text:
@testset "1.2.2.3 (d+e x^2)^m (a+b x^2+c x^4)^p" begin
    (A, B, a, b, c, d, e, f, g, p, q, x, ) = @variables A B a b c d e f g p q x

    #= ::Package:: =#

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

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

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

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

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

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

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

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

    @test_int [(2 + 3*x^2)/(4 - 9*x^4), x, 2, atanh(sqrt(3/2)*x)/sqrt(6)]
    @test_int [(2 - 3*x^2)/(4 - 9*x^4), x, 2, atan(sqrt(3/2)*x)/sqrt(6)]

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

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

    @test_int [(5 + 2*x^2)/(-1 + x^4), x, 3, -((3*atan(x))/2) - (7*atanh(x))/2]

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

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

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

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

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

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

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

    #= Following*pairs*of*integrands*are*equal. =#
    @test_int [sqrt(1 + c^2*x^2)/sqrt(1 - c^2*x^2), x, 1, Elliptic.E(asin(c*x), -1)/c]
    @test_int [(1 + c^2*x^2)/sqrt(1 - c^4*x^4), x, 2, Elliptic.E(asin(c*x), -1)/c]

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

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

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

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

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

    @test_int [(d + e*x^2)/(d^2 + b*x^2 + e^2*x^4), x, 5, -(atan((sqrt(-b + 2*d*e) - 2*e*x)/sqrt(b + 2*d*e))/sqrt(b + 2*d*e)) + atan((sqrt(-b + 2*d*e) + 2*e*x)/sqrt(b + 2*d*e))/sqrt(b + 2*d*e)]
    @test_int [(d + e*x^2)/(d^2 + f*x^2 + e^2*x^4), x, 5, -(atan((sqrt(2*d*e - f) - 2*e*x)/sqrt(2*d*e + f))/sqrt(2*d*e + f)) + atan((sqrt(2*d*e - f) + 2*e*x)/sqrt(2*d*e + f))/sqrt(2*d*e + f)]

    @test_int [(d + e*x^2)/(d^2 - b*x^2 + e^2*x^4), x, 5, atanh((sqrt(b + 2*d*e) - 2*e*x)/sqrt(b - 2*d*e))/sqrt(b - 2*d*e) - atanh((sqrt(b + 2*d*e) + 2*e*x)/sqrt(b - 2*d*e))/sqrt(b - 2*d*e)]
    @test_int [(d + e*x^2)/(d^2 - f*x^2 + e^2*x^4), x, 5, -(atan((sqrt(2*d*e + f) - 2*e*x)/sqrt(2*d*e - f))/sqrt(2*d*e - f)) + atan((sqrt(2*d*e + f) + 2*e*x)/sqrt(2*d*e - f))/sqrt(2*d*e - f)]

    @test_int [(d - e*x^2)/(d^2 + b*x^2 + e^2*x^4), x, 3, -(log(d - sqrt(-b + 2*d*e)*x + e*x^2)/(2*sqrt(-b + 2*d*e))) + log(d + sqrt(-b + 2*d*e)*x + e*x^2)/(2*sqrt(-b + 2*d*e))]
    @test_int [(d - e*x^2)/(d^2 + f*x^2 + e^2*x^4), x, 3, -(log(d - sqrt(2*d*e - f)*x + e*x^2)/(2*sqrt(2*d*e - f))) + log(d + sqrt(2*d*e - f)*x + e*x^2)/(2*sqrt(2*d*e - f))]

    @test_int [(d - e*x^2)/(d^2 - b*x^2 + e^2*x^4), x, 3, -(log(d - sqrt(b + 2*d*e)*x + e*x^2)/(2*sqrt(b + 2*d*e))) + log(d + sqrt(b + 2*d*e)*x + e*x^2)/(2*sqrt(b + 2*d*e))]
    @test_int [(d - e*x^2)/(d^2 - f*x^2 + e^2*x^4), x, 3, -(log(d - sqrt(2*d*e + f)*x + e*x^2)/(2*sqrt(2*d*e + f))) + log(d + sqrt(2*d*e + f)*x + e*x^2)/(2*sqrt(2*d*e + f))]

    @test_int [(d - e*x^2)/(c*d^2/e^2 + b*x^2 + c*x^4), x, 3, -((e^(3/2)*log(sqrt(c)*d - sqrt(e)*sqrt(2*c*d - b*e)*x + sqrt(c)*e*x^2))/(2*sqrt(c)*sqrt(2*c*d - b*e))) + (e^(3/2)*log(sqrt(c)*d + sqrt(e)*sqrt(2*c*d - b*e)*x + sqrt(c)*e*x^2))/(2*sqrt(c)*sqrt(2*c*d - b*e))]
    @test_int [(d + e*x^2)/(c*d^2/e^2 + b*x^2 + c*x^4), x, 5, -((e^(3/2)*atan((sqrt(2*c*d - b*e) - 2*sqrt(c)*sqrt(e)*x)/sqrt(2*c*d + b*e)))/(sqrt(c)*sqrt(2*c*d + b*e))) + (e^(3/2)*atan((sqrt(2*c*d - b*e) + 2*sqrt(c)*sqrt(e)*x)/sqrt(2*c*d + b*e)))/(sqrt(c)*sqrt(2*c*d + b*e))]
    @test_int [(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + x^4)), x, 6, -((e^(3/2)*atan((sqrt(2*c*d - b*e) - 2*sqrt(c)*sqrt(e)*x)/sqrt(2*c*d + b*e)))/(sqrt(c)*sqrt(2*c*d + b*e))) + (e^(3/2)*atan((sqrt(2*c*d - b*e) + 2*sqrt(c)*sqrt(e)*x)/sqrt(2*c*d + b*e)))/(sqrt(c)*sqrt(2*c*d + b*e))]

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

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

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

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

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

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

    @test_int [(1 + x^2)/(1 + 5*x^2 + x^4), x, 3, atan(sqrt(2/(5 + sqrt(21)))*x)/sqrt(7) + atan(sqrt((1/2)*(5 + sqrt(21)))*x)/sqrt(7)]
    @test_int [(1 + x^2)/(1 + 4*x^2 + x^4), x, 3, atan(x/sqrt(2 - sqrt(3)))/sqrt(6) + atan(x/sqrt(2 + sqrt(3)))/sqrt(6)]
    @test_int [(1 + x^2)/(1 + 3*x^2 + x^4), x, 3, atan(sqrt(2/(3 + sqrt(5)))*x)/sqrt(5) + atan(sqrt((1/2)*(3 + sqrt(5)))*x)/sqrt(5)]
    @test_int [(1 + x^2)/(1 + 2*x^2 + x^4), x, 2, atan(x)]
    @test_int [(1 + x^2)/(1 + 1*x^2 + x^4), x, 5, -(atan((1 - 2*x)/sqrt(3))/sqrt(3)) + atan((1 + 2*x)/sqrt(3))/sqrt(3)]
    @test_int [(1 + x^2)/(1 + 0*x^2 + x^4), x, 5, -(atan(1 - sqrt(2)*x)/sqrt(2)) + atan(1 + sqrt(2)*x)/sqrt(2)]
    @test_int [(1 + x^2)/(1 - 1*x^2 + x^4), x, 5, -atan(sqrt(3) - 2*x) + atan(sqrt(3) + 2*x)]
    @test_int [(1 + x^2)/(1 - 2*x^2 + x^4), x, 2, x/(1 - x^2)]
    @test_int [(1 + x^2)/(1 - 3*x^2 + x^4), x, 7, (1/2)*log(1 - sqrt(5) - 2*x) + (1/2)*log(1 + sqrt(5) - 2*x) - (1/2)*log(1 - sqrt(5) + 2*x) - (1/2)*log(1 + sqrt(5) + 2*x)]
    @test_int [(1 + x^2)/(1 - 4*x^2 + x^4), x, 5, atanh(sqrt(3) - sqrt(2)*x)/sqrt(2) - atanh(sqrt(3) + sqrt(2)*x)/sqrt(2)]
    @test_int [(1 + x^2)/(1 - 5*x^2 + x^4), x, 5, atanh((sqrt(7) - 2*x)/sqrt(3))/sqrt(3) - atanh((sqrt(7) + 2*x)/sqrt(3))/sqrt(3)]

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

    @test_int [(1 - x^2)/(1 + 5*x^2 + x^4), x, 3, -(atan(sqrt(2/(5 + sqrt(21)))*x)/sqrt(3)) + atan(sqrt((1/2)*(5 + sqrt(21)))*x)/sqrt(3)]
    @test_int [(1 - x^2)/(1 + 4*x^2 + x^4), x, 3, atan(x/sqrt(2 - sqrt(3)))/sqrt(2) - atan(x/sqrt(2 + sqrt(3)))/sqrt(2)]
    @test_int [(1 - x^2)/(1 + 3*x^2 + x^4), x, 3, -atan(sqrt(2/(3 + sqrt(5)))*x) + atan(sqrt((1/2)*(3 + sqrt(5)))*x)]
    @test_int [(1 - x^2)/(1 + 2*x^2 + x^4), x, 2, x/(1 + x^2)]
    @test_int [(1 - x^2)/(1 + 1*x^2 + x^4), x, 3, (-(1/2))*log(1 - x + x^2) + (1/2)*log(1 + x + x^2)]
    @test_int [(1 - x^2)/(1 + 0*x^2 + x^4), x, 3, -(log(1 - sqrt(2)*x + x^2)/(2*sqrt(2))) + log(1 + sqrt(2)*x + x^2)/(2*sqrt(2))]
    @test_int [(1 - x^2)/(1 - 1*x^2 + x^4), x, 3, -(log(1 - sqrt(3)*x + x^2)/(2*sqrt(3))) + log(1 + sqrt(3)*x + x^2)/(2*sqrt(3))]
    @test_int [(1 - x^2)/(1 - 2*x^2 + x^4), x, 3, atanh(x)]
    @test_int [(1 - x^2)/(1 - 3*x^2 + x^4), x, 5, -(atanh((1 - 2*x)/sqrt(5))/sqrt(5)) + atanh((1 + 2*x)/sqrt(5))/sqrt(5)]
    @test_int [(1 - x^2)/(1 - 4*x^2 + x^4), x, 5, -(atanh((1 - sqrt(2)*x)/sqrt(3))/sqrt(6)) + atanh((1 + sqrt(2)*x)/sqrt(3))/sqrt(6)]
    @test_int [(1 - x^2)/(1 - 5*x^2 + x^4), x, 5, -(atanh((sqrt(3) - 2*x)/sqrt(7))/sqrt(7)) + atanh((sqrt(3) + 2*x)/sqrt(7))/sqrt(7)]

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

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

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

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

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

    @test_int [(3 + 2*x^2)/(1 - 2*x^2 + x^4), x, 3, (5*x)/(2*(1 - x^2)) + atanh(x)/2]

    @test_int [(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4), x, 3, (5*atanh(x))/2 - (7/2)*sqrt(3/5)*atanh(sqrt(3/5)*x)]
    @test_int [(d + e*x^2)/(5 - 8*x^2 + 3*x^4), x, 3, (1/2)*(d + e)*atanh(x) - ((3*d + 5*e)*atanh(sqrt(3/5)*x))/(2*sqrt(15))]

    @test_int [(3 + x^2)/(1 + 3*x^2 + x^4), x, 3, (-(1/10))*sqrt(180 - 80*sqrt(5))*atan(sqrt(2/(3 + sqrt(5)))*x) + ((3 + sqrt(5))^(3/2)*atan(sqrt((1/2)*(3 + sqrt(5)))*x))/(2*sqrt(10))]

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

    @test_int [(a + b*x^2)/(2 + x^2 + x^4), x, 9, (-(1/2))*sqrt((1/14)*(-1 + 2*sqrt(2)))*(a + sqrt(2)*b)*atan((sqrt(-1 + 2*sqrt(2)) - 2*x)/sqrt(1 + 2*sqrt(2))) + (1/2)*sqrt((1/14)*(-1 + 2*sqrt(2)))*(a + sqrt(2)*b)*atan((sqrt(-1 + 2*sqrt(2)) + 2*x)/sqrt(1 + 2*sqrt(2))) - ((a - sqrt(2)*b)*log(sqrt(2) - sqrt(-1 + 2*sqrt(2))*x + x^2))/(4*sqrt(2*(-1 + 2*sqrt(2)))) + ((a - sqrt(2)*b)*log(sqrt(2) + sqrt(-1 + 2*sqrt(2))*x + x^2))/(4*sqrt(2*(-1 + 2*sqrt(2))))]
    @test_int [(a + b*x^2)/(2 + x^2 + x^4)^2, x, 10, (x*(3*a + 2*b - (a - 4*b)*x^2))/(28*(2 + x^2 + x^4)) - (1/56)*sqrt((1/14)*(-1 + 2*sqrt(2)))*((11 - sqrt(2))*a - (2 - 4*sqrt(2))*b)*atan((sqrt(-1 + 2*sqrt(2)) - 2*x)/sqrt(1 + 2*sqrt(2))) + (1/56)*sqrt((1/14)*(-1 + 2*sqrt(2)))*((11 - sqrt(2))*a - (2 - 4*sqrt(2))*b)*atan((sqrt(-1 + 2*sqrt(2)) + 2*x)/sqrt(1 + 2*sqrt(2))) - ((11*a + sqrt(2)*(a - 4*b) - 2*b)*log(sqrt(2) - sqrt(-1 + 2*sqrt(2))*x + x^2))/(112*sqrt(2*(-1 + 2*sqrt(2)))) + (((11 + sqrt(2))*a - 2*(b + 2*sqrt(2)*b))*log(sqrt(2) + sqrt(-1 + 2*sqrt(2))*x + x^2))/(112*sqrt(2*(-1 + 2*sqrt(2))))]

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

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

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

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

    @test_int [(A + B*x^2)/(a - sqrt(a*c)*x^2 + c*x^4), x, 9, -(((sqrt(a)*B + A*sqrt(c))*atan((sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c)) - 2*sqrt(c)*x)/sqrt(2*sqrt(a)*sqrt(c) - sqrt(a*c))))/(2*sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c) - sqrt(a*c)))) + ((sqrt(a)*B + A*sqrt(c))*atan((sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c)) + 2*sqrt(c)*x)/sqrt(2*sqrt(a)*sqrt(c) - sqrt(a*c))))/(2*sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c) - sqrt(a*c))) - ((A - (sqrt(a)*B)/sqrt(c))*log(sqrt(a) - sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c))*x + sqrt(c)*x^2))/(4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c))) + ((A - (sqrt(a)*B)/sqrt(c))*log(sqrt(a) + sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c))*x + sqrt(c)*x^2))/(4*sqrt(a)*sqrt(2*sqrt(a)*sqrt(c) + sqrt(a*c)))]
    @test_int [(A + B*x^2)/(a - sqrt(a)*sqrt(c)*x^2 + c*x^4), x, 9, -(((sqrt(a)*B + A*sqrt(c))*atan(sqrt(3) - (2*c^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*c^(3/4))) + ((sqrt(a)*B + A*sqrt(c))*atan(sqrt(3) + (2*c^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*c^(3/4)) - ((A - (sqrt(a)*B)/sqrt(c))*log(sqrt(a) - sqrt(3)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(3)*a^(3/4)*c^(1/4)) + ((A - (sqrt(a)*B)/sqrt(c))*log(sqrt(a) + sqrt(3)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(3)*a^(3/4)*c^(1/4))]

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

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

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

    @test_int [(3 - x^2)/sqrt(3 + 1*x^2 - x^4), x, 4, (-sqrt((1/2)*(-1 + sqrt(13))))*Elliptic.E(asin(sqrt(2/(1 + sqrt(13)))*x), (1/6)*(-7 - sqrt(13))) + sqrt(7 + 2*sqrt(13))*Elliptic.F(asin(sqrt(2/(1 + sqrt(13)))*x), (1/6)*(-7 - sqrt(13)))]
    @test_int [(3 - x^2)/sqrt(3 + 2*x^2 - x^4), x, 5, -Elliptic.E(asin(x/sqrt(3)), -3) + 4*Elliptic.F(asin(x/sqrt(3)), -3)]
    @test_int [(3 - x^2)/sqrt(3 + 3*x^2 - x^4), x, 4, (-sqrt((1/2)*(-3 + sqrt(21))))*Elliptic.E(asin(sqrt(2/(3 + sqrt(21)))*x), (1/2)*(-5 - sqrt(21))) + sqrt(9 + 2*sqrt(21))*Elliptic.F(asin(sqrt(2/(3 + sqrt(21)))*x), (1/2)*(-5 - sqrt(21)))]

    @test_int [(3 - x^2)/sqrt(3 - 1*x^2 - x^4), x, 4, (-sqrt((1/2)*(1 + sqrt(13))))*Elliptic.E(asin(sqrt(2/(-1 + sqrt(13)))*x), (1/6)*(-7 + sqrt(13))) + sqrt(5 + 2*sqrt(13))*Elliptic.F(asin(sqrt(2/(-1 + sqrt(13)))*x), (1/6)*(-7 + sqrt(13)))]
    @test_int [(3 - x^2)/sqrt(3 - 2*x^2 - x^4), x, 4, (-sqrt(3))*Elliptic.E(asin(x), -(1/3)) + 2*sqrt(3)*Elliptic.F(asin(x), -(1/3))]
    @test_int [(3 - x^2)/sqrt(3 - 3*x^2 - x^4), x, 4, (-sqrt((1/2)*(3 + sqrt(21))))*Elliptic.E(asin(sqrt(2/(-3 + sqrt(21)))*x), (1/2)*(-5 + sqrt(21))) + sqrt(3 + 2*sqrt(21))*Elliptic.F(asin(sqrt(2/(-3 + sqrt(21)))*x), (1/2)*(-5 + sqrt(21)))]

