text:
@testset "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p" begin
    (A, B, a, b, c, d, e, f, m, p, q, x, ) = @variables A B a b c d e f m p q x

    #= ::Package:: =#

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

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

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

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

    @test_int [x^3*(d + e*x^2)*(a + c*x^4)^5, x, 3, (1/4)*a^5*d*x^4 + (1/6)*a^5*e*x^6 + (5/8)*a^4*c*d*x^8 + (1/2)*a^4*c*e*x^10 + (5/6)*a^3*c^2*d*x^12 + (5/7)*a^3*c^2*e*x^14 + (5/8)*a^2*c^3*d*x^16 + (5/9)*a^2*c^3*e*x^18 + (1/4)*a*c^4*d*x^20 + (5/22)*a*c^4*e*x^22 + (1/24)*c^5*d*x^24 + (1/26)*c^5*e*x^26]
    @test_int [x^2*(d + e*x^2)*(a + c*x^4)^5, x, 2, (1/3)*a^5*d*x^3 + (1/5)*a^5*e*x^5 + (5/7)*a^4*c*d*x^7 + (5/9)*a^4*c*e*x^9 + (10/11)*a^3*c^2*d*x^11 + (10/13)*a^3*c^2*e*x^13 + (2/3)*a^2*c^3*d*x^15 + (10/17)*a^2*c^3*e*x^17 + (5/19)*a*c^4*d*x^19 + (5/21)*a*c^4*e*x^21 + (1/23)*c^5*d*x^23 + (1/25)*c^5*e*x^25]
    @test_int [x^1*(d + e*x^2)*(a + c*x^4)^5, x, 4, (1/2)*a^5*d*x^2 + (5/6)*a^4*c*d*x^6 + a^3*c^2*d*x^10 + (5/7)*a^2*c^3*d*x^14 + (5/18)*a*c^4*d*x^18 + (1/22)*c^5*d*x^22 + (e*(a + c*x^4)^6)/(24*c)]
    @test_int [x^0*(d + e*x^2)*(a + c*x^4)^5, x, 2, a^5*d*x + (1/3)*a^5*e*x^3 + a^4*c*d*x^5 + (5/7)*a^4*c*e*x^7 + (10/9)*a^3*c^2*d*x^9 + (10/11)*a^3*c^2*e*x^11 + (10/13)*a^2*c^3*d*x^13 + (2/3)*a^2*c^3*e*x^15 + (5/17)*a*c^4*d*x^17 + (5/19)*a*c^4*e*x^19 + (1/21)*c^5*d*x^21 + (1/23)*c^5*e*x^23]
    @test_int [1/x^1*(d + e*x^2)*(a + c*x^4)^5, x, 3, (1/2)*a^5*e*x^2 + (5/4)*a^4*c*d*x^4 + (5/6)*a^4*c*e*x^6 + (5/4)*a^3*c^2*d*x^8 + a^3*c^2*e*x^10 + (5/6)*a^2*c^3*d*x^12 + (5/7)*a^2*c^3*e*x^14 + (5/16)*a*c^4*d*x^16 + (5/18)*a*c^4*e*x^18 + (1/20)*c^5*d*x^20 + (1/22)*c^5*e*x^22 + a^5*d*log(x)]
    @test_int [1/x^2*(d + e*x^2)*(a + c*x^4)^5, x, 2, -((a^5*d)/x) + a^5*e*x + (5/3)*a^4*c*d*x^3 + a^4*c*e*x^5 + (10/7)*a^3*c^2*d*x^7 + (10/9)*a^3*c^2*e*x^9 + (10/11)*a^2*c^3*d*x^11 + (10/13)*a^2*c^3*e*x^13 + (1/3)*a*c^4*d*x^15 + (5/17)*a*c^4*e*x^17 + (1/19)*c^5*d*x^19 + (1/21)*c^5*e*x^21]
    @test_int [1/x^3*(d + e*x^2)*(a + c*x^4)^5, x, 3, -((a^5*d)/(2*x^2)) + (5/2)*a^4*c*d*x^2 + (5/4)*a^4*c*e*x^4 + (5/3)*a^3*c^2*d*x^6 + (5/4)*a^3*c^2*e*x^8 + a^2*c^3*d*x^10 + (5/6)*a^2*c^3*e*x^12 + (5/14)*a*c^4*d*x^14 + (5/16)*a*c^4*e*x^16 + (1/18)*c^5*d*x^18 + (1/20)*c^5*e*x^20 + a^5*e*log(x)]

