)*(a+b*x^2+c*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d*x)^(3/2)*sqrt(a + b*x^2 + c*x^4), x, 2, (2*(d*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(1/2), -(1/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d*x)^(1/2)*sqrt(a + b*x^2 + c*x^4), x, 2, (2*(d*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(1/2), -(1/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [1/(d*x)^(1/2)*sqrt(a + b*x^2 + c*x^4), x, 2, (2*sqrt(d*x)*sqrt(a + b*x^2 + c*x^4)*AppellF1(1/4, -(1/2), -(1/2), 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [1/(d*x)^(3/2)*sqrt(a + b*x^2 + c*x^4), x, 2, -((2*sqrt(a + b*x^2 + c*x^4)*AppellF1(-(1/4), -(1/2), -(1/2), 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(d*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))]

    @test_int [(d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*a*(d*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(3/2), -(3/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d*x)^(1/2)*(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*a*(d*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(3/2), -(3/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [1/(d*x)^(1/2)*(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*a*sqrt(d*x)*sqrt(a + b*x^2 + c*x^4)*AppellF1(1/4, -(3/2), -(3/2), 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [1/(d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2), x, 2, -((2*a*sqrt(a + b*x^2 + c*x^4)*AppellF1(-(1/4), -(3/2), -(3/2), 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(d*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d*x)^(3/2)/sqrt(a + b*x^2 + c*x^4), x, 2, (2*(d*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 1/2, 1/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d*x)^(1/2)/sqrt(a + b*x^2 + c*x^4), x, 2, (2*(d*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 1/2, 1/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(d*x)^(1/2)/sqrt(a + b*x^2 + c*x^4), x, 2, (2*sqrt(d*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/4, 1/2, 1/2, 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(d*x)^(3/2)/sqrt(a + b*x^2 + c*x^4), x, 2, -((2*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/4), 1/2, 1/2, 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*sqrt(d*x)*sqrt(a + b*x^2 + c*x^4)))]

    @test_int [(d*x)^(3/2)/(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*(d*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 3/2, 3/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*a*d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d*x)^(1/2)/(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*(d*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 3/2, 3/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*a*d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(d*x)^(1/2)/(a + b*x^2 + c*x^4)^(3/2), x, 2, (2*sqrt(d*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/4, 3/2, 3/2, 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*d*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/(d*x)^(3/2)/(a + b*x^2 + c*x^4)^(3/2), x, 2, -((2*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/4), 3/2, 3/2, 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*d*sqrt(d*x)*sqrt(a + b*x^2 + c*x^4)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p*with*m*symbolic=#

    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^3, x, 2, (a^3*(d*x)^(1 + m))/(d*(1 + m)) + (3*a^2*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (3*a*(b^2 + a*c)*(d*x)^(5 + m))/(d^5*(5 + m)) + (b*(b^2 + 6*a*c)*(d*x)^(7 + m))/(d^7*(7 + m)) + (3*c*(b^2 + a*c)*(d*x)^(9 + m))/(d^9*(9 + m)) + (3*b*c^2*(d*x)^(11 + m))/(d^11*(11 + m)) + (c^3*(d*x)^(13 + m))/(d^13*(13 + m))]
    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^2, x, 2, (a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + ((b^2 + 2*a*c)*(d*x)^(5 + m))/(d^5*(5 + m)) + (2*b*c*(d*x)^(7 + m))/(d^7*(7 + m)) + (c^2*(d*x)^(9 + m))/(d^9*(9 + m))]
    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^1, x, 2, (a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(3 + m))/(d^3*(3 + m)) + (c*(d*x)^(5 + m))/(d^5*(5 + m))]
    @test_int [(d*x)^m/(a + b*x^2 + c*x^4)^1, x, 3, (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)) - (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m))]
    @test_int [(d*x)^m/(a + b*x^2 + c*x^4)^2, x, 4, ((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*d*(a + b*x^2 + c*x^4)) + (c*(b^2*(1 - m) + b*sqrt(b^2 - 4*a*c)*(1 - m) - 4*a*c*(3 - m))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))))/(2*a*(b^2 - 4*a*c)^(3/2)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)) - (c*(b^2*(1 - m) - b*sqrt(b^2 - 4*a*c)*(1 - m) - 4*a*c*(3 - m))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(2*a*(b^2 - 4*a*c)^(3/2)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m))]

    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^(3/2), x, 2, (a*(d*x)^(1 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((1 + m)/2, -(3/2), -(3/2), (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^(1/2), x, 2, ((d*x)^(1 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((1 + m)/2, -(1/2), -(1/2), (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d*x)^m/(a + b*x^2 + c*x^4)^(1/2), x, 2, ((d*x)^(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((1 + m)/2, 1/2, 1/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(d*(1 + m)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d*x)^m/(a + b*x^2 + c*x^4)^(3/2), x, 2, ((d*x)^(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((1 + m)/2, 3/2, 3/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*d*(1 + m)*sqrt(a + b*x^2 + c*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^2+c*x^4)^p*with*p*symbolic=#

    @test_int [(d*x)^m*(a + b*x^2 + c*x^4)^p, x, 2, ((d*x)^(1 + m)*(a + b*x^2 + c*x^4)^p*AppellF1((1 + m)/2, -p, -p, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p*(d*(1 + m)))]