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

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

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

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

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

    @test_int [(d + e*x^2)^4*(a + c*x^4), x, 2, a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*e^2)*x^7)/7 + (e^2*(6*c*d^2 + a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)/13]
    @test_int [(d + e*x^2)^3*(a + c*x^4), x, 2, a*d^3*x + a*d^2*e*x^3 + (d*(c*d^2 + 3*a*e^2)*x^5)/5 + (e*(3*c*d^2 + a*e^2)*x^7)/7 + (c*d*e^2*x^9)/3 + (c*e^3*x^11)/11]
    @test_int [(d + e*x^2)^2*(a + c*x^4), x, 2, a*d^2*x + (2*a*d*e*x^3)/3 + ((c*d^2 + a*e^2)*x^5)/5 + (2*c*d*e*x^7)/7 + (c*e^2*x^9)/9]
    @test_int [(d + e*x^2)^1*(a + c*x^4), x, 2, a*d*x + (a*e*x^3)/3 + (c*d*x^5)/5 + (c*e*x^7)/7]
    @test_int [(a + c*x^4)/(d + e*x^2)^1, x, 3, -((c*d*x)/e^2) + (c*x^3)/(3*e) + ((c*d^2 + a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*e^(5/2))]
    @test_int [(a + c*x^4)/(d + e*x^2)^2, x, 3, (c*x)/e^2 + ((a + (c*d^2)/e^2)*x)/(2*d*(d + e*x^2)) - ((3*c*d^2 - a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(5/2))]
    @test_int [(a + c*x^4)/(d + e*x^2)^3, x, 3, ((a + (c*d^2)/e^2)*x)/(4*d*(d + e*x^2)^2) + (((3*a)/d^2 - (5*c)/e^2)*x)/(8*(d + e*x^2)) + (3*(c*d^2 + a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(5/2)*e^(5/2))]
    @test_int [(a + c*x^4)/(d + e*x^2)^4, x, 4, ((a + (c*d^2)/e^2)*x)/(6*d*(d + e*x^2)^3) + (((5*a)/d^2 - (7*c)/e^2)*x)/(24*(d + e*x^2)^2) + (((5*a)/d^2 + c/e^2)*x)/(16*d*(d + e*x^2)) + ((c*d^2 + 5*a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(16*d^(7/2)*e^(5/2))]

    @test_int [(d + e*x^2)^3*(a + c*x^4)^2, x, 2, a^2*d^3*x + a^2*d^2*e*x^3 + (a*d*(2*c*d^2 + 3*a*e^2)*x^5)/5 + (a*e*(6*c*d^2 + a*e^2)*x^7)/7 + (c*d*(c*d^2 + 6*a*e^2)*x^9)/9 + (c*e*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (3*c^2*d*e^2*x^13)/13 + (c^2*e^3*x^15)/15]
    @test_int [(d + e*x^2)^2*(a + c*x^4)^2, x, 2, a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9 + (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13]
    @test_int [(d + e*x^2)^1*(a + c*x^4)^2, x, 2, a^2*d*x + (a^2*e*x^3)/3 + (2*a*c*d*x^5)/5 + (2*a*c*e*x^7)/7 + (c^2*d*x^9)/9 + (c^2*e*x^11)/11]
    @test_int [(d + e*x^2)^0*(a + c*x^4)^2, x, 2, a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9]
    @test_int [(a + c*x^4)^2/(d + e*x^2)^1, x, 3, -((c*d*(c*d^2 + 2*a*e^2)*x)/e^4) + (c*(c*d^2 + 2*a*e^2)*x^3)/(3*e^3) - (c^2*d*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 + a*e^2)^2*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*e^(9/2))]
    @test_int [(a + c*x^4)^2/(d + e*x^2)^2, x, 4, (c*(3*c*d^2 + 2*a*e^2)*x)/e^4 - (2*c^2*d*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((7*c*d^2 - a*e^2)*(c*d^2 + a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(9/2))]
    @test_int [(a + c*x^4)^2/(d + e*x^2)^3, x, 5, -((3*c^2*d*x)/e^4) + (c^2*x^3)/(3*e^3) + ((c*d^2 + a*e^2)^2*x)/(4*d*e^4*(d + e*x^2)^2) + ((3*a^2 - (13*c^2*d^4)/e^4 - (10*a*c*d^2)/e^2)*x)/(8*d^2*(d + e*x^2)) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(5/2)*e^(9/2))]
    @test_int [(a + c*x^4)^2/(d + e*x^2)^4, x, 5, (c^2*x)/e^4 + ((c*d^2 + a*e^2)^2*x)/(6*d*e^4*(d + e*x^2)^3) + ((5*a^2 - (19*c^2*d^4)/e^4 - (14*a*c*d^2)/e^2)*x)/(24*d^2*(d + e*x^2)^2) + ((5*a^2 + (29*c^2*d^4)/e^4 + (2*a*c*d^2)/e^2)*x)/(16*d^3*(d + e*x^2)) - ((35*c^2*d^4 - 2*a*c*d^2*e^2 - 5*a^2*e^4)*atan((sqrt(e)*x)/sqrt(d)))/(16*d^(7/2)*e^(9/2))]
    @test_int [(a + c*x^4)^2/(d + e*x^2)^5, x, 5, ((c*d^2 + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) + ((7*a^2 - (25*c^2*d^4)/e^4 - (18*a*c*d^2)/e^2)*x)/(48*d^2*(d + e*x^2)^3) + ((35*a^2 + (163*c^2*d^4)/e^4 + (6*a*c*d^2)/e^2)*x)/(192*d^3*(d + e*x^2)^2) - ((93*c^2*d^4 - 6*a*c*d^2*e^2 - 35*a^2*e^4)*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*atan((sqrt(e)*x)/sqrt(d)))/(128*d^(9/2)*e^(9/2))]

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

    @test_int [(d + e*x^2)^4/(a + c*x^4), x, 11, (e^2*(6*c*d^2 - a*e^2)*x)/c^2 + (4*d*e^3*x^3)/(3*c) + (e^4*x^5)/(5*c) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(a)*sqrt(c)*d*e*(c*d^2 - a*e^2))*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(9/4)) + ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(a)*sqrt(c)*d*e*(c*d^2 - a*e^2))*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(9/4)) - ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(a)*sqrt(c)*d*e*(c*d^2 - a*e^2))*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(9/4)) + ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(a)*sqrt(c)*d*e*(c*d^2 - a*e^2))*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(9/4))]
    @test_int [(d + e*x^2)^3/(a + c*x^4), x, 11, (3*d*e^2*x)/c + (e^3*x^3)/(3*c) - ((sqrt(c)*d*(c*d^2 - 3*a*e^2) + sqrt(a)*e*(3*c*d^2 - a*e^2))*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(7/4)) + ((sqrt(c)*d*(c*d^2 - 3*a*e^2) + sqrt(a)*e*(3*c*d^2 - a*e^2))*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(7/4)) - ((sqrt(c)*d*(c*d^2 - 3*a*e^2) - sqrt(a)*e*(3*c*d^2 - a*e^2))*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(7/4)) + ((sqrt(c)*d*(c*d^2 - 3*a*e^2) - sqrt(a)*e*(3*c*d^2 - a*e^2))*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(7/4))]
    @test_int [(d + e*x^2)^2/(a + c*x^4), x, 11, (e^2*x)/c - ((c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(5/4)) + ((c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(5/4)) - ((c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(5/4)) + ((c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(5/4))]
    @test_int [(d + e*x^2)^1/(a + c*x^4), x, 9, -((sqrt(c)*d + sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(3/4)) + ((sqrt(c)*d + sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*c^(3/4)) - ((sqrt(c)*d - sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(3/4)) + ((sqrt(c)*d - sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*c^(3/4))]
    @test_int [(d + e*x^2)^0/(a + c*x^4), x, 9, -atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4))/(2*sqrt(2)*a^(3/4)*c^(1/4)) + atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4))/(2*sqrt(2)*a^(3/4)*c^(1/4)) - log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2)/(4*sqrt(2)*a^(3/4)*c^(1/4)) + log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2)/(4*sqrt(2)*a^(3/4)*c^(1/4))]
    @test_int [1/((d + e*x^2)^1*(a + c*x^4)), x, 12, (e^(3/2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*(c*d^2 + a*e^2)) - (c^(1/4)*(sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)) + (c^(1/4)*(sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)) - (c^(1/4)*(sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)) + (c^(1/4)*(sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2))]
    @test_int [1/((d + e*x^2)^2*(a + c*x^4)), x, 14, (e^2*x)/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + (2*c*sqrt(d)*e^(3/2)*atan((sqrt(e)*x)/sqrt(d)))/(c*d^2 + a*e^2)^2 + (e^(3/2)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*(c*d^2 + a*e^2)) - (c^(3/4)*(c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*(c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2)]

    @test_int [(d + e*x^2)^3/(a + c*x^4)^2, x, 11, -((e^3*x^3)/(c*(a + c*x^4))) + (x*(d*(c*d^2 - 3*a*e^2) + 3*e*(c*d^2 + a*e^2)*x^2))/(4*a*c*(a + c*x^4)) - (3*(sqrt(c)*d + sqrt(a)*e)*(c*d^2 + a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(7/4)) + (3*(sqrt(c)*d + sqrt(a)*e)*(c*d^2 + a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(7/4)) - (3*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(7/4)) + (3*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(7/4))]
    @test_int [(d + e*x^2)^2/(a + c*x^4)^2, x, 11, -(e^2*x)/(3*c*(a + c*x^4)) + (x*(3*c*d^2 + a*e^2 + 6*c*d*e*x^2))/(12*a*c*(a + c*x^4)) - ((3*c*d^2 + 2*sqrt(a)*sqrt(c)*d*e + a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(5/4)) + ((3*c*d^2 + 2*sqrt(a)*sqrt(c)*d*e + a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(5/4)) - ((3*c*d^2 - 2*sqrt(a)*sqrt(c)*d*e + a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(5/4)) + ((3*c*d^2 - 2*sqrt(a)*sqrt(c)*d*e + a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(5/4))]
    @test_int [(d + e*x^2)^1/(a + c*x^4)^2, x, 10, (x*(d + e*x^2))/(4*a*(a + c*x^4)) - ((3*sqrt(c)*d + sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(3/4)) + ((3*sqrt(c)*d + sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(3/4)) - ((3*sqrt(c)*d - sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(3/4)) + ((3*sqrt(c)*d - sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(3/4))]
    @test_int [(d + e*x^2)^0/(a + c*x^4)^2, x, 10, x/(4*a*(a + c*x^4)) - (3*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(1/4)) + (3*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*c^(1/4)) - (3*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(1/4)) + (3*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*c^(1/4))]
    @test_int [1/((d + e*x^2)^1*(a + c*x^4)^2), x, 22, (c*x*(d - e*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*(c*d^2 + a*e^2)^2) - (c^(1/4)*e^2*(sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) - (c^(1/4)*(3*sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)) + (c^(1/4)*e^2*(sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(1/4)*(3*sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)) - (c^(1/4)*e^2*(sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) - (c^(1/4)*(3*sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)) + (c^(1/4)*e^2*(sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(1/4)*(3*sqrt(c)*d + sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2))]
    @test_int [1/((d + e*x^2)^2*(a + c*x^4)^2), x, 24, (e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2))/(4*a*(c*d^2 + a*e^2)^2*(a + c*x^4)) + (4*c*sqrt(d)*e^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(c*d^2 + a*e^2)^3 + (e^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - 3*a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^2 - 4*sqrt(a)*sqrt(c)*d*e - a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - 3*a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 + 4*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - 3*a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^2 + 4*sqrt(a)*sqrt(c)*d*e - a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*sqrt(a)*sqrt(c)*d*e - 3*a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(7/4)*(c*d^2 + a*e^2)^2)]

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

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

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

    @test_int [(d + e*x^2)^4/sqrt(a + c*x^4), x, 6, (e^2*(42*c*d^2 - 5*a*e^2)*x*sqrt(a + c*x^4))/(21*c^2) + (4*d*e^3*x^3*sqrt(a + c*x^4))/(5*c) + (e^4*x^5*sqrt(a + c*x^4))/(7*c) + (4*d*e*(5*c*d^2 - 3*a*e^2)*x*sqrt(a + c*x^4))/(5*c^(3/2)*(sqrt(a) + sqrt(c)*x^2)) - (4*a^(1/4)*d*e*(5*c*d^2 - 3*a*e^2)*(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))/(5*c^(7/4)*sqrt(a + c*x^4)) + ((105*c^2*d^4 + 420*sqrt(a)*c^(3/2)*d^3*e - 210*a*c*d^2*e^2 - 252*a^(3/2)*sqrt(c)*d*e^3 + 25*a^2*e^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))/(210*a^(1/4)*c^(9/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x^2)^3/sqrt(a + c*x^4), x, 5, (d*e^2*x*sqrt(a + c*x^4))/c + (e^3*x^3*sqrt(a + c*x^4))/(5*c) + (3*e*(5*c*d^2 - a*e^2)*x*sqrt(a + c*x^4))/(5*c^(3/2)*(sqrt(a) + sqrt(c)*x^2)) - (3*a^(1/4)*e*(5*c*d^2 - a*e^2)*(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))/(5*c^(7/4)*sqrt(a + c*x^4)) + (a^(1/4)*(15*c*d^2*e - 3*a*e^3 + (5*sqrt(c)*d*(c*d^2 - a*e^2))/sqrt(a))*(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))/(10*c^(7/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x^2)^2/sqrt(a + c*x^4), x, 4, (e^2*x*sqrt(a + c*x^4))/(3*c) + (2*d*e*x*sqrt(a + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (2*a^(1/4)*d*e*(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)) + ((3*c*d^2 + 6*sqrt(a)*sqrt(c)*d*e - a*e^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))/(6*a^(1/4)*c^(5/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x^2)^1/sqrt(a + c*x^4), x, 3, (e*x*sqrt(a + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*e*(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(c)*d)/sqrt(a) + e)*(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 [1/((d + e*x^2)^1*sqrt(a + c*x^4)), x, 3, (sqrt(e)*atan((sqrt(c*d^2 + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + c*x^4))))/(2*sqrt(d)*sqrt(c*d^2 + a*e^2)) + (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^(1/4)*(sqrt(c)*d - sqrt(a)*e)*sqrt(a + c*x^4)) - (a^(3/4)*((sqrt(c)*d)/sqrt(a) + e)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*c^(1/4)*d*(c*d^2 - a*e^2)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x^2)^2*sqrt(a + c*x^4)), x, 6, -((sqrt(c)*e*x*sqrt(a + c*x^4))/(2*d*(c*d^2 + a*e^2)*(sqrt(a) + sqrt(c)*x^2))) + (e^2*x*sqrt(a + c*x^4))/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + (sqrt(e)*(3*c*d^2 + a*e^2)*atan((sqrt(c*d^2 + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + c*x^4))))/(4*d^(3/2)*(c*d^2 + a*e^2)^(3/2)) + (a^(1/4)*c^(1/4)*e*(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))/(2*d*(c*d^2 + a*e^2)*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^(1/4)*d*(sqrt(c)*d - sqrt(a)*e)*sqrt(a + c*x^4)) - ((sqrt(c)*d + sqrt(a)*e)*(3*c*d^2 + a*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*c^(1/4)*d^2*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + a*e^2)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x^2)^3*sqrt(a + c*x^4)), x, 7, -((3*sqrt(c)*e*(3*c*d^2 + a*e^2)*x*sqrt(a + c*x^4))/(8*d^2*(c*d^2 + a*e^2)^2*(sqrt(a) + sqrt(c)*x^2))) + (e^2*x*sqrt(a + c*x^4))/(4*d*(c*d^2 + a*e^2)*(d + e*x^2)^2) + (3*e^2*(3*c*d^2 + a*e^2)*x*sqrt(a + c*x^4))/(8*d^2*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (3*sqrt(e)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*atan((sqrt(c*d^2 + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + c*x^4))))/(16*d^(5/2)*(c*d^2 + a*e^2)^(5/2)) + (3*a^(1/4)*c^(1/4)*e*(3*c*d^2 + a*e^2)*(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))/(8*d^2*(c*d^2 + a*e^2)^2*sqrt(a + c*x^4)) + (c^(1/4)*(4*c*d^2 - sqrt(a)*sqrt(c)*d*e + 3*a*e^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))/(8*a^(1/4)*d^2*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + a*e^2)*sqrt(a + c*x^4)) - (3*(sqrt(c)*d + sqrt(a)*e)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(32*a^(1/4)*c^(1/4)*d^3*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + a*e^2)^2*sqrt(a + c*x^4))]