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

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

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

    @test_int [x^5*(2 + 3*x^2)*sqrt(5 + x^4), x, 5, (-5*x^2*sqrt(5 + x^4))/8 + (3*x^4*(5 + x^4)^(3/2))/10 - ((4 - x^2)*(5 + x^4)^(3/2))/4 - (25*asinh(x^2/sqrt(5)))/8]
    @test_int [x^3*(2 + 3*x^2)*sqrt(5 + x^4), x, 4, (-15*x^2*sqrt(5 + x^4))/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 - (75*asinh(x^2/sqrt(5)))/16]
    @test_int [x^1*(2 + 3*x^2)*sqrt(5 + x^4), x, 4, (x^2*sqrt(5 + x^4))/2 + (5 + x^4)^(3/2)/2 + (5*asinh(x^2/sqrt(5)))/2]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^1, x, 7, ((4 + 3*x^2)*sqrt(5 + x^4))/4 + (15*asinh(x^2/sqrt(5)))/4 - sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5))]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^3, x, 7, -((2 - 3*x^2)*sqrt(5 + x^4))/(2*x^2) + asinh(x^2/sqrt(5)) - (3*sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5)))/2]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^5, x, 7, -(((1 + 3*x^2)*sqrt(5 + x^4))/(2*x^4)) + (3/2)*asinh(x^2/sqrt(5)) - atanh(sqrt(5 + x^4)/sqrt(5))/(2*sqrt(5))]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^7, x, 6, -((3*sqrt(5 + x^4))/(4*x^4)) - (5 + x^4)^(3/2)/(15*x^6) - (3*atanh(sqrt(5 + x^4)/sqrt(5)))/(4*sqrt(5))]

    @test_int [x^4*(2 + 3*x^2)*sqrt(5 + x^4), x, 6, (20*x*sqrt(5 + x^4))/21 + (2*x^3*sqrt(5 + x^4))/3 - (10*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (x^5*(6 + 7*x^2)*sqrt(5 + x^4))/21 + (10*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) - (5*5^(1/4)*(21 + 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(21*sqrt(5 + x^4))]
    @test_int [x^2*(2 + 3*x^2)*sqrt(5 + x^4), x, 5, (10*x*sqrt(5 + x^4))/7 + (4*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (x^3*(14 + 15*x^2)*sqrt(5 + x^4))/35 - (4*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (5^(1/4)*(14 - 5*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(7*sqrt(5 + x^4))]
    @test_int [x^0*(2 + 3*x^2)*sqrt(5 + x^4), x, 4, (6*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (x*(10 + 9*x^2)*sqrt(5 + x^4))/15 - (6*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (5^(1/4)*(9 + 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(3*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^2, x, 4, -(((2 - x^2)*sqrt(5 + x^4))/x) + (4*x*sqrt(5 + x^4))/(sqrt(5) + x^2) - (4*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (5^(1/4)*(2 + sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4)]
    @test_int [(2 + 3*x^2)*sqrt(5 + x^4)/x^4, x, 5, (-6*sqrt(5 + x^4))/x - ((2 - 9*x^2)*sqrt(5 + x^4))/(3*x^3) + (6*x*sqrt(5 + x^4))/(sqrt(5) + x^2) - (6*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + ((2 + 9*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(3*5^(1/4)*sqrt(5 + x^4))]