    @test_int [(a + b*x^2 + c*x^4)^p*x^7, x, 4, If(13>=8, (x^4*(a + b*x^2 + c*x^4)^(1 + p))/(4*c*(2 + p)) + ((b^2*(2 + p)*(3 + p) - 2*a*c*(3 + 2*p) - 2*b*c*(1 + p)*(3 + p)*x^2)*(a + b*x^2 + c*x^4)^(1 + p))/(8*c^3*(1 + p)*(2 + p)*(3 + 2*p)) - (2^(-2 + p)*b*(6*a*c - b^2*(3 + p))*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(c^3*sqrt(b^2 - 4*a*c)*(1 + p)*(3 + 2*p)), (x^4*(a + b*x^2 + c*x^4)^(1 + p))/(4*c*(2 + p)) + ((b^2*(2 + p)*(3 + p) - 2*a*c*(3 + 2*p) - 2*b*c*(1 + p)*(3 + p)*x^2)*(a + b*x^2 + c*x^4)^(1 + p))/(8*c^3*(2 + p)*(3 + 5*p + 2*p^2)) - (2^(-2 + p)*b*(6*a*c - b^2*(3 + p))*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(c^3*sqrt(b^2 - 4*a*c)*(1 + p)*(3 + 2*p)))]
    @test_int [(a + b*x^2 + c*x^4)^p*x^5, x, 4, If(13>=8, -((b*(2 + p)*(a + b*x^2 + c*x^4)^(1 + p))/(4*c^2*(1 + p)*(3 + 2*p))) + (x^2*(a + b*x^2 + c*x^4)^(1 + p))/(2*c*(3 + 2*p)) + (2^(-1 + p)*(2*a*c - b^2*(2 + p))*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(c^2*sqrt(b^2 - 4*a*c)*(1 + p)*(3 + 2*p)), -((b*(2 + p)*(a + b*x^2 + c*x^4)^(1 + p))/(4*c^2*(3 + 5*p + 2*p^2))) + (x^2*(a + b*x^2 + c*x^4)^(1 + p))/(2*c*(3 + 2*p)) + (2^(-1 + p)*(2*a*c - b^2*(2 + p))*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(c^2*sqrt(b^2 - 4*a*c)*(1 + p)*(3 + 2*p)))]
    @test_int [(a + b*x^2 + c*x^4)^p*x^3, x, 3, (a + b*x^2 + c*x^4)^(1 + p)/(4*c*(1 + p)) + (2^(-1 + p)*b*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(c*sqrt(b^2 - 4*a*c)*(1 + p))]
    @test_int [(a + b*x^2 + c*x^4)^p*x^1, x, 2, -((2^p*(-((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/sqrt(b^2 - 4*a*c)))^(-1 - p)*(a + b*x^2 + c*x^4)^(1 + p)*HypergeometricFunctions._₂F₁(-p, 1 + p, 2 + p, (b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(2*sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*(1 + p)))]
    @test_int [(a + b*x^2 + c*x^4)^p/x^1, x, 3, (4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1(-2*p, -p, -p, 1 - 2*p, -((b - sqrt(b^2 - 4*a*c))/(2*c*x^2)), -((b + sqrt(b^2 - 4*a*c))/(2*c*x^2))))/(((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*((b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*p)]
    @test_int [(a + b*x^2 + c*x^4)^p/x^3, x, 3, -((2^(-1 + 2*p)*(a + b*x^2 + c*x^4)^p*AppellF1(1 - 2*p, -p, -p, 2*(1 - p), -((b - sqrt(b^2 - 4*a*c))/(2*c*x^2)), -((b + sqrt(b^2 - 4*a*c))/(2*c*x^2))))/(((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*((b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*((1 - 2*p)*x^2)))]
    @test_int [(a + b*x^2 + c*x^4)^p/x^5, x, 3, -((4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1(2*(1 - p), -p, -p, 3 - 2*p, -((b - sqrt(b^2 - 4*a*c))/(2*c*x^2)), -((b + sqrt(b^2 - 4*a*c))/(2*c*x^2))))/(((b - sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*((b + sqrt(b^2 - 4*a*c) + 2*c*x^2)/(c*x^2))^p*((1 - p)*x^4)))]

    @test_int [(a + b*x^2 + c*x^4)^p*x^4, x, 2, ((1/5)*x^5*(a + b*x^2 + c*x^4)^p*AppellF1(5/2, -p, -p, 7/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p)]
    @test_int [(a + b*x^2 + c*x^4)^p*x^2, x, 2, ((1/3)*x^3*(a + b*x^2 + c*x^4)^p*AppellF1(3/2, -p, -p, 5/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p)]
    @test_int [(a + b*x^2 + c*x^4)^p*x^0, x, 2, (x*(a + b*x^2 + c*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p)]
    @test_int [(a + b*x^2 + c*x^4)^p/x^2, x, 2, -(((a + b*x^2 + c*x^4)^p*AppellF1(-(1/2), -p, -p, 1/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p*x))]
    @test_int [(a + b*x^2 + c*x^4)^p/x^4, x, 2, -(((a + b*x^2 + c*x^4)^p*AppellF1(-(3/2), -p, -p, -(1/2), -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))^p*(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))^p*(3*x^3)))]
    end

ext: jl

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.jl