    @test_int [(d + e*x^2)^3/sqrt(a - c*x^4), x, 8, -((d*e^2*x*sqrt(a - c*x^4))/c) - (e^3*x^3*sqrt(a - c*x^4))/(5*c) + (3*a^(3/4)*e*(5*c*d^2 + a*e^2)*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(5*c^(7/4)*sqrt(a - c*x^4)) + (a^(3/4)*((5*sqrt(c)*d*(c*d^2 + a*e^2))/sqrt(a) - 3*e*(5*c*d^2 + a*e^2))*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(5*c^(7/4)*sqrt(a - c*x^4))]
    @test_int [(d + e*x^2)^2/sqrt(a - c*x^4), x, 7, -((e^2*x*sqrt(a - c*x^4))/(3*c)) + (2*a^(3/4)*d*e*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(3/4)*sqrt(a - c*x^4)) + (a^(1/4)*(3*c*d^2 - 6*sqrt(a)*sqrt(c)*d*e + a*e^2)*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(3*c^(5/4)*sqrt(a - c*x^4))]
    @test_int [(d + e*x^2)^1/sqrt(a - c*x^4), x, 6, (a^(3/4)*e*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(3/4)*sqrt(a - c*x^4)) + (a^(3/4)*((sqrt(c)*d)/sqrt(a) - e)*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(3/4)*sqrt(a - c*x^4))]
    @test_int [1/((d + e*x^2)^1*sqrt(a - c*x^4)), x, 2, (a^(1/4)*sqrt(1 - (c*x^4)/a)*Elliptic.Pi(-((sqrt(a)*e)/(sqrt(c)*d)), asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(1/4)*d*sqrt(a - c*x^4))]
    @test_int [1/((d + e*x^2)^2*sqrt(a - c*x^4)), x, 10, -((e^2*x*sqrt(a - c*x^4))/(2*d*(c*d^2 - a*e^2)*(d + e*x^2))) - (a^(3/4)*c^(1/4)*e*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(2*d*(c*d^2 - a*e^2)*sqrt(a - c*x^4)) - (a^(1/4)*c^(1/4)*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(2*d*(sqrt(c)*d + sqrt(a)*e)*sqrt(a - c*x^4)) + (a^(1/4)*(3*c*d^2 - a*e^2)*sqrt(1 - (c*x^4)/a)*Elliptic.Pi(-((sqrt(a)*e)/(sqrt(c)*d)), asin((c^(1/4)*x)/a^(1/4)), -1))/(2*c^(1/4)*d^2*(c*d^2 - a*e^2)*sqrt(a - c*x^4))]
    @test_int [1/((d + e*x^2)^3*sqrt(a - c*x^4)), x, 11, -((e^2*x*sqrt(a - c*x^4))/(4*d*(c*d^2 - a*e^2)*(d + e*x^2)^2)) - (3*e^2*(3*c*d^2 - a*e^2)*x*sqrt(a - c*x^4))/(8*d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)) - (3*a^(3/4)*c^(1/4)*e*(3*c*d^2 - a*e^2)*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(8*d^2*(c*d^2 - a*e^2)^2*sqrt(a - c*x^4)) - (a^(1/4)*c^(1/4)*(7*c*d^2 - 2*sqrt(a)*sqrt(c)*d*e - 3*a*e^2)*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(8*d^2*(sqrt(c)*d + sqrt(a)*e)*(c*d^2 - a*e^2)*sqrt(a - c*x^4)) + (3*a^(1/4)*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(1 - (c*x^4)/a)*Elliptic.Pi(-((sqrt(a)*e)/(sqrt(c)*d)), asin((c^(1/4)*x)/a^(1/4)), -1))/(8*c^(1/4)*d^3*(c*d^2 - a*e^2)^2*sqrt(a - c*x^4))]
    @test_int [1/((d + e*x^2)^4*sqrt(a - c*x^4)), x, 12, -((e^2*x*sqrt(a - c*x^4))/(6*d*(c*d^2 - a*e^2)*(d + e*x^2)^3)) - (5*e^2*(3*c*d^2 - a*e^2)*x*sqrt(a - c*x^4))/(24*d^2*(c*d^2 - a*e^2)^2*(d + e*x^2)^2) - (e^2*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*x*sqrt(a - c*x^4))/(16*d^3*(c*d^2 - a*e^2)^3*(d + e*x^2)) - (a^(3/4)*c^(1/4)*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(1 - (c*x^4)/a)*Elliptic.E(asin((c^(1/4)*x)/a^(1/4)), -1))/(16*d^3*(c*d^2 - a*e^2)^3*sqrt(a - c*x^4)) - (a^(1/4)*c^(1/4)*(57*c^2*d^4 - 30*sqrt(a)*c^(3/2)*d^3*e - 32*a*c*d^2*e^2 + 10*a^(3/2)*sqrt(c)*d*e^3 + 15*a^2*e^4)*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(48*d^3*(sqrt(c)*d - sqrt(a)*e)^2*(sqrt(c)*d + sqrt(a)*e)^3*sqrt(a - c*x^4)) + (a^(1/4)*(35*c^3*d^6 - 7*a*c^2*d^4*e^2 + 17*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(1 - (c*x^4)/a)*Elliptic.Pi(-((sqrt(a)*e)/(sqrt(c)*d)), asin((c^(1/4)*x)/a^(1/4)), -1))/(16*c^(1/4)*d^4*(c*d^2 - a*e^2)^3*sqrt(a - c*x^4))]

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

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

    @test_int [(d + e*x^2)/sqrt(-a - c*x^4), x, 3, -((e*x*sqrt(-a - c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2))) - (a^(1/4)*e*(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(c)*d)/sqrt(a) + e)*(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 [1/((d + e*x^2)*sqrt(-a - c*x^4)), x, 3, (sqrt(e)*atan((sqrt((-c)*d^2 - a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(-a - c*x^4))))/(2*sqrt(d)*sqrt((-c)*d^2 - a*e^2)) + (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^(1/4)*(sqrt(c)*d - sqrt(a)*e)*sqrt(-a - c*x^4)) - (a^(3/4)*((sqrt(c)*d)/sqrt(a) + e)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*c^(1/4)*d*(c*d^2 - a*e^2)*sqrt(-a - c*x^4))]

    @test_int [1/((a + b*x^2)*sqrt(4 - 5*x^4)), x, 2, Elliptic.Pi(-((2*b)/(sqrt(5)*a)), asin((5^(1/4)*x)/sqrt(2)), -1)/(sqrt(2)*5^(1/4)*a)]
    @test_int [1/((a + b*x^2)*sqrt(4 + 5*x^4)), x, 3, (sqrt(b)*atan((sqrt(5*a^2 + 4*b^2)*x)/(sqrt(a)*sqrt(b)*sqrt(4 + 5*x^4))))/(2*sqrt(a)*sqrt(5*a^2 + 4*b^2)) + (5^(1/4)*(sqrt(5)*a + 2*b)*(2 + sqrt(5)*x^2)*sqrt((4 + 5*x^4)/(2 + sqrt(5)*x^2)^2)*Elliptic.F(2*atan((5^(1/4)*x)/sqrt(2)), 1/2))/(2*sqrt(2)*(5*a^2 - 4*b^2)*sqrt(4 + 5*x^4)) - ((sqrt(5)*a + 2*b)^2*(2 + sqrt(5)*x^2)*sqrt((4 + 5*x^4)/(2 + sqrt(5)*x^2)^2)*Elliptic.Pi(-((sqrt(5)*a - 2*b)^2/(8*sqrt(5)*a*b)), 2*atan((5^(1/4)*x)/sqrt(2)), 1/2))/(4*sqrt(2)*5^(1/4)*a*(5*a^2 - 4*b^2)*sqrt(4 + 5*x^4))]

    @test_int [1/((a + b*x^2)*sqrt(4 - d*x^4)), x, 1, Elliptic.Pi(-((2*b)/(a*sqrt(d))), asin((d^(1/4)*x)/sqrt(2)), -1)/(sqrt(2)*a*d^(1/4))]
    @test_int [1/((a + b*x^2)*sqrt(4 + d*x^4)), x, 3, (sqrt(b)*atan((sqrt(4*b^2 + a^2*d)*x)/(sqrt(a)*sqrt(b)*sqrt(4 + d*x^4))))/(2*sqrt(a)*sqrt(4*b^2 + a^2*d)) - (d^(1/4)*(2 + sqrt(d)*x^2)*sqrt((4 + d*x^4)/(2 + sqrt(d)*x^2)^2)*Elliptic.F(2*atan((d^(1/4)*x)/sqrt(2)), 1/2))/(2*sqrt(2)*(2*b - a*sqrt(d))*sqrt(4 + d*x^4)) + ((2*b + a*sqrt(d))*(2 + sqrt(d)*x^2)*sqrt((4 + d*x^4)/(2 + sqrt(d)*x^2)^2)*Elliptic.Pi(-((2*b - a*sqrt(d))^2/(8*a*b*sqrt(d))), 2*atan((d^(1/4)*x)/sqrt(2)), 1/2))/(4*sqrt(2)*a*(2*b - a*sqrt(d))*d^(1/4)*sqrt(4 + d*x^4))]

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

    @test_int [sqrt(a + b*x^2)/sqrt(1 - x^4), x, -1, (a*sqrt(1 - x^2)*sqrt((a*(1 + x^2))/(a + b*x^2))*Elliptic.Pi(b/(a + b), asin((sqrt(a + b)*x)/sqrt(a + b*x^2)), -((a - b)/(a + b))))/(sqrt(a + b)*sqrt(1 + x^2)*sqrt((a*(1 - x^2))/(a + b*x^2)))]

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

    @test_int [(c + e*x^2)^q*(a + b*x^4)^p, x, 0, Unintegrable((c + e*x^2)^q*(a + b*x^4)^p, x)]

    @test_int [(c + e*x^2)^3*(a + b*x^4)^p, x, 9, (e^3*x^3*(a + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c^3*x*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p - (e*(a*e^2 - b*c^2*(7 + 4*p))*x^3*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, -((b*x^4)/a)))/((1 + (b*x^4)/a)^p*(b*(7 + 4*p))) + ((3/5)*c*e^2*x^5*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(5/4, -p, 9/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p]
    @test_int [(c + e*x^2)^2*(a + b*x^4)^p, x, 7, (e^2*x*(a + b*x^4)^(1 + p))/(b*(5 + 4*p)) - ((a*e^2 - b*c^2*(5 + 4*p))*x*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, -((b*x^4)/a)))/((1 + (b*x^4)/a)^p*(b*(5 + 4*p))) + ((2/3)*c*e*x^3*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p]
    @test_int [(c + e*x^2)^1*(a + b*x^4)^p, x, 6, (c*x*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p + ((1/3)*e*x^3*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p]
    @test_int [(c + e*x^2)^0*(a + b*x^4)^p, x, 2, (x*(a + b*x^4)^p*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, -((b*x^4)/a)))/(1 + (b*x^4)/a)^p]
    @test_int [(a + b*x^4)^p/(c + e*x^2)^1, x, 6, (x*(a + b*x^4)^p*AppellF1(1/4, -p, 1, 5/4, -((b*x^4)/a), (e^2*x^4)/c^2))/((1 + (b*x^4)/a)^p*c) - (e*x^3*(a + b*x^4)^p*AppellF1(3/4, -p, 1, 7/4, -((b*x^4)/a), (e^2*x^4)/c^2))/((1 + (b*x^4)/a)^p*(3*c^2))]
    @test_int [(a + b*x^4)^p/(c + e*x^2)^2, x, 8, (x*(a + b*x^4)^p*AppellF1(1/4, -p, 2, 5/4, -((b*x^4)/a), (e^2*x^4)/c^2))/((1 + (b*x^4)/a)^p*c^2) - (2*e*x^3*(a + b*x^4)^p*AppellF1(3/4, -p, 2, 7/4, -((b*x^4)/a), (e^2*x^4)/c^2))/((1 + (b*x^4)/a)^p*(3*c^3)) + (e^2*x^5*(a + b*x^4)^p*AppellF1(5/4, -p, 2, 9/4, -((b*x^4)/a), (e^2*x^4)/c^2))/((1 + (b*x^4)/a)^p*(5*c^4))]

    @test_int [(1 - x^2)^3*(1 + b*x^4)^p, x, 6, -((x^3*(1 + b*x^4)^(1 + p))/(b*(7 + 4*p))) + x*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, (-b)*x^4) + ((1 - b*(7 + 4*p))*x^3*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, (-b)*x^4))/(b*(7 + 4*p)) + (3/5)*x^5*HypergeometricFunctions._₂F₁(5/4, -p, 9/4, (-b)*x^4)]
    @test_int [(1 - x^2)^2*(1 + b*x^4)^p, x, 5, (x*(1 + b*x^4)^(1 + p))/(b*(5 + 4*p)) - ((1 - b*(5 + 4*p))*x*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, (-b)*x^4))/(b*(5 + 4*p)) - (2/3)*x^3*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, (-b)*x^4)]
    @test_int [(1 - x^2)^1*(1 + b*x^4)^p, x, 4, x*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, (-b)*x^4) - (1/3)*x^3*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, (-b)*x^4)]
    @test_int [(1 - x^2)^0*(1 + b*x^4)^p, x, 1, x*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, (-b)*x^4)]
    @test_int [(1 + b*x^4)^p/(1 - x^2)^1, x, 4, x*AppellF1(1/4, 1, -p, 5/4, x^4, (-b)*x^4) + (1/3)*x^3*AppellF1(3/4, 1, -p, 7/4, x^4, (-b)*x^4)]
    @test_int [(1 + b*x^4)^p/(1 - x^2)^2, x, 5, x*AppellF1(1/4, 2, -p, 5/4, x^4, (-b)*x^4) + (2/3)*x^3*AppellF1(3/4, 2, -p, 7/4, x^4, (-b)*x^4) + (1/5)*x^5*AppellF1(5/4, 2, -p, 9/4, x^4, (-b)*x^4)]
    @test_int [(1 + b*x^4)^p/(1 - x^2)^3, x, 6, x*AppellF1(1/4, 3, -p, 5/4, x^4, (-b)*x^4) + x^3*AppellF1(3/4, 3, -p, 7/4, x^4, (-b)*x^4) + (3/5)*x^5*AppellF1(5/4, 3, -p, 9/4, x^4, (-b)*x^4) + (1/7)*x^7*AppellF1(7/4, 3, -p, 11/4, x^4, (-b)*x^4)]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x^2)^q*(d^2-e^2*x^4)^p=#

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

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

    @test_int [(d + e*x^2)^4/(d^2 - e^2*x^4), x, 4, -7*d^2*x - (4/3)*d*e*x^3 - (e^2*x^5)/5 + (8*d^(5/2)*atanh((sqrt(e)*x)/sqrt(d)))/sqrt(e)]
    @test_int [(d + e*x^2)^3/(d^2 - e^2*x^4), x, 4, -3*d*x - (e*x^3)/3 + (4*d^(3/2)*atanh((sqrt(e)*x)/sqrt(d)))/sqrt(e)]
    @test_int [(d + e*x^2)^2/(d^2 - e^2*x^4), x, 3, -x + (2*sqrt(d)*atanh((sqrt(e)*x)/sqrt(d)))/sqrt(e)]
    @test_int [(d + e*x^2)^1/(d^2 - e^2*x^4), x, 2, atanh((sqrt(e)*x)/sqrt(d))/(sqrt(d)*sqrt(e))]
    @test_int [1/((d + e*x^2)^1*(d^2 - e^2*x^4)), x, 5, x/(4*d^2*(d + e*x^2)) + atan((sqrt(e)*x)/sqrt(d))/(2*d^(5/2)*sqrt(e)) + atanh((sqrt(e)*x)/sqrt(d))/(4*d^(5/2)*sqrt(e))]
    @test_int [1/((d + e*x^2)^2*(d^2 - e^2*x^4)), x, 6, x/(8*d^2*(d + e*x^2)^2) + (5*x)/(16*d^3*(d + e*x^2)) + (7*atan((sqrt(e)*x)/sqrt(d)))/(16*d^(7/2)*sqrt(e)) + atanh((sqrt(e)*x)/sqrt(d))/(8*d^(7/2)*sqrt(e))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x^2)^(q/2)*(d^2-e^2*x^4)^p=#

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

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

    @test_int [(d + e*x^2)^(3/2)/(d^2 - e^2*x^4), x, 6, -(atanh((sqrt(e)*x)/sqrt(d + e*x^2))/sqrt(e)) + (sqrt(2)*atanh((sqrt(2)*sqrt(e)*x)/sqrt(d + e*x^2)))/sqrt(e)]
    @test_int [(d + e*x^2)^(1/2)/(d^2 - e^2*x^4), x, 3, atanh((sqrt(2)*sqrt(e)*x)/sqrt(d + e*x^2))/(sqrt(2)*d*sqrt(e))]
    @test_int [1/((d + e*x^2)^(1/2)*(d^2 - e^2*x^4)), x, 4, x/(2*d^2*sqrt(d + e*x^2)) + atanh((sqrt(2)*sqrt(e)*x)/sqrt(d + e*x^2))/(2*sqrt(2)*d^2*sqrt(e))]
    @test_int [1/((d + e*x^2)^(3/2)*(d^2 - e^2*x^4)), x, 6, x/(6*d^2*(d + e*x^2)^(3/2)) + (7*x)/(12*d^3*sqrt(d + e*x^2)) + atanh((sqrt(2)*sqrt(e)*x)/sqrt(d + e*x^2))/(4*sqrt(2)*d^3*sqrt(e))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x^2)^(q/2)*(d^2-e^2*x^4)^(p/2)=#

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

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

    @test_int [(a + b*x^2)^(5/2)/sqrt(a^2 - b^2*x^4), x, 5, -((9*a*x*(a - b*x^2)*sqrt(a + b*x^2))/(8*sqrt(a^2 - b^2*x^4))) - (x*(a - b*x^2)*(a + b*x^2)^(3/2))/(4*sqrt(a^2 - b^2*x^4)) + (19*a^2*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a - b*x^2)))/(8*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [(a + b*x^2)^(3/2)/sqrt(a^2 - b^2*x^4), x, 4, -((x*(a - b*x^2)*sqrt(a + b*x^2))/(2*sqrt(a^2 - b^2*x^4))) + (3*a*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a - b*x^2)))/(2*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [(a + b*x^2)^(1/2)/sqrt(a^2 - b^2*x^4), x, 3, (sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(b)*x)/sqrt(a - b*x^2)))/(sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a + b*x^2)^(1/2)*sqrt(a^2 - b^2*x^4)), x, 3, (sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(2)*sqrt(b)*x)/sqrt(a - b*x^2)))/(sqrt(2)*a*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a + b*x^2)^(3/2)*sqrt(a^2 - b^2*x^4)), x, 4, (x*(a - b*x^2))/(4*a^2*sqrt(a + b*x^2)*sqrt(a^2 - b^2*x^4)) + (3*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(2)*sqrt(b)*x)/sqrt(a - b*x^2)))/(4*sqrt(2)*a^2*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a + b*x^2)^(5/2)*sqrt(a^2 - b^2*x^4)), x, 6, (x*(a - b*x^2))/(8*a^2*(a + b*x^2)^(3/2)*sqrt(a^2 - b^2*x^4)) + (9*x*(a - b*x^2))/(32*a^3*sqrt(a + b*x^2)*sqrt(a^2 - b^2*x^4)) + (19*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atan((sqrt(2)*sqrt(b)*x)/sqrt(a - b*x^2)))/(32*sqrt(2)*a^3*sqrt(b)*sqrt(a^2 - b^2*x^4))]

    @test_int [(a - b*x^2)^(5/2)/sqrt(a^2 - b^2*x^4), x, 5, -((9*a*x*sqrt(a - b*x^2)*(a + b*x^2))/(8*sqrt(a^2 - b^2*x^4))) - (x*(a - b*x^2)^(3/2)*(a + b*x^2))/(4*sqrt(a^2 - b^2*x^4)) + (19*a^2*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [(a - b*x^2)^(3/2)/sqrt(a^2 - b^2*x^4), x, 4, -((x*sqrt(a - b*x^2)*(a + b*x^2))/(2*sqrt(a^2 - b^2*x^4))) + (3*a*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [(a - b*x^2)^(1/2)/sqrt(a^2 - b^2*x^4), x, 3, (sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a - b*x^2)^(1/2)*sqrt(a^2 - b^2*x^4)), x, 3, (sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(2)*sqrt(b)*x)/sqrt(a + b*x^2)))/(sqrt(2)*a*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a - b*x^2)^(3/2)*sqrt(a^2 - b^2*x^4)), x, 4, (x*(a + b*x^2))/(4*a^2*sqrt(a - b*x^2)*sqrt(a^2 - b^2*x^4)) + (3*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(2)*sqrt(b)*x)/sqrt(a + b*x^2)))/(4*sqrt(2)*a^2*sqrt(b)*sqrt(a^2 - b^2*x^4))]
    @test_int [1/((a - b*x^2)^(5/2)*sqrt(a^2 - b^2*x^4)), x, 6, (x*(a + b*x^2))/(8*a^2*(a - b*x^2)^(3/2)*sqrt(a^2 - b^2*x^4)) + (9*x*(a + b*x^2))/(32*a^3*sqrt(a - b*x^2)*sqrt(a^2 - b^2*x^4)) + (19*sqrt(a - b*x^2)*sqrt(a + b*x^2)*atanh((sqrt(2)*sqrt(b)*x)/sqrt(a + b*x^2)))/(32*sqrt(2)*a^3*sqrt(b)*sqrt(a^2 - b^2*x^4))]

    @test_int [sqrt(-1 + x^2)/sqrt(-1 + x^4), x, 2, (sqrt(-1 + x^2)*sqrt(1 + x^2)*asinh(x))/sqrt(-1 + x^4)]
    @test_int [sqrt(1 + x^2)/sqrt(-1 + x^4), x, 3, -((sqrt(-1 + x^4)*asin(x))/sqrt(1 - x^4)), (sqrt(-1 + x^2)*sqrt(1 + x^2)*atanh(x/sqrt(-1 + x^2)))/sqrt(-1 + x^4)]

    @test_int [(-sqrt(-1 + x^2) + sqrt(1 + x^2))/sqrt(-1 + x^4), x, 7, -((sqrt(-1 + x^4)*asin(x))/(sqrt(1 - x^2)*sqrt(1 + x^2))) + (sqrt(-1 + x^2)*sqrt(-1 + x^4)*asinh(x))/((1 - x^2)*sqrt(1 + x^2)), -((sqrt(-1 + x^2)*sqrt(1 + x^2)*asinh(x))/sqrt(-1 + x^4)) + (sqrt(-1 + x^2)*sqrt(1 + x^2)*atanh(x/sqrt(-1 + x^2)))/sqrt(-1 + x^4)]

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

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

    @test_int [(d + e*x^2)^4/(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4), x, 4, ((7*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*x)/c^3 + (e*(4*c*d - b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((2*c*d - b*e)^3*atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e)))/(c^(7/2)*sqrt(e)*sqrt(c*d - b*e))]
    @test_int [(d + e*x^2)^3/(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4), x, 4, ((3*c*d - b*e)*x)/c^2 + (e*x^3)/(3*c) - ((2*c*d - b*e)^2*atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e)))/(c^(5/2)*sqrt(e)*sqrt(c*d - b*e))]
    @test_int [(d + e*x^2)^2/(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4), x, 3, x/c - ((2*c*d - b*e)*atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e)))/(c^(3/2)*sqrt(e)*sqrt(c*d - b*e))]
    @test_int [(d + e*x^2)^1/(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4), x, 2, -(atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e))/(sqrt(c)*sqrt(e)*sqrt(c*d - b*e)))]
    @test_int [1/((d + e*x^2)^1*(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4)), x, 5, -(x/(2*d*(2*c*d - b*e)*(d + e*x^2))) - ((4*c*d - b*e)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*sqrt(e)*(2*c*d - b*e)^2) - (c^(3/2)*atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e)))/(sqrt(e)*sqrt(c*d - b*e)*(2*c*d - b*e)^2)]
    @test_int [1/((d + e*x^2)^2*(-c*d^2 + b*d*e + b*e^2*x^2 + c*e^2*x^4)), x, 6, -(x/(4*d*(2*c*d - b*e)*(d + e*x^2)^2)) - ((10*c*d - 3*b*e)*x)/(8*d^2*(2*c*d - b*e)^2*(d + e*x^2)) - ((28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(5/2)*sqrt(e)*(2*c*d - b*e)^3) - (c^(5/2)*atanh((sqrt(c)*sqrt(e)*x)/sqrt(c*d - b*e)))/(sqrt(e)*sqrt(c*d - b*e)*(2*c*d - b*e)^3)]

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

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

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

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

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

    @test_int [(1 + x^2)^3*sqrt(1 + x^2 + x^4), x, 6, (26*x*sqrt(1 + x^2 + x^4))/(45*(1 + x^2)) + (2/45)*x*(7 + 6*x^2)*sqrt(1 + x^2 + x^4) + (1/3)*x*(1 + x^2 + x^4)^(3/2) + (1/9)*x^3*(1 + x^2 + x^4)^(3/2) - (26*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(45*sqrt(1 + x^2 + x^4)) + (7*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(15*sqrt(1 + x^2 + x^4))]
    @test_int [(1 + x^2)^2*sqrt(1 + x^2 + x^4), x, 5, (2*x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + (2/21)*x*(4 + 3*x^2)*sqrt(1 + x^2 + x^4) + (1/7)*x*(1 + x^2 + x^4)^(3/2) - (2*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4)) + (4*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(7*sqrt(1 + x^2 + x^4))]
    @test_int [(1 + x^2)^1*sqrt(1 + x^2 + x^4), x, 4, (3*x*sqrt(1 + x^2 + x^4))/(5*(1 + x^2)) + (1/5)*x*(2 + x^2)*sqrt(1 + x^2 + x^4) - (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(5*sqrt(1 + x^2 + x^4)) + (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(5*sqrt(1 + x^2 + x^4))]
    @test_int [sqrt(1 + x^2 + x^4)/(1 + x^2)^1, x, 8, (x*sqrt(1 + x^2 + x^4))/(1 + x^2) + (1/2)*atan(x/sqrt(1 + x^2 + x^4)) - ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/sqrt(1 + x^2 + x^4) + (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]
    @test_int [sqrt(1 + x^2 + x^4)/(1 + x^2)^2, x, 1, ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(2*sqrt(1 + x^2 + x^4))]
    @test_int [sqrt(1 + x^2 + x^4)/(1 + x^2)^3, x, 23, (x*sqrt(1 + x^2 + x^4))/(4*(1 + x^2)^2) + (1/4)*atan(x/sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]
    @test_int [sqrt(1 + x^2 + x^4)/(1 + x^2)^4, x, 26, (x*sqrt(1 + x^2 + x^4))/(6*(1 + x^2)^3) + (x*sqrt(1 + x^2 + x^4))/(6*(1 + x^2)^2) + (1/4)*atan(x/sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4)) - ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(8*sqrt(1 + x^2 + x^4))]

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

    @test_int [(1 + x^2)^3/sqrt(1 + x^2 + x^4), x, 5, (11/15)*x*sqrt(1 + x^2 + x^4) + (1/5)*x^3*sqrt(1 + x^2 + x^4) + (14*x*sqrt(1 + x^2 + x^4))/(15*(1 + x^2)) - (14*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(15*sqrt(1 + x^2 + x^4)) + (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(5*sqrt(1 + x^2 + x^4))]
    @test_int [(1 + x^2)^2/sqrt(1 + x^2 + x^4), x, 4, (1/3)*x*sqrt(1 + x^2 + x^4) + (4*x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) - (4*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/sqrt(1 + x^2 + x^4)]
    @test_int [(1 + x^2)^1/sqrt(1 + x^2 + x^4), x, 3, (x*sqrt(1 + x^2 + x^4))/(1 + x^2) - ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/sqrt(1 + x^2 + x^4) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/sqrt(1 + x^2 + x^4)]
    @test_int [1/((1 + x^2)^1*sqrt(1 + x^2 + x^4)), x, 4, (1/2)*atan(x/sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]
    @test_int [1/((1 + x^2)^2*sqrt(1 + x^2 + x^4)), x, 8, (1/2)*atan(x/sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(2*sqrt(1 + x^2 + x^4)) - ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]
    @test_int [1/((1 + x^2)^3*sqrt(1 + x^2 + x^4)), x, 9, (x*sqrt(1 + x^2 + x^4))/(4*(1 + x^2)^2) + (1/4)*atan(x/sqrt(1 + x^2 + x^4)) + (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4)) - ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(2*sqrt(1 + x^2 + x^4))]

    @test_int [(1 + x^2)^3/(1 + x^2 + x^4)^(3/2), x, 4, -((x*(1 - x^2))/(3*sqrt(1 + x^2 + x^4))) + (2*x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) - (2*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/sqrt(1 + x^2 + x^4)]
    @test_int [(1 + x^2)^2/(1 + x^2 + x^4)^(3/2), x, 2, (x*(1 + 2*x^2))/(3*sqrt(1 + x^2 + x^4)) - (2*x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + (2*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4))]
    @test_int [(1 + x^2)^1/(1 + x^2 + x^4)^(3/2), x, 2, (x*(2 + x^2))/(3*sqrt(1 + x^2 + x^4)) - (x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4))]
    @test_int [1/((1 + x^2)^1*(1 + x^2 + x^4)^(3/2)), x, 9, -((x*(1 + 2*x^2))/(3*sqrt(1 + x^2 + x^4))) + (2*x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + (1/2)*atan(x/sqrt(1 + x^2 + x^4)) - (2*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(3*sqrt(1 + x^2 + x^4)) + (3*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]
    @test_int [1/((1 + x^2)^2*(1 + x^2 + x^4)^(3/2)), x, 16, -((x*(2 + x^2))/(3*sqrt(1 + x^2 + x^4))) + (x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + atan(x/sqrt(1 + x^2 + x^4)) + ((1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(6*sqrt(1 + x^2 + x^4))]
    @test_int [1/((1 + x^2)^3*(1 + x^2 + x^4)^(3/2)), x, 23, -((x*(1 - x^2))/(3*sqrt(1 + x^2 + x^4))) + (x*sqrt(1 + x^2 + x^4))/(4*(1 + x^2)^2) - (x*sqrt(1 + x^2 + x^4))/(3*(1 + x^2)) + (3/4)*atan(x/sqrt(1 + x^2 + x^4)) + (19*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.E(2*atan(x), 1/4))/(12*sqrt(1 + x^2 + x^4)) - (5*(1 + x^2)*sqrt((1 + x^2 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/4))/(4*sqrt(1 + x^2 + x^4))]

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

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

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

    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^4, x, 2, a*d^4*x + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^5)/5 + (2*d*e*(2*c*d^2 + e*(3*b*d + 2*a*e))*x^7)/7 + (e^2*(6*c*d^2 + e*(4*b*d + a*e))*x^9)/9 + (e^3*(4*c*d + b*e)*x^11)/11 + (c*e^4*x^13)/13]
    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^3, x, 2, a*d^3*x + (d^2*(b*d + 3*a*e)*x^3)/3 + (d*(c*d^2 + 3*e*(b*d + a*e))*x^5)/5 + (e*(3*c*d^2 + e*(3*b*d + a*e))*x^7)/7 + (e^2*(3*c*d + b*e)*x^9)/9 + (c*e^3*x^11)/11]
    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^2, x, 2, a*d^2*x + (d*(b*d + 2*a*e)*x^3)/3 + ((c*d^2 + e*(2*b*d + a*e))*x^5)/5 + (e*(2*c*d + b*e)*x^7)/7 + (c*e^2*x^9)/9]
    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^1, x, 2, a*d*x + ((b*d + a*e)*x^3)/3 + ((c*d + b*e)*x^5)/5 + (c*e*x^7)/7]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^1, x, 3, -(((c*d - b*e)*x)/e^2) + (c*x^3)/(3*e) + ((c*d^2 - b*d*e + a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^2, x, 3, (c*x)/e^2 + ((a + (d*(c*d - b*e))/e^2)*x)/(2*d*(d + e*x^2)) - ((3*c*d^2 - e*(b*d + a*e))*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 3, ((a + (d*(c*d - b*e))/e^2)*x)/(4*d*(d + e*x^2)^2) - ((5*c*d^2 - e*(b*d + 3*a*e))*x)/(8*d^2*e^2*(d + e*x^2)) + ((3*c*d^2 + e*(b*d + 3*a*e))*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(5/2)*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^4, x, 4, ((a + (d*(c*d - b*e))/e^2)*x)/(6*d*(d + e*x^2)^3) - ((7*c*d^2 - e*(b*d + 5*a*e))*x)/(24*d^2*e^2*(d + e*x^2)^2) + ((c*d^2 + e*(b*d + 5*a*e))*x)/(16*d^3*e^2*(d + e*x^2)) + ((c*d^2 + e*(b*d + 5*a*e))*atan((sqrt(e)*x)/sqrt(d)))/(16*d^(7/2)*e^(5/2))]

    @test_int [(a + b*x^2 + c*x^4)^2*(d + e*x^2)^3, x, 2, a^2*d^3*x + (1/3)*a*d^2*(2*b*d + 3*a*e)*x^3 + (1/5)*d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^5 + (1/7)*(2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^7 + (1/9)*(c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^9 + (1/11)*e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11 + (1/13)*c*e^2*(3*c*d + 2*b*e)*x^13 + (1/15)*c^2*e^3*x^15]
    @test_int [(a + b*x^2 + c*x^4)^2*(d + e*x^2)^2, x, 2, a^2*d^2*x + (2/3)*a*d*(b*d + a*e)*x^3 + (1/5)*(b^2*d^2 + 4*a*b*d*e + a*(2*c*d^2 + a*e^2))*x^5 + (2/7)*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^7 + (1/9)*(c^2*d^2 + b^2*e^2 + 2*c*e*(2*b*d + a*e))*x^9 + (2/11)*c*e*(c*d + b*e)*x^11 + (1/13)*c^2*e^2*x^13]
    @test_int [(a + b*x^2 + c*x^4)^2*(d + e*x^2)^1, x, 2, a^2*d*x + (1/3)*a*(2*b*d + a*e)*x^3 + (1/5)*(b^2*d + 2*a*c*d + 2*a*b*e)*x^5 + (1/7)*(2*b*c*d + b^2*e + 2*a*c*e)*x^7 + (1/9)*c*(c*d + 2*b*e)*x^9 + (1/11)*c^2*e*x^11]
    @test_int [(a + b*x^2 + c*x^4)^2*(d + e*x^2)^0, 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/(d + e*x^2)^1, x, 3, -(((c*d - b*e)*(c*d^2 - e*(b*d - 2*a*e))*x)/e^4) + ((c^2*d^2 + b^2*e^2 - 2*c*e*(b*d - a*e))*x^3)/(3*e^3) - (c*(c*d - 2*b*e)*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 - b*d*e + a*e^2)^2*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*e^(9/2))]
    @test_int [(a + b*x^2 + c*x^4)^2/(d + e*x^2)^2, x, 4, ((3*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d - a*e))*x)/e^4 - (2*c*(c*d - b*e)*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 - b*d*e + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((c*d^2 - b*d*e + a*e^2)*(7*c*d^2 - e*(3*b*d + a*e))*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(9/2))]
    @test_int [(a + b*x^2 + c*x^4)^2/(d + e*x^2)^3, x, 5, -((c*(3*c*d - 2*b*e)*x)/e^4) + (c^2*x^3)/(3*e^3) + ((c*d^2 - b*d*e + a*e^2)^2*x)/(4*d*e^4*(d + e*x^2)^2) - ((13*c*d^2 - 5*b*d*e - 3*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c^2*d^4 - 6*c*d^2*e*(5*b*d - a*e) + e^2*(3*b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2))*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(5/2)*e^(9/2))]
    @test_int [(a + b*x^2 + c*x^4)^2/(d + e*x^2)^4, x, 5, (c^2*x)/e^4 + ((c*d^2 - b*d*e + a*e^2)^2*x)/(6*d*e^4*(d + e*x^2)^3) - ((19*c*d^2 - 7*b*d*e - 5*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(24*d^2*e^4*(d + e*x^2)^2) + ((29*c^2*d^4 - 2*c*d^2*e*(11*b*d - a*e) + e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*x)/(16*d^3*e^4*(d + e*x^2)) - ((35*c^2*d^4 - 2*c*d^2*e*(5*b*d + a*e) - e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*atan((sqrt(e)*x)/sqrt(d)))/(16*d^(7/2)*e^(9/2))]
    @test_int [(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5, x, 5, ((c*d^2 - b*d*e + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 9*b*d*e - 7*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 - 2*c*d^2*e*(59*b*d - 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(192*d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 2*c*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*atan((sqrt(e)*x)/sqrt(d)))/(128*d^(9/2)*e^(9/2))]

    #= Following*integrands*are*equal: =#
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^2, x, 3, (c*x)/e^2 + ((a + (d*(c*d - b*e))/e^2)*x)/(2*d*(d + e*x^2)) - ((3*c*d^2 - e*(b*d + a*e))*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(5/2))]
    @test_int [(a + x^2*(b + c*x^2))/(d + e*x^2)^2, x, 3, (c*x)/e^2 + ((a + (d*(c*d - b*e))/e^2)*x)/(2*d*(d + e*x^2)) - ((3*c*d^2 - e*(b*d + a*e))*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*e^(5/2))]

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

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

    @test_int [(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2, x, 4, (x*(c*(b^2*d^3 - 2*a*d*(c*d^2 - 3*a*e^2) - (a*b*e*(3*c*d^2 + a*e^2))/c) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((a*b^3*e^3 + 6*a*c*(2*c*d + sqrt(b^2 - 4*a*c)*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*sqrt(b^2 - 4*a*c)*e^3) - b*c*(a*e^2*(3*sqrt(b^2 - 4*a*c)*d + 8*a*e) + c*d^2*(sqrt(b^2 - 4*a*c)*d + 12*a*e)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((a*b^3*e^3 + 6*a*c*(2*c*d - sqrt(b^2 - 4*a*c)*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 - a*sqrt(b^2 - 4*a*c)*e^3) + b*c*(c*d^2*(sqrt(b^2 - 4*a*c)*d - 12*a*e) + a*e^2*(3*sqrt(b^2 - 4*a*c)*d - 8*a*e)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [(d + e*x^2)^2/(a + b*x^2 + c*x^4)^2, x, 4, (x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b*c*d^2 - 4*a*c*d*e + a*b*e^2 + (8*a*b*c*d*e + b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b*c*d^2 - 4*a*c*d*e + a*b*e^2 - (8*a*b*c*d*e + b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*sqrt(c)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [(d + e*x^2)^1/(a + b*x^2 + c*x^4)^2, x, 4, (x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (sqrt(c)*(b*d - 2*a*e + (b^2*d - 12*a*c*d + 4*a*b*e)/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)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(b*d - 2*a*e - (b^2*d - 12*a*c*d + 4*a*b*e)/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)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [(d + e*x^2)^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/((d + e*x^2)^1*(a + b*x^2 + c*x^4)^2), x, 10, (x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^2))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x^2 + c*x^4)) - (sqrt(c)*e^2*(e - (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(b - sqrt(b^2 - 4*a*c))*(c*d^2 - b*d*e + a*e^2)^2) + (sqrt(c)*(b*c*d - b^2*e + 2*a*c*e + (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/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)*sqrt(b - sqrt(b^2 - 4*a*c))*(c*d^2 - b*d*e + a*e^2)) - (sqrt(c)*e^2*(e + (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(b + sqrt(b^2 - 4*a*c))*(c*d^2 - b*d*e + a*e^2)^2) + (sqrt(c)*(b*c*d - b^2*e + 2*a*c*e - (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/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)*sqrt(b + sqrt(b^2 - 4*a*c))*(c*d^2 - b*d*e + a*e^2)) + (e^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*(c*d^2 - b*d*e + a*e^2)^2)]
    @test_int [1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2), x, 12, (e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) + (x*(a*b*c*e*(2*c*d - b*e) + (b^2 - 2*a*c)*(c^2*d^2 + b^2*e^2 - c*e*(2*b*d + a*e)) - c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^2))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 + c*x^4)) + (sqrt(2)*sqrt(c)*e^2*(3*c^2*d^2 + b*(b + sqrt(b^2 - 4*a*c))*e^2 - c*e*(3*b*d + 2*sqrt(b^2 - 4*a*c)*d + a*e))*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))*(c*d^2 - b*d*e + a*e^2)^3) + (sqrt(c)*(b^4*e^2 - b^3*e*(2*c*d - sqrt(b^2 - 4*a*c)*e) - 4*a*c^2*(3*c*d^2 - e*(sqrt(b^2 - 4*a*c)*d + 3*a*e)) + b^2*c*(c*d^2 - e*(2*sqrt(b^2 - 4*a*c)*d + 9*a*e)) - b*c*(3*a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d + 16*a*e)))*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))*(c*d^2 - b*d*e + a*e^2)^2) - (sqrt(2)*sqrt(c)*e^2*(3*c^2*d^2 + b*(b - sqrt(b^2 - 4*a*c))*e^2 - c*e*(3*b*d - 2*sqrt(b^2 - 4*a*c)*d + a*e))*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))*(c*d^2 - b*d*e + a*e^2)^3) - (sqrt(c)*(b^4*e^2 - b^3*e*(2*c*d + sqrt(b^2 - 4*a*c)*e) + b*c*(3*a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d - 16*a*e)) + b^2*c*(c*d^2 + e*(2*sqrt(b^2 - 4*a*c)*d - 9*a*e)) - 4*a*c^2*(3*c*d^2 + e*(sqrt(b^2 - 4*a*c)*d - 3*a*e)))*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))*(c*d^2 - b*d*e + a*e^2)^2) + (2*e^(7/2)*(2*c*d - b*e)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(d)*(c*d^2 - b*d*e + a*e^2)^3) + (e^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(2*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^2)]

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

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

    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^(5/2), x, 7, (d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*sqrt(d + e*x^2))/(256*e^2) + (d*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*x*(d + e*x^2)^(7/2))/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(256*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^(3/2), x, 6, (d*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*sqrt(d + e*x^2))/(128*e^2) + ((3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*(d + e*x^2)^(3/2))/(192*e^2) - ((3*c*d - 8*b*e)*x*(d + e*x^2)^(5/2))/(48*e^2) + (c*x^3*(d + e*x^2)^(5/2))/(8*e) + (d^2*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(128*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)*(d + e*x^2)^(1/2), x, 5, ((c*d^2 - 2*b*d*e + 8*a*e^2)*x*sqrt(d + e*x^2))/(16*e^2) - ((c*d - 2*b*e)*x*(d + e*x^2)^(3/2))/(8*e^2) + (c*x^3*(d + e*x^2)^(3/2))/(6*e) + (d*(c*d^2 - 2*b*d*e + 8*a*e^2)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(16*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(1/2), x, 4, -(((3*c*d - 4*b*e)*x*sqrt(d + e*x^2))/(8*e^2)) + (c*x^3*sqrt(d + e*x^2))/(4*e) + ((3*c*d^2 - 4*b*d*e + 8*a*e^2)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(8*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(3/2), x, 4, ((a + (d*(c*d - b*e))/e^2)*x)/(d*sqrt(d + e*x^2)) + (c*x*sqrt(d + e*x^2))/(2*e^2) - ((3*c*d - 2*b*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*e^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(5/2), x, 4, ((a + (d*(c*d - b*e))/e^2)*x)/(3*d*(d + e*x^2)^(3/2)) - ((4*c*d^2 - e*(b*d + 2*a*e))*x)/(3*d^2*e^2*sqrt(d + e*x^2)) + (c*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/e^(5/2)]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(7/2), x, 4, (a*x)/(d*(d + e*x^2)^(5/2)) + ((b*d + 4*a*e)*x^3)/(3*d^2*(d + e*x^2)^(5/2)) + ((3*c*d^2 + 2*e*(b*d + 4*a*e))*x^5)/(15*d^3*(d + e*x^2)^(5/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2), x, 5, (a*x)/(d*(d + e*x^2)^(7/2)) + ((b*d + 6*a*e)*x^3)/(3*d^2*(d + e*x^2)^(7/2)) + ((3*c*d^2 + 4*e*(b*d + 6*a*e))*x^5)/(15*d^3*(d + e*x^2)^(7/2)) + (2*e*(3*c*d^2 + 4*e*(b*d + 6*a*e))*x^7)/(105*d^4*(d + e*x^2)^(7/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2), x, 6, (a*x)/(d*(d + e*x^2)^(9/2)) + ((b*d + 8*a*e)*x^3)/(3*d^2*(d + e*x^2)^(9/2)) + ((c*d^2 + 2*e*(b*d + 8*a*e))*x^5)/(5*d^3*(d + e*x^2)^(9/2)) + (4*e*(c*d^2 + 2*e*(b*d + 8*a*e))*x^7)/(35*d^4*(d + e*x^2)^(9/2)) + (8*e^2*(c*d^2 + 2*e*(b*d + 8*a*e))*x^9)/(315*d^5*(d + e*x^2)^(9/2))]
    @test_int [(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2), x, 7, (a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) + ((3*c*d^2 + 8*e*(b*d + 10*a*e))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (8*e^2*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^11)/(3465*d^6*(d + e*x^2)^(11/2))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(7+5*x^2)^q*(2+3*x^2+x^4)^(p/2)=#

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

    @test_int [sqrt(2 + 3*x^2 + x^4)*(7 + 5*x^2)^3, x, 6, (577*x*(2 + x^2))/(3*sqrt(2 + 3*x^2 + x^4)) + (1/21)*x*(2608 + 757*x^2)*sqrt(2 + 3*x^2 + x^4) + (275/7)*x*(2 + 3*x^2 + x^4)^(3/2) + (125/9)*x^3*(2 + 3*x^2 + x^4)^(3/2) - (577*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4)) + (2945*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(21*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)*(7 + 5*x^2)^2, x, 5, (31*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + (1/21)*x*(407 + 114*x^2)*sqrt(2 + 3*x^2 + x^4) + (25/7)*x*(2 + 3*x^2 + x^4)^(3/2) - (31*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (472*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(21*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)*(7 + 5*x^2)^1, x, 4, (5*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + (1/3)*x*(10 + 3*x^2)*sqrt(2 + 3*x^2 + x^4) - (5*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (11*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)*(7 + 5*x^2)^0, x, 4, (x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + (1/3)*x*sqrt(2 + 3*x^2 + x^4) - (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (2*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)/(7 + 5*x^2)^1, x, 8, (x*(2 + x^2))/(5*sqrt(2 + 3*x^2 + x^4)) - (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(5*sqrt(2 + 3*x^2 + x^4)) + ((1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.F(atan(x), 1/2))/(5*sqrt(2 + 3*x^2 + x^4)) + (3*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.Pi(2/7, atan(x), 1/2))/(35*sqrt(2)*sqrt(2 + 3*x^2 + x^4)), (x*(2 + x^2))/(5*sqrt(2 + 3*x^2 + x^4)) - (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(5*sqrt(2 + 3*x^2 + x^4)) - (3*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(25*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (4*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(25*sqrt(2 + 3*x^2 + x^4)) + (3*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(35*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)/(7 + 5*x^2)^2, x, 8, -((x*(2 + x^2))/(70*sqrt(2 + 3*x^2 + x^4))) + (x*sqrt(2 + 3*x^2 + x^4))/(14*(7 + 5*x^2)) + ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(35*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (3*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(140*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - ((2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(980*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [sqrt(2 + 3*x^2 + x^4)/(7 + 5*x^2)^3, x, 25, -((11*x*(2 + x^2))/(11760*sqrt(2 + 3*x^2 + x^4))) + (x*sqrt(2 + 3*x^2 + x^4))/(28*(7 + 5*x^2)^2) + (11*x*sqrt(2 + 3*x^2 + x^4))/(2352*(7 + 5*x^2)) + (11*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(5880*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (81*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(7840*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (1201*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(164640*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]

    @test_int [(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^3, x, 7, (20884*x*(2 + x^2))/(65*sqrt(2 + 3*x^2 + x^4)) + (x*(1032541 + 297911*x^2)*sqrt(2 + 3*x^2 + x^4))/5005 + (x*(208212 + 65345*x^2)*(2 + 3*x^2 + x^4)^(3/2))/3003 + (3825/143)*x*(2 + 3*x^2 + x^4)^(5/2) + (125/13)*x^3*(2 + 3*x^2 + x^4)^(5/2) - (20884*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(65*sqrt(2 + 3*x^2 + x^4)) + (1171349*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(5005*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^2, x, 6, (742*x*(2 + x^2))/(15*sqrt(2 + 3*x^2 + x^4)) + (x*(36783 + 10643*x^2)*sqrt(2 + 3*x^2 + x^4))/1155 + (1/693)*x*(7281 + 2240*x^2)*(2 + 3*x^2 + x^4)^(3/2) + (25/11)*x*(2 + 3*x^2 + x^4)^(5/2) - (742*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(15*sqrt(2 + 3*x^2 + x^4)) + (13879*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(385*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^1, x, 5, (116*x*(2 + x^2))/(15*sqrt(2 + 3*x^2 + x^4)) + (1/105)*x*(519 + 149*x^2)*sqrt(2 + 3*x^2 + x^4) + (1/63)*x*(108 + 35*x^2)*(2 + 3*x^2 + x^4)^(3/2) - (116*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(15*sqrt(2 + 3*x^2 + x^4)) + (197*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(35*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^0, x, 5, (6*x*(2 + x^2))/(5*sqrt(2 + 3*x^2 + x^4)) + (1/35)*x*(29 + 9*x^2)*sqrt(2 + 3*x^2 + x^4) + (1/7)*x*(2 + 3*x^2 + x^4)^(3/2) - (6*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(5*sqrt(2 + 3*x^2 + x^4)) + (31*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(35*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^1, x, 13, (24*x*(2 + x^2))/(125*sqrt(2 + 3*x^2 + x^4)) + (1/75)*x*(11 + 3*x^2)*sqrt(2 + 3*x^2 + x^4) - (24*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(125*sqrt(2 + 3*x^2 + x^4)) + (56*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(375*sqrt(2 + 3*x^2 + x^4)) - (9*sqrt(2)*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(875*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^2, x, 21, (9*x*(2 + x^2))/(175*sqrt(2 + 3*x^2 + x^4)) + (1/75)*x*sqrt(2 + 3*x^2 + x^4) - (3*x*sqrt(2 + 3*x^2 + x^4))/(175*(7 + 5*x^2)) - (9*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(175*sqrt(2 + 3*x^2 + x^4)) + (59*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.F(atan(x), 1/2))/(1050*sqrt(2 + 3*x^2 + x^4)) + (9*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.Pi(2/7, atan(x), 1/2))/(2450*sqrt(2 + 3*x^2 + x^4)), (9*x*(2 + x^2))/(175*sqrt(2 + 3*x^2 + x^4)) + (1/75)*x*sqrt(2 + 3*x^2 + x^4) - (3*x*sqrt(2 + 3*x^2 + x^4))/(175*(7 + 5*x^2)) - (9*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(175*sqrt(2 + 3*x^2 + x^4)) + (81*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(8750*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (44*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(1875*sqrt(2 + 3*x^2 + x^4)) - (39*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(12250*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4)) + (3*sqrt(2)*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(875*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(2 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^3, x, 27, (3*x*(2 + x^2))/(392*sqrt(2 + 3*x^2 + x^4)) - (3*x*sqrt(2 + 3*x^2 + x^4))/(350*(7 + 5*x^2)^2) + (17*x*sqrt(2 + 3*x^2 + x^4))/(9800*(7 + 5*x^2)) - (3*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.E(atan(x), 1/2))/(196*sqrt(2 + 3*x^2 + x^4)) + (5*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.F(atan(x), 1/2))/(784*sqrt(2 + 3*x^2 + x^4)) + (141*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(27440*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4)), (3*x*(2 + x^2))/(392*sqrt(2 + 3*x^2 + x^4)) - (3*x*sqrt(2 + 3*x^2 + x^4))/(350*(7 + 5*x^2)^2) + (17*x*sqrt(2 + 3*x^2 + x^4))/(9800*(7 + 5*x^2)) - (39*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(24500*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (6*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(875*sqrt(2 + 3*x^2 + x^4)) + (5*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(784*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (141*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(27440*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]

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

    @test_int [(7 + 5*x^2)^3/sqrt(2 + 3*x^2 + x^4), x, 5, (135*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + 75*x*sqrt(2 + 3*x^2 + x^4) + 25*x^3*sqrt(2 + 3*x^2 + x^4) - (135*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (193*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^2/sqrt(2 + 3*x^2 + x^4), x, 4, (20*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + (25/3)*x*sqrt(2 + 3*x^2 + x^4) - (20*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (97*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^1/sqrt(2 + 3*x^2 + x^4), x, 3, (5*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) - (5*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (7*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^0/sqrt(2 + 3*x^2 + x^4), x, 1, ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^1*sqrt(2 + 3*x^2 + x^4)), x, 4, ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(2*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (5*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(14*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^2*sqrt(2 + 3*x^2 + x^4)), x, 9, (5*x*(2 + x^2))/(84*sqrt(2 + 3*x^2 + x^4)) - (25*x*sqrt(2 + 3*x^2 + x^4))/(84*(7 + 5*x^2)) - (5*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(42*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (9*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(56*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (65*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(1176*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^3*sqrt(2 + 3*x^2 + x^4)), x, 10, (65*x*(2 + x^2))/(4704*sqrt(2 + 3*x^2 + x^4)) - (25*x*sqrt(2 + 3*x^2 + x^4))/(168*(7 + 5*x^2)^2) - (325*x*sqrt(2 + 3*x^2 + x^4))/(4704*(7 + 5*x^2)) - (65*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(2352*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (631*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(9408*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (2525*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(65856*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]

    @test_int [(7 + 5*x^2)^5/(2 + 3*x^2 + x^4)^(3/2), x, 6, (7679*x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4)) - (x*(115 + 179*x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (5000/3)*x*sqrt(2 + 3*x^2 + x^4) + 625*x^3*sqrt(2 + 3*x^2 + x^4) - (7679*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (15383*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^4/(2 + 3*x^2 + x^4)^(3/2), x, 5, (637*x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (x*(145 + 113*x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (625/3)*x*sqrt(2 + 3*x^2 + x^4) - (637*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (1067*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^3/(2 + 3*x^2 + x^4)^(3/2), x, 4, (x*(5 - 11*x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (261*x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4)) - (261*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (169*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2), x, 4, -((17*x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4))) + (x*(25 + 17*x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (17*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (6*sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4)]
    @test_int [(7 + 5*x^2)^1/(2 + 3*x^2 + x^4)^(3/2), x, 4, -((x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4))) + (x*(5 + x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^0/(2 + 3*x^2 + x^4)^(3/2), x, 4, -((3*x*(2 + x^2))/(2*sqrt(2 + 3*x^2 + x^4))) + (x*(5 + 3*x^2))/(2*sqrt(2 + 3*x^2 + x^4)) + (3*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4)]
    @test_int [1/((7 + 5*x^2)^1*(2 + 3*x^2 + x^4)^(3/2)), x, 9, x/(6*sqrt(2 + 3*x^2 + x^4)) + (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4)) - (9*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.F(atan(x), 1/2))/(4*sqrt(2 + 3*x^2 + x^4)) + (125*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.Pi(2/7, atan(x), 1/2))/(84*sqrt(2)*sqrt(2 + 3*x^2 + x^4)), -((x*(2 + x^2))/(3*sqrt(2 + 3*x^2 + x^4))) + (x*(5 + 2*x^2))/(6*sqrt(2 + 3*x^2 + x^4)) + (sqrt(2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(3*sqrt(2 + 3*x^2 + x^4)) - (9*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(4*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (125*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(84*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^2*(2 + 3*x^2 + x^4)^(3/2)), x, 19, -((31*x*(2 + x^2))/(56*sqrt(2 + 3*x^2 + x^4))) + (x*(20 + 11*x^2))/(36*sqrt(2 + 3*x^2 + x^4)) + (625*x*sqrt(2 + 3*x^2 + x^4))/(504*(7 + 5*x^2)) + (31*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(28*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (463*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(336*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (375*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(784*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^3*(2 + 3*x^2 + x^4)^(3/2)), x, 29, -((5797*x*(2 + x^2))/(28224*sqrt(2 + 3*x^2 + x^4))) + (x*(50 + 23*x^2))/(216*sqrt(2 + 3*x^2 + x^4)) + (625*x*sqrt(2 + 3*x^2 + x^4))/(1008*(7 + 5*x^2)^2) + (41875*x*sqrt(2 + 3*x^2 + x^4))/(84672*(7 + 5*x^2)) + (5797*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(14112*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) - (49907*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(56448*sqrt(2)*sqrt(2 + 3*x^2 + x^4)) + (192625*(2 + x^2)*Elliptic.Pi(2/7, atan(x), 1/2))/(395136*sqrt(2)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(7+5*x^2)^q*(2+x^2-x^4)^(p/2)=#

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

    @test_int [sqrt(2 + x^2 - x^4)*(7 + 5*x^2)^4, x, 8, (1/231)*x*(177953 + 717372*x^2)*sqrt(2 + x^2 - x^4) - (116100/77)*x*(2 + x^2 - x^4)^(3/2) - (14500/33)*x^3*(2 + x^2 - x^4)^(3/2) - (625/11)*x^5*(2 + x^2 - x^4)^(3/2) + (3764813/231)*Elliptic.E(asin(x/sqrt(2)), -2) - (539419/77)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)*(7 + 5*x^2)^3, x, 7, (1/63)*x*(5956 + 14691*x^2)*sqrt(2 + x^2 - x^4) - (1825/21)*x*(2 + x^2 - x^4)^(3/2) - (125/9)*x^3*(2 + x^2 - x^4)^(3/2) + (79411/63)*Elliptic.E(asin(x/sqrt(2)), -2) - (8735/21)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)*(7 + 5*x^2)^2, x, 6, (1/21)*x*(275 + 354*x^2)*sqrt(2 + x^2 - x^4) - (25/7)*x*(2 + x^2 - x^4)^(3/2) + (2045/21)*Elliptic.E(asin(x/sqrt(2)), -2) - (79/7)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)*(7 + 5*x^2)^1, x, 5, x*(2 + x^2)*sqrt(2 + x^2 - x^4) + 7*Elliptic.E(asin(x/sqrt(2)), -2) + 3*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)*(7 + 5*x^2)^0, x, 5, (1/3)*x*sqrt(2 + x^2 - x^4) + (1/3)*Elliptic.E(asin(x/sqrt(2)), -2) + Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)/(7 + 5*x^2)^1, x, 7, (-(1/5))*Elliptic.E(asin(x/sqrt(2)), -2) + (17/25)*Elliptic.F(asin(x/sqrt(2)), -2) - (34/175)*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2)]
    @test_int [sqrt(2 + x^2 - x^4)/(7 + 5*x^2)^2, x, 7, (x*sqrt(2 + x^2 - x^4))/(14*(7 + 5*x^2)) + (1/70)*Elliptic.E(asin(x/sqrt(2)), -2) - (6/175)*Elliptic.F(asin(x/sqrt(2)), -2) + (99*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/2450]
    @test_int [sqrt(2 + x^2 - x^4)/(7 + 5*x^2)^3, x, 21, (x*sqrt(2 + x^2 - x^4))/(28*(7 + 5*x^2)^2) - (31*x*sqrt(2 + x^2 - x^4))/(13328*(7 + 5*x^2)) - (31*Elliptic.E(asin(x/sqrt(2)), -2))/66640 - (269*Elliptic.F(asin(x/sqrt(2)), -2))/166600 + (16601*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/2332400]

    @test_int [(2 + x^2 - x^4)^(3/2)*(7 + 5*x^2)^4, x, 9, (3*x*(2193559 + 7837383*x^2)*sqrt(2 + x^2 - x^4))/5005 - (x*(69817 - 1581440*x^2)*(2 + x^2 - x^4)^(3/2))/1001 - (132300/143)*x*(2 + x^2 - x^4)^(5/2) - (11750/39)*x^3*(2 + x^2 - x^4)^(5/2) - (125/3)*x^5*(2 + x^2 - x^4)^(5/2) + (124141422*Elliptic.E(asin(x/sqrt(2)), -2))/5005 - (50794416*Elliptic.F(asin(x/sqrt(2)), -2))/5005]
    @test_int [(2 + x^2 - x^4)^(3/2)*(7 + 5*x^2)^3, x, 8, (x*(2512273 + 5712051*x^2)*sqrt(2 + x^2 - x^4))/15015 + (x*(33792 + 374045*x^2)*(2 + x^2 - x^4)^(3/2))/3003 - (7825/143)*x*(2 + x^2 - x^4)^(5/2) - (125/13)*x^3*(2 + x^2 - x^4)^(5/2) + (31072528*Elliptic.E(asin(x/sqrt(2)), -2))/15015 - (3199778*Elliptic.F(asin(x/sqrt(2)), -2))/5005]
    @test_int [(2 + x^2 - x^4)^(3/2)*(7 + 5*x^2)^2, x, 7, (1/495)*x*(11497 + 14889*x^2)*sqrt(2 + x^2 - x^4) + (1/99)*x*(363 + 920*x^2)*(2 + x^2 - x^4)^(3/2) - (25/11)*x*(2 + x^2 - x^4)^(5/2) + (85942/495)*Elliptic.E(asin(x/sqrt(2)), -2) - (3392/165)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(2 + x^2 - x^4)^(3/2)*(7 + 5*x^2)^1, x, 6, (1/315)*x*(1087 + 669*x^2)*sqrt(2 + x^2 - x^4) + (1/63)*x*(48 + 35*x^2)*(2 + x^2 - x^4)^(3/2) + (4432/315)*Elliptic.E(asin(x/sqrt(2)), -2) + (418/105)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(2 + x^2 - x^4)^(3/2)*(7 + 5*x^2)^0, x, 6, (1/35)*x*(19 + 3*x^2)*sqrt(2 + x^2 - x^4) + (1/7)*x*(2 + x^2 - x^4)^(3/2) + (34/35)*Elliptic.E(asin(x/sqrt(2)), -2) + (48/35)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^1, x, 13, (1/75)*x*(13 - 3*x^2)*sqrt(2 + x^2 - x^4) + (92/375)*Elliptic.E(asin(x/sqrt(2)), -2) - (178/625)*Elliptic.F(asin(x/sqrt(2)), -2) + (1156*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/4375]
    @test_int [(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^2, x, 21, (-(1/75))*x*sqrt(2 + x^2 - x^4) - (17*x*sqrt(2 + x^2 - x^4))/(175*(7 + 5*x^2)) - (97/525)*Elliptic.E(asin(x/sqrt(2)), -2) + (458/875)*Elliptic.F(asin(x/sqrt(2)), -2) - (1241*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/6125]
    @test_int [(2 + x^2 - x^4)^(3/2)/(7 + 5*x^2)^3, x, 27, -((17*x*sqrt(2 + x^2 - x^4))/(350*(7 + 5*x^2)^2)) + (563*x*sqrt(2 + x^2 - x^4))/(9800*(7 + 5*x^2)) + (191*Elliptic.E(asin(x/sqrt(2)), -2))/9800 - (1251*Elliptic.F(asin(x/sqrt(2)), -2))/24500 + (9879*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/343000]

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

    @test_int [(7 + 5*x^2)^3/sqrt(2 + x^2 - x^4), x, 6, (-(625/3))*x*sqrt(2 + x^2 - x^4) - 25*x^3*sqrt(2 + x^2 - x^4) + (3905/3)*Elliptic.E(asin(x/sqrt(2)), -2) - 542*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^2/sqrt(2 + x^2 - x^4), x, 5, (-(25/3))*x*sqrt(2 + x^2 - x^4) + (260/3)*Elliptic.E(asin(x/sqrt(2)), -2) - 21*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^1/sqrt(2 + x^2 - x^4), x, 4, 5*Elliptic.E(asin(x/sqrt(2)), -2) + 2*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^0/sqrt(2 + x^2 - x^4), x, 2, Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [1/((7 + 5*x^2)^1*sqrt(2 + x^2 - x^4)), x, 2, (1/7)*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2)]
    @test_int [1/((7 + 5*x^2)^2*sqrt(2 + x^2 - x^4)), x, 8, -((25*x*sqrt(2 + x^2 - x^4))/(476*(7 + 5*x^2))) - (5/476)*Elliptic.E(asin(x/sqrt(2)), -2) - (1/238)*Elliptic.F(asin(x/sqrt(2)), -2) + (167*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/3332]
    @test_int [1/((7 + 5*x^2)^3*sqrt(2 + x^2 - x^4)), x, 9, -((25*x*sqrt(2 + x^2 - x^4))/(952*(7 + 5*x^2)^2)) - (12525*x*sqrt(2 + x^2 - x^4))/(453152*(7 + 5*x^2)) - (2505*Elliptic.E(asin(x/sqrt(2)), -2))/453152 - (263*Elliptic.F(asin(x/sqrt(2)), -2))/226576 + (58915*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/3172064]

    @test_int [(7 + 5*x^2)^5/(2 + x^2 - x^4)^(3/2), x, 7, (x*(1419985 + 1419793*x^2))/(18*sqrt(2 + x^2 - x^4)) + (27500/3)*x*sqrt(2 + x^2 - x^4) + 625*x^3*sqrt(2 + x^2 - x^4) - (3482293/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (627857/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^4/(2 + x^2 - x^4)^(3/2), x, 6, (x*(83585 + 83489*x^2))/(18*sqrt(2 + x^2 - x^4)) + (625/3)*x*sqrt(2 + x^2 - x^4) - (165239/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (31921/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^3/(2 + x^2 - x^4)^(3/2), x, 5, (x*(4945 + 4897*x^2))/(18*sqrt(2 + x^2 - x^4)) - (7147/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (1763/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^2/(2 + x^2 - x^4)^(3/2), x, 5, (x*(305 + 281*x^2))/(18*sqrt(2 + x^2 - x^4)) - (281/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (139/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^1/(2 + x^2 - x^4)^(3/2), x, 5, (x*(25 + 13*x^2))/(18*sqrt(2 + x^2 - x^4)) - (13/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (17/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [(7 + 5*x^2)^0/(2 + x^2 - x^4)^(3/2), x, 5, (x*(5 - x^2))/(18*sqrt(2 + x^2 - x^4)) + (1/18)*Elliptic.E(asin(x/sqrt(2)), -2) + (1/6)*Elliptic.F(asin(x/sqrt(2)), -2)]
    @test_int [1/((7 + 5*x^2)^1*(2 + x^2 - x^4)^(3/2)), x, 8, (x*(35 - 16*x^2))/(306*sqrt(2 + x^2 - x^4)) + (8/153)*Elliptic.E(asin(x/sqrt(2)), -2) + (1/102)*Elliptic.F(asin(x/sqrt(2)), -2) - (25/238)*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2)]
    @test_int [1/((7 + 5*x^2)^2*(2 + x^2 - x^4)^(3/2)), x, 17, (x*(580 - 287*x^2))/(10404*sqrt(2 + x^2 - x^4)) + (625*x*sqrt(2 + x^2 - x^4))/(16184*(7 + 5*x^2)) + (5143*Elliptic.E(asin(x/sqrt(2)), -2))/145656 + (89*Elliptic.F(asin(x/sqrt(2)), -2))/24276 - (10825*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/113288]
    @test_int [1/((7 + 5*x^2)^3*(2 + x^2 - x^4)^(3/2)), x, 26, (x*(9830 - 4909*x^2))/(353736*sqrt(2 + x^2 - x^4)) + (625*x*sqrt(2 + x^2 - x^4))/(32368*(7 + 5*x^2)^2) + (645625*x*sqrt(2 + x^2 - x^4))/(15407168*(7 + 5*x^2)) + (3086453*Elliptic.E(asin(x/sqrt(2)), -2))/138664512 + (60409*Elliptic.F(asin(x/sqrt(2)), -2))/23110752 - (6898575*Elliptic.Pi(-(10/7), asin(x/sqrt(2)), -2))/107850176]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(7+5*x^2)^q*(4+3*x^2+x^4)^(p/2)=#

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

    @test_int [sqrt(4 + 3*x^2 + x^4)*(7 + 5*x^2)^4, x, 7, (51665*x*sqrt(4 + 3*x^2 + x^4))/(33*(2 + x^2)) + (1/33)*x*(18727 + 4516*x^2)*sqrt(4 + 3*x^2 + x^4) + (3050/11)*x*(4 + 3*x^2 + x^4)^(3/2) + (23500/99)*x^3*(4 + 3*x^2 + x^4)^(3/2) + (625/11)*x^5*(4 + 3*x^2 + x^4)^(3/2) - (51665*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(33*sqrt(4 + 3*x^2 + x^4)) + (33159*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(11*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)*(7 + 5*x^2)^3, x, 6, (4717*x*sqrt(4 + 3*x^2 + x^4))/(21*(2 + x^2)) + (1/21)*x*(1708 + 407*x^2)*sqrt(4 + 3*x^2 + x^4) + (275/7)*x*(4 + 3*x^2 + x^4)^(3/2) + (125/9)*x^3*(4 + 3*x^2 + x^4)^(3/2) - (4717*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(21*sqrt(4 + 3*x^2 + x^4)) + (1301*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(3*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)*(7 + 5*x^2)^2, x, 5, (319*x*sqrt(4 + 3*x^2 + x^4))/(7*(2 + x^2)) + (1/7)*x*(119 + 38*x^2)*sqrt(4 + 3*x^2 + x^4) + (25/7)*x*(4 + 3*x^2 + x^4)^(3/2) - (319*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(7*sqrt(4 + 3*x^2 + x^4)) + (81*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)*(7 + 5*x^2)^1, x, 4, (9*x*sqrt(4 + 3*x^2 + x^4))/(2 + x^2) + (1/3)*x*(10 + 3*x^2)*sqrt(4 + 3*x^2 + x^4) - (9*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4) + (49*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(3*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)*(7 + 5*x^2)^0, x, 4, (1/3)*x*sqrt(4 + 3*x^2 + x^4) + (x*sqrt(4 + 3*x^2 + x^4))/(2 + x^2) - (sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4) + (7*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(3*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)/(7 + 5*x^2)^1, x, 7, (x*sqrt(4 + 3*x^2 + x^4))/(5*(2 + x^2)) + (1/5)*sqrt(11/35)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - (sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(5*sqrt(4 + 3*x^2 + x^4)) + (9*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(25*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (11*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(75*sqrt(4 + 3*x^2 + x^4)) + (187*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(525*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)/(7 + 5*x^2)^2, x, 7, -((x*sqrt(4 + 3*x^2 + x^4))/(70*(2 + x^2))) + (x*sqrt(4 + 3*x^2 + x^4))/(14*(7 + 5*x^2)) + (51*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/(280*sqrt(385)) + ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(35*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(35*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (289*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(9800*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [sqrt(4 + 3*x^2 + x^4)/(7 + 5*x^2)^3, x, 18, -((139*x*sqrt(4 + 3*x^2 + x^4))/(86240*(2 + x^2))) + (x*sqrt(4 + 3*x^2 + x^4))/(28*(7 + 5*x^2)^2) + (139*x*sqrt(4 + 3*x^2 + x^4))/(17248*(7 + 5*x^2)) + (14999*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/(344960*sqrt(385)) + (139*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(43120*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (23*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(2940*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (254983*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(36220800*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]

    @test_int [(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^4, x, 8, (12665086*x*sqrt(4 + 3*x^2 + x^4))/(2145*(2 + x^2)) + (7*x*(661429 + 174989*x^2)*sqrt(4 + 3*x^2 + x^4))/2145 + (x*(452001 + 131080*x^2)*(4 + 3*x^2 + x^4)^(3/2))/1287 + (92150/429)*x*(4 + 3*x^2 + x^4)^(5/2) + (2250/13)*x^3*(4 + 3*x^2 + x^4)^(5/2) + (125/3)*x^5*(4 + 3*x^2 + x^4)^(5/2) - (12665086*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(2145*sqrt(4 + 3*x^2 + x^4)) + (2383556*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(429*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^3, x, 7, (4525662*x*sqrt(4 + 3*x^2 + x^4))/(5005*(2 + x^2)) + (x*(1653701 + 435441*x^2)*sqrt(4 + 3*x^2 + x^4))/5005 + (x*(53504 + 15365*x^2)*(4 + 3*x^2 + x^4)^(3/2))/1001 + (3825/143)*x*(4 + 3*x^2 + x^4)^(5/2) + (125/13)*x^3*(4 + 3*x^2 + x^4)^(5/2) - (4525662*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(5005*sqrt(4 + 3*x^2 + x^4)) + (121826*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(143*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^2, x, 6, (175346*x*sqrt(4 + 3*x^2 + x^4))/(1155*(2 + x^2)) + (x*(64533 + 18253*x^2)*sqrt(4 + 3*x^2 + x^4))/1155 + (1/693)*x*(6831 + 2240*x^2)*(4 + 3*x^2 + x^4)^(3/2) + (25/11)*x*(4 + 3*x^2 + x^4)^(5/2) - (175346*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(1155*sqrt(4 + 3*x^2 + x^4)) + (4628*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(33*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^1, x, 5, (2798*x*sqrt(4 + 3*x^2 + x^4))/(105*(2 + x^2)) + (1/105)*x*(1029 + 289*x^2)*sqrt(4 + 3*x^2 + x^4) + (1/63)*x*(108 + 35*x^2)*(4 + 3*x^2 + x^4)^(3/2) - (2798*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(105*sqrt(4 + 3*x^2 + x^4)) + (74*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(3*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^0, x, 5, (138*x*sqrt(4 + 3*x^2 + x^4))/(35*(2 + x^2)) + (1/35)*x*(49 + 9*x^2)*sqrt(4 + 3*x^2 + x^4) + (1/7)*x*(4 + 3*x^2 + x^4)^(3/2) - (138*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(35*sqrt(4 + 3*x^2 + x^4)) + (4*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4)]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^1, x, 12, (94*x*sqrt(4 + 3*x^2 + x^4))/(125*(2 + x^2)) + (1/75)*x*(11 + 3*x^2)*sqrt(4 + 3*x^2 + x^4) + (44/125)*sqrt(11/35)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - (94*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(125*sqrt(4 + 3*x^2 + x^4)) + (54*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(125*sqrt(4 + 3*x^2 + x^4)) + (4114*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(13125*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^2, x, 19, (1/75)*x*sqrt(4 + 3*x^2 + x^4) + (4*x*sqrt(4 + 3*x^2 + x^4))/(175*(2 + x^2)) + (22*x*sqrt(4 + 3*x^2 + x^4))/(175*(7 + 5*x^2)) + (13/350)*sqrt(11/35)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - (4*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(175*sqrt(4 + 3*x^2 + x^4)) + (4*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(175*sqrt(4 + 3*x^2 + x^4)) + (2431*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(36750*sqrt(2)*sqrt(4 + 3*x^2 + x^4)), (1/75)*x*sqrt(4 + 3*x^2 + x^4) + (4*x*sqrt(4 + 3*x^2 + x^4))/(175*(2 + x^2)) + (22*x*sqrt(4 + 3*x^2 + x^4))/(175*(7 + 5*x^2)) + (13/350)*sqrt(11/35)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - (4*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(175*sqrt(4 + 3*x^2 + x^4)) + (4*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(175*sqrt(4 + 3*x^2 + x^4)) + (6919*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(183750*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (187*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(13125*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(4 + 3*x^2 + x^4)^(3/2)/(7 + 5*x^2)^3, x, 22, (9*x*sqrt(4 + 3*x^2 + x^4))/(1960*(2 + x^2)) + (11*x*sqrt(4 + 3*x^2 + x^4))/(175*(7 + 5*x^2)^2) + (167*x*sqrt(4 + 3*x^2 + x^4))/(9800*(7 + 5*x^2)) + (1347*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/(7840*sqrt(385)) + (111*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(24500*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (6*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(875*sqrt(4 + 3*x^2 + x^4)) - (817*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(91875*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (22*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(13125*sqrt(4 + 3*x^2 + x^4)) + (7633*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(274400*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]

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

    @test_int [(7 + 5*x^2)^3/sqrt(4 + 3*x^2 + x^4), x, 5, 75*x*sqrt(4 + 3*x^2 + x^4) + 25*x^3*sqrt(4 + 3*x^2 + x^4) - (15*x*sqrt(4 + 3*x^2 + x^4))/(2 + x^2) + (15*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4) + (13*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(2*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^2/sqrt(4 + 3*x^2 + x^4), x, 4, (25/3)*x*sqrt(4 + 3*x^2 + x^4) + (20*x*sqrt(4 + 3*x^2 + x^4))/(2 + x^2) - (20*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4) + (167*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(6*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^1/sqrt(4 + 3*x^2 + x^4), x, 3, (5*x*sqrt(4 + 3*x^2 + x^4))/(2 + x^2) - (5*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/sqrt(4 + 3*x^2 + x^4) + (17*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(2*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^0/sqrt(4 + 3*x^2 + x^4), x, 1, ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(2*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^1*sqrt(4 + 3*x^2 + x^4)), x, 3, (1/4)*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(6*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (17*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(84*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^2*sqrt(4 + 3*x^2 + x^4)), x, 6, -((5*x*sqrt(4 + 3*x^2 + x^4))/(616*(2 + x^2))) + (25*x*sqrt(4 + 3*x^2 + x^4))/(616*(7 + 5*x^2)) + (37*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/2464 + (5*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(308*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(42*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (629*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(51744*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^3*sqrt(4 + 3*x^2 + x^4)), x, 7, -((555*x*sqrt(4 + 3*x^2 + x^4))/(758912*(2 + x^2))) + (25*x*sqrt(4 + 3*x^2 + x^4))/(1232*(7 + 5*x^2)^2) + (2775*x*sqrt(4 + 3*x^2 + x^4))/(758912*(7 + 5*x^2)) - (3285*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/3035648 + (555*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(379456*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(8624*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (18615*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(21249536*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]

    @test_int [(7 + 5*x^2)^5/(4 + 3*x^2 + x^4)^(3/2), x, 6, (x*(99493 + 45779*x^2))/(28*sqrt(4 + 3*x^2 + x^4)) + (5000/3)*x*sqrt(4 + 3*x^2 + x^4) + 625*x^3*sqrt(4 + 3*x^2 + x^4) - (220779*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) + (220779*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (130729*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(12*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^4/(4 + 3*x^2 + x^4)^(3/2), x, 5, (x*(2719 - 4023*x^2))/(28*sqrt(4 + 3*x^2 + x^4)) + (625/3)*x*sqrt(4 + 3*x^2 + x^4) + (14523*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) - (14523*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (4243*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(12*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^3/(4 + 3*x^2 + x^4)^(3/2), x, 4, -((x*(2323 + 949*x^2))/(28*sqrt(4 + 3*x^2 + x^4))) + (4449*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) - (4449*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (973*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(4*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^2/(4 + 3*x^2 + x^4)^(3/2), x, 4, -((x*(9 - 113*x^2))/(28*sqrt(4 + 3*x^2 + x^4))) - (113*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) + (113*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (9*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(4*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^1/(4 + 3*x^2 + x^4)^(3/2), x, 4, (x*(53 + 19*x^2))/(28*sqrt(4 + 3*x^2 + x^4)) - (19*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) + (19*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (3*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(4*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [(7 + 5*x^2)^0/(4 + 3*x^2 + x^4)^(3/2), x, 4, -((x*(1 + 3*x^2))/(28*sqrt(4 + 3*x^2 + x^4))) + (3*x*sqrt(4 + 3*x^2 + x^4))/(28*(2 + x^2)) - (3*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(14*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(4*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^1*(4 + 3*x^2 + x^4)^(3/2)), x, 8, -((x*(13 + 4*x^2))/(308*sqrt(4 + 3*x^2 + x^4))) + (x*sqrt(4 + 3*x^2 + x^4))/(77*(2 + x^2)) + (25/176)*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)) - (sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(77*sqrt(4 + 3*x^2 + x^4)) - ((2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(12*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (425*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(3696*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^2*(4 + 3*x^2 + x^4)^(3/2)), x, 15, (x*(24 + 37*x^2))/(13552*sqrt(4 + 3*x^2 + x^4)) - (199*x*sqrt(4 + 3*x^2 + x^4))/(27104*(2 + x^2)) + (625*x*sqrt(4 + 3*x^2 + x^4))/(27104*(7 + 5*x^2)) + (575*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/108416 + (199*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(13552*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (2*sqrt(2)*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(231*sqrt(4 + 3*x^2 + x^4)) + (9775*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(2276736*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]
    @test_int [1/((7 + 5*x^2)^3*(4 + 3*x^2 + x^4)^(3/2)), x, 22, (x*(548 + 139*x^2))/(596288*sqrt(4 + 3*x^2 + x^4)) - (18159*x*sqrt(4 + 3*x^2 + x^4))/(33392128*(2 + x^2)) + (625*x*sqrt(4 + 3*x^2 + x^4))/(54208*(7 + 5*x^2)^2) + (51875*x*sqrt(4 + 3*x^2 + x^4))/(33392128*(7 + 5*x^2)) - (529425*sqrt(5/77)*atan((2*sqrt(11/35)*x)/sqrt(4 + 3*x^2 + x^4)))/133568512 + (18159*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.E(2*atan(x/sqrt(2)), 1/8))/(16696064*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) + (843*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.F(2*atan(x/sqrt(2)), 1/8))/(379456*sqrt(2)*sqrt(4 + 3*x^2 + x^4)) - (3000075*(2 + x^2)*sqrt((4 + 3*x^2 + x^4)/(2 + x^2)^2)*Elliptic.Pi(-(9/280), 2*atan(x/sqrt(2)), 1/8))/(934979584*sqrt(2)*sqrt(4 + 3*x^2 + x^4))]

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

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

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

    @test_int [(d + e*x^2)^3/sqrt(a + b*x^2 + c*x^4), x, 5, (e^2*(15*c*d - 4*b*e)*x*sqrt(a + b*x^2 + c*x^4))/(15*c^2) + (e^3*x^3*sqrt(a + b*x^2 + c*x^4))/(5*c) + (e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e))*x*sqrt(a + b*x^2 + c*x^4))/(15*c^(5/2)*(sqrt(a) + sqrt(c)*x^2)) - (1/(15*c^(11/4)*sqrt(a + b*x^2 + c*x^4)))*(a^(1/4)*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e))*(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))))) + (1/(30*c^(11/4)*sqrt(a + b*x^2 + c*x^4)))*(a^(1/4)*((sqrt(c)*(15*c^2*d^3 - 15*a*c*d*e^2 + 4*a*b*e^3))/sqrt(a) + e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)))*(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)))))]
    @test_int [(d + e*x^2)^2/sqrt(a + b*x^2 + c*x^4), x, 4, (e^2*x*sqrt(a + b*x^2 + c*x^4))/(3*c) + (2*e*(3*c*d - b*e)*x*sqrt(a + b*x^2 + c*x^4))/(3*c^(3/2)*(sqrt(a) + sqrt(c)*x^2)) - (2*a^(1/4)*e*(3*c*d - b*e)*(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)) + (1/(6*c^(7/4)*sqrt(a + b*x^2 + c*x^4)))*(a^(1/4)*(2*e*(3*c*d - b*e) + (sqrt(c)*(3*c*d^2 - a*e^2))/sqrt(a))*(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)))))]
    @test_int [(d + e*x^2)^1/sqrt(a + b*x^2 + c*x^4), x, 3, (e*x*sqrt(a + b*x^2 + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*e*(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(c)*d)/sqrt(a) + e)*(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 [1/((d + e*x^2)^1*sqrt(a + b*x^2 + c*x^4)), x, 3, (sqrt(e)*atan((sqrt(c*d^2 - b*d*e + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt(d)*sqrt(c*d^2 - b*d*e + a*e^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.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(1/4)*(sqrt(c)*d - sqrt(a)*e)*sqrt(a + b*x^2 + c*x^4)) - (a^(3/4)*((sqrt(c)*d)/sqrt(a) + e)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(4*c^(1/4)*d*(c*d^2 - a*e^2)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [1/((d + e*x^2)^2*sqrt(a + b*x^2 + c*x^4)), x, 6, -((sqrt(c)*e*x*sqrt(a + b*x^2 + c*x^4))/(2*d*(c*d^2 - b*d*e + a*e^2)*(sqrt(a) + sqrt(c)*x^2))) + (e^2*x*sqrt(a + b*x^2 + c*x^4))/(2*d*(c*d^2 - b*d*e + a*e^2)*(d + e*x^2)) + (sqrt(e)*(3*c*d^2 - e*(2*b*d - a*e))*atan((sqrt(c*d^2 - b*d*e + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + b*x^2 + c*x^4))))/(4*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^(3/2)) + (a^(1/4)*c^(1/4)*e*(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)))))/(2*d*(c*d^2 - b*d*e + a*e^2)*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^(1/4)*d*(sqrt(c)*d - sqrt(a)*e)*sqrt(a + b*x^2 + c*x^4)) - ((sqrt(c)*d + sqrt(a)*e)*(3*c*d^2 - e*(2*b*d - a*e))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(8*a^(1/4)*c^(1/4)*d^2*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))]
    #= [1/((d + e*x^2)^3*sqrt(a + b*x^2 + c*x^4)), x, 0, 0] =#

    @test_int [(d + e*x^2)^3/sqrt(a + b*x^2 - c*x^4), x, 6, -((e^2*(15*c*d + 4*b*e)*x*sqrt(a + b*x^2 - c*x^4))/(15*c^2)) - (e^3*x^3*sqrt(a + b*x^2 - c*x^4))/(5*c) - ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3*a*e))*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))))/(30*sqrt(2)*c^(7/2)*sqrt(a + b*x^2 - c*x^4)) + ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*((2*c*(15*c^2*d^3 + 15*a*c*d*e^2 + 4*a*b*e^3))/(b - sqrt(b^2 + 4*a*c)) + e*(45*c^2*d^2 + 8*b^2*e^2 + 3*c*e*(10*b*d + 3*a*e)))*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))))/(30*sqrt(2)*c^(7/2)*sqrt(a + b*x^2 - c*x^4))]
    @test_int [(d + e*x^2)^2/sqrt(a + b*x^2 - c*x^4), x, 5, -((e^2*x*sqrt(a + b*x^2 - c*x^4))/(3*c)) - ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*e*(3*c*d + b*e)*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))*(3*c^2*d^2 + b*(b - sqrt(b^2 + 4*a*c))*e^2 + c*e*(3*b*d - 3*sqrt(b^2 + 4*a*c)*d + a*e))*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 [(d + e*x^2)^1/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))*e*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))) + (sqrt(b + sqrt(b^2 + 4*a*c))*(2*c*d + (b - sqrt(b^2 + 4*a*c))*e)*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 [1/((d + e*x^2)^1*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.Pi(-(((b + sqrt(b^2 + 4*a*c))*e)/(2*c*d)), 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)*d*sqrt(a + b*x^2 - c*x^4))]
    @test_int [1/((d + e*x^2)^2*sqrt(a + b*x^2 - c*x^4)), x, 8, -((e^2*x*sqrt(a + b*x^2 - c*x^4))/(2*d*(c*d^2 + b*d*e - a*e^2)*(d + e*x^2))) + ((b - sqrt(b^2 + 4*a*c))*sqrt(b + sqrt(b^2 + 4*a*c))*e*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))))/(4*sqrt(2)*sqrt(c)*d*(c*d^2 + e*(b*d - a*e))*sqrt(a + b*x^2 - c*x^4)) - (sqrt(b + sqrt(b^2 + 4*a*c))*(2*c*d + (b - sqrt(b^2 + 4*a*c))*e)*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))))/(4*sqrt(2)*sqrt(c)*d*(c*d^2 + e*(b*d - a*e))*sqrt(a + b*x^2 - c*x^4)) + (sqrt(b + sqrt(b^2 + 4*a*c))*(3*c*d^2 + e*(2*b*d - a*e))*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.Pi(-(((b + sqrt(b^2 + 4*a*c))*e)/(2*c*d)), 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)*sqrt(c)*d^2*(c*d^2 + e*(b*d - a*e))*sqrt(a + b*x^2 - c*x^4))]
    #= [1/((d + e*x^2)^3*sqrt(a + b*x^2 - c*x^4)), x, 0, 0] =#

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

    @test_int [(d + e*x^2)/sqrt(-a + b*x^2 - c*x^4), x, 3, -((e*x*sqrt(-a + b*x^2 - c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2))) - (a^(1/4)*e*(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(c)*d)/sqrt(a) + e)*(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 [1/((d + e*x^2)*sqrt(-a + b*x^2 - c*x^4)), x, 3, If(13>=8, (sqrt(e)*atan((sqrt((-c)*d^2 - e*(b*d + a*e))*x)/(sqrt(d)*sqrt(e)*sqrt(-a + b*x^2 - c*x^4))))/(2*sqrt(d)*sqrt((-c)*d^2 - e*(b*d + a*e))) + (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^(1/4)*(sqrt(c)*d - sqrt(a)*e)*sqrt(-a + b*x^2 - c*x^4)) - (a^(3/4)*((sqrt(c)*d)/sqrt(a) + e)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a - b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 + b/(sqrt(a)*sqrt(c)))))/(4*c^(1/4)*d*(c*d^2 - a*e^2)*sqrt(-a + b*x^2 - c*x^4)), (sqrt(e)*atan((sqrt((-c)*d^2 - b*d*e - a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(-a + b*x^2 - c*x^4))))/(2*sqrt(d)*sqrt((-c)*d^2 - b*d*e - a*e^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.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 + b/(sqrt(a)*sqrt(c)))))/(2*a^(1/4)*(sqrt(c)*d - sqrt(a)*e)*sqrt(-a + b*x^2 - c*x^4)) - (a^(3/4)*((sqrt(c)*d)/sqrt(a) + e)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a - b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi(-((sqrt(c)*d - sqrt(a)*e)^2/(4*sqrt(a)*sqrt(c)*d*e)), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 + b/(sqrt(a)*sqrt(c)))))/(4*c^(1/4)*d*(c*d^2 - a*e^2)*sqrt(-a + b*x^2 - c*x^4)))]

    @test_int [(d + e*x^2)^3/sqrt(2 + 3*x^2 + x^4), x, 5, (3*e*(5*d^2 - 10*d*e + 6*e^2)*x*(2 + x^2))/(5*sqrt(2 + 3*x^2 + x^4)) + (1/5)*(5*d - 4*e)*e^2*x*sqrt(2 + 3*x^2 + x^4) + (1/5)*e^3*x^3*sqrt(2 + 3*x^2 + x^4) - (3*sqrt(2)*e*(5*d^2 - 10*d*e + 6*e^2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(5*sqrt(2 + 3*x^2 + x^4)) + ((5*d^3 - 10*d*e^2 + 8*e^3)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(5*sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(d + e*x^2)^2/sqrt(2 + 3*x^2 + x^4), x, 4, (2*(d - e)*e*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) + (1/3)*e^2*x*sqrt(2 + 3*x^2 + x^4) - (2*sqrt(2)*(d - e)*e*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + ((3*d^2 - 2*e^2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(3*sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [(d + e*x^2)^1/sqrt(2 + 3*x^2 + x^4), x, 3, (e*x*(2 + x^2))/sqrt(2 + 3*x^2 + x^4) - (sqrt(2)*e*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/sqrt(2 + 3*x^2 + x^4) + (d*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((d + e*x^2)^1*sqrt(2 + 3*x^2 + x^4)), x, 4, ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*(d - e)*sqrt(2 + 3*x^2 + x^4)) - (e*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.Pi(1 - e/d, atan(x), 1/2))/(sqrt(2)*d*(d - e)*sqrt(2 + 3*x^2 + x^4)), ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(sqrt(2)*(d - e)*sqrt(2 + 3*x^2 + x^4)) - (e*(2 + x^2)*Elliptic.Pi(1 - e/d, atan(x), 1/2))/(sqrt(2)*d*(d - e)*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]
    @test_int [1/((d + e*x^2)^2*sqrt(2 + 3*x^2 + x^4)), x, 9, -((e*x*(2 + x^2))/(2*d*(d^2 - 3*d*e + 2*e^2)*sqrt(2 + 3*x^2 + x^4))) + (e^2*x*sqrt(2 + 3*x^2 + x^4))/(2*d*(d^2 - 3*d*e + 2*e^2)*(d + e*x^2)) + (e*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*d*(d - 2*e)*(d - e)*sqrt(2 + 3*x^2 + x^4)) + ((2*d - e)*(1 + x^2)*sqrt((2 + x^2)/(2 + 2*x^2))*Elliptic.F(atan(x), 1/2))/(2*d*(d - e)^2*sqrt(2 + 3*x^2 + x^4)) - (e*(3*d^2 - 6*d*e + 2*e^2)*(2 + x^2)*Elliptic.Pi(1 - e/d, atan(x), 1/2))/(2*sqrt(2)*d^2*(d - 2*e)*(d - e)^2*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4)), -((e*x*(2 + x^2))/(2*d*(d^2 - 3*d*e + 2*e^2)*sqrt(2 + 3*x^2 + x^4))) + (e^2*x*sqrt(2 + 3*x^2 + x^4))/(2*d*(d^2 - 3*d*e + 2*e^2)*(d + e*x^2)) + (e*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.E(atan(x), 1/2))/(sqrt(2)*d*(d - 2*e)*(d - e)*sqrt(2 + 3*x^2 + x^4)) - ((1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(2*sqrt(2)*(d - 2*e)*(d - e)*sqrt(2 + 3*x^2 + x^4)) + ((3*d^2 - 6*d*e + 2*e^2)*(1 + x^2)*sqrt((2 + x^2)/(1 + x^2))*Elliptic.F(atan(x), 1/2))/(2*sqrt(2)*d*(d - 2*e)*(d - e)^2*sqrt(2 + 3*x^2 + x^4)) - (e*(3*d^2 - 6*d*e + 2*e^2)*(2 + x^2)*Elliptic.Pi(1 - e/d, atan(x), 1/2))/(2*sqrt(2)*d^2*(d - 2*e)*(d - e)^2*sqrt((2 + x^2)/(1 + x^2))*sqrt(2 + 3*x^2 + x^4))]

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

    @test_int [(c + e*x^2)^q*(a + c*x^2 + b*x^4)^p, x, 0, Unintegrable((c + e*x^2)^q*(a + c*x^2 + b*x^4)^p, x)]

    @test_int [(c + e*x^2)^3*(a + c*x^2 + b*x^4)^p, x, 8, If(13>=8, (c*e^2*(21*b - 5*e + 12*b*p - 2*e*p)*x*(a + c*x^2 + b*x^4)^(1 + p))/(b^2*(5 + 4*p)*(7 + 4*p)) + (e^3*x^3*(a + c*x^2 + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c*(a*e^3*(5 + 2*p) - 3*a*b*e^2*(7 + 4*p) + b^2*c^2*(35 + 48*p + 16*p^2))*x*(a + c*x^2 + b*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p*(b^2*(5 + 4*p)*(7 + 4*p))) + ((1/(3*b^2*(5 + 4*p)*(7 + 4*p)))*e*(c^2*e^2*(15 + 16*p + 4*p^2) + 3*b^2*c^2*(35 + 48*p + 16*p^2) - 3*b*e*(a*e*(5 + 4*p) + c^2*(21 + 26*p + 8*p^2)))*x^3*(a + c*x^2 + b*x^4)^p*AppellF1(3/2, -p, -p, 5/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p), -((c*e^2*(e*(5 + 2*p) - 3*b*(7 + 4*p))*x*(a + c*x^2 + b*x^4)^(1 + p))/(b^2*(35 + 48*p + 16*p^2))) + (e^3*x^3*(a + c*x^2 + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c*(a*e^3*(5 + 2*p) - 3*a*b*e^2*(7 + 4*p) + b^2*c^2*(35 + 48*p + 16*p^2))*x*(a + c*x^2 + b*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p*(b^2*(5 + 4*p)*(7 + 4*p))) + (1/(3*b^2*(5 + 4*p)*(7 + 4*p)))*((e*(c^2*e^2*(15 + 16*p + 4*p^2) + 3*b^2*c^2*(35 + 48*p + 16*p^2) - 3*b*e*(a*e*(5 + 4*p) + c^2*(21 + 26*p + 8*p^2)))*x^3*(a + c*x^2 + b*x^4)^p*AppellF1(3/2, -p, -p, 5/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p)))]
    @test_int [(c + e*x^2)^2*(a + c*x^2 + b*x^4)^p, x, 7, (e^2*x*(a + c*x^2 + b*x^4)^(1 + p))/(b*(5 + 4*p)) - ((a*e^2 - b*c^2*(5 + 4*p))*x*(a + c*x^2 + b*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p*(b*(5 + 4*p))) + (c*e*(10*b - 3*e + 8*b*p - 2*e*p)*x^3*(a + c*x^2 + b*x^4)^p*AppellF1(3/2, -p, -p, 5/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p*(3*b*(5 + 4*p)))]
    @test_int [(c + e*x^2)^1*(a + c*x^2 + b*x^4)^p, x, 6, (c*x*(a + c*x^2 + b*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p) + ((1/3)*e*x^3*(a + c*x^2 + b*x^4)^p*AppellF1(3/2, -p, -p, 5/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p)]
    @test_int [(c + e*x^2)^0*(a + c*x^2 + b*x^4)^p, x, 2, (x*(a + c*x^2 + b*x^4)^p*AppellF1(1/2, -p, -p, 3/2, -((2*b*x^2)/(c - sqrt(-4*a*b + c^2))), -((2*b*x^2)/(c + sqrt(-4*a*b + c^2)))))/((1 + (2*b*x^2)/(c - sqrt(-4*a*b + c^2)))^p*(1 + (2*b*x^2)/(c + sqrt(-4*a*b + c^2)))^p)]
    @test_int [(a + c*x^2 + b*x^4)^p/(c + e*x^2)^1, x, 0, Unintegrable((a + c*x^2 + b*x^4)^p/(c + e*x^2), x)]
    @test_int [(a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x, 0, Unintegrable((a + c*x^2 + b*x^4)^p/(c + e*x^2)^2, x)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(f+g*x)^r*(a+b*x^2+c*x^4)^p=#

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

    @test_int [(f + g*x)/((d + e*x)*sqrt(a + c*x^4)), x, 8, ((e*f - d*g)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*sqrt((-c)*d^4 - a*e^4)) - ((e*f - d*g)*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(2*sqrt(c*d^4 + a*e^4)) + ((sqrt(c)*d*f + sqrt(a)*e*g)*(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(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4)) - ((sqrt(c)*d^2 - sqrt(a)*e^2)*(e*f - d*g)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d*e*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4))]

    @test_int [(f + g*x)/((d + e*x)*sqrt(-a + c*x^4)), x, 10, ((e*f - d*g)*atanh((a*e^2 - c*d^2*x^2)/(sqrt(c*d^4 - a*e^4)*sqrt(-a + c*x^4))))/(2*sqrt(c*d^4 - a*e^4)) + (a^(1/4)*g*sqrt(1 - (c*x^4)/a)*Elliptic.F(asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(1/4)*e*sqrt(-a + c*x^4)) + (a^(1/4)*(e*f - d*g)*sqrt(1 - (c*x^4)/a)*Elliptic.Pi((sqrt(a)*e^2)/(sqrt(c)*d^2), asin((c^(1/4)*x)/a^(1/4)), -1))/(c^(1/4)*d*e*sqrt(-a + c*x^4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(f+g*x)^r*(a+b*x^2+c*x^4)^p*with*e^2*f^2+4*d*e*f*g+d^2*g^2=0, 4*c*d*e*f+2*c*d^2*g-b*e^2*g=0*and*4*a*e^5*f+c*d^5*g+15*a*d*e^4*g=0=#

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

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

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(f+g*x)^r*(a+b*x^2+c*x^4)^p=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(f+g*x)^r*(a+b*x^2+c*x^4)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(f+g*x)^r*(a+b*x^2+c*x^4)^(p/2)=#

    @test_int [(f + g*x)/((d + e*x)*sqrt(a + b*x^2 + c*x^4)), x, 8, If(13>=8, ((e*f - d*g)*atan((sqrt((-c)*d^4 - b*d^2*e^2 - a*e^4)*x)/(d*e*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt((-c)*d^4 - e^2*(b*d^2 + a*e^2))) - ((e*f - d*g)*atanh((b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*sqrt(c*d^4 + b*d^2*e^2 + a*e^4)*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt(c*d^4 + b*d^2*e^2 + a*e^4)) + ((sqrt(c)*d*f + sqrt(a)*e*g)*(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(c)*d^2 + sqrt(a)*e^2)*sqrt(a + b*x^2 + c*x^4)) - ((sqrt(c)*d^2 - sqrt(a)*e^2)*(e*f - d*g)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(4*a^(1/4)*c^(1/4)*d*e*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + b*x^2 + c*x^4)), ((e*f - d*g)*atan((sqrt((-c)*d^4 - b*d^2*e^2 - a*e^4)*x)/(d*e*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt((-c)*d^4 - b*d^2*e^2 - a*e^4)) - ((e*f - d*g)*atanh((b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*sqrt(c*d^4 + b*d^2*e^2 + a*e^4)*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt(c*d^4 + b*d^2*e^2 + a*e^4)) + ((sqrt(c)*d*f + sqrt(a)*e*g)*(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(c)*d^2 + sqrt(a)*e^2)*sqrt(a + b*x^2 + c*x^4)) - ((sqrt(c)*d^2 - sqrt(a)*e^2)*(e*f - d*g)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(4*a^(1/4)*c^(1/4)*d*e*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + b*x^2 + c*x^4)))]

    @test_int [(f + g*x)/((d + e*x)*sqrt(-a + b*x^2 + c*x^4)), x, 10, -(((e*f - d*g)*atanh((b*d^2 - 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*sqrt(c*d^4 + b*d^2*e^2 - a*e^4)*sqrt(-a + b*x^2 + c*x^4))))/(2*sqrt(c*d^4 + b*d^2*e^2 - a*e^4))) + (sqrt(b + sqrt(b^2 + 4*a*c))*g*(1 + (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))*Elliptic.F(atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 + 4*a*c))), -((2*sqrt(b^2 + 4*a*c))/(b - sqrt(b^2 + 4*a*c)))))/(sqrt(2)*sqrt(c)*e*sqrt((1 + (2*c*x^2)/(b - sqrt(b^2 + 4*a*c)))/(1 + (2*c*x^2)/(b + sqrt(b^2 + 4*a*c))))*sqrt(-a + b*x^2 + c*x^4)) + (sqrt(-b + sqrt(b^2 + 4*a*c))*(e*f - d*g)*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.Pi(-(((b - sqrt(b^2 + 4*a*c))*e^2)/(2*c*d^2)), 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)*d*e*sqrt(-a + b*x^2 + c*x^4))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^m (a+b x^2+c x^4)^p.jl