    @test_int [x^5*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 6, (-(25/16))*x^2*sqrt(5 + x^4) - (5/24)*x^2*(5 + x^4)^(3/2) + (3/14)*x^4*(5 + x^4)^(5/2) - (1/42)*(18 - 7*x^2)*(5 + x^4)^(5/2) - (125/16)*asinh(x^2/sqrt(5))]
    @test_int [x^3*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 5, (-(75/32))*x^2*sqrt(5 + x^4) - (5/16)*x^2*(5 + x^4)^(3/2) + (1/20)*(4 + 5*x^2)*(5 + x^4)^(5/2) - (375/32)*asinh(x^2/sqrt(5))]
    @test_int [x^1*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 5, (15/8)*x^2*sqrt(5 + x^4) + (1/4)*x^2*(5 + x^4)^(3/2) + (3/10)*(5 + x^4)^(5/2) + (75/8)*asinh(x^2/sqrt(5))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^1, x, 8, (5/16)*(16 + 9*x^2)*sqrt(5 + x^4) + (1/24)*(8 + 9*x^2)*(5 + x^4)^(3/2) + (225/16)*asinh(x^2/sqrt(5)) - 5*sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^3, x, 8, (3/2)*(5 + x^2)*sqrt(5 + x^4) - ((2 - x^2)*(5 + x^4)^(3/2))/(2*x^2) + (15/2)*asinh(x^2/sqrt(5)) - (15/2)*sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^5, x, 8, -((3*(15 - 2*x^2)*sqrt(5 + x^4))/(4*x^2)) - ((2 - 3*x^2)*(5 + x^4)^(3/2))/(4*x^4) + (45/4)*asinh(x^2/sqrt(5)) - (3/2)*sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^7, x, 8, -(((4 - 9*x^2)*sqrt(5 + x^4))/(4*x^2)) - ((4 + 9*x^2)*(5 + x^4)^(3/2))/(12*x^6) + asinh(x^2/sqrt(5)) - (9/4)*sqrt(5)*atanh(sqrt(5 + x^4)/sqrt(5))]

    @test_int [x^4*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 7, (200/77)*x*sqrt(5 + x^4) + (20/13)*x^3*sqrt(5 + x^4) - (300*x*sqrt(5 + x^4))/(13*(sqrt(5) + x^2)) + (10*x^5*(78 + 77*x^2)*sqrt(5 + x^4))/1001 + (1/143)*x^5*(26 + 33*x^2)*(5 + x^4)^(3/2) + (300*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(13*sqrt(5 + x^4)) - (50*5^(1/4)*(231 + 26*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(1001*sqrt(5 + x^4))]
    @test_int [x^2*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 6, (300/77)*x*sqrt(5 + x^4) + (40*x*sqrt(5 + x^4))/(3*(sqrt(5) + x^2)) + (2/231)*x^3*(154 + 135*x^2)*sqrt(5 + x^4) + (1/99)*x^3*(22 + 27*x^2)*(5 + x^4)^(3/2) - (40*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(3*sqrt(5 + x^4)) + (10*5^(1/4)*(154 - 45*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(231*sqrt(5 + x^4))]
    @test_int [x^0*(2 + 3*x^2)*(5 + x^4)^(3/2), x, 5, (20*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (2/7)*x*(10 + 7*x^2)*sqrt(5 + x^4) + (1/21)*x*(6 + 7*x^2)*(5 + x^4)^(3/2) - (20*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (10*5^(1/4)*(7 + 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(7*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^2, x, 5, (24*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (6/35)*x*(25 + 14*x^2)*sqrt(5 + x^4) - ((14 - 3*x^2)*(5 + x^4)^(3/2))/(7*x) - (24*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (6*5^(1/4)*(14 + 5*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(7*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)*(5 + x^4)^(3/2)/x^4, x, 5, -((2*(27 - 2*x^2)*sqrt(5 + x^4))/(3*x)) + (36*x*sqrt(5 + x^4))/(sqrt(5) + x^2) - ((10 - 9*x^2)*(5 + x^4)^(3/2))/(15*x^3) - (36*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (2*5^(1/4)*(27 + 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(3*sqrt(5 + x^4))]

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

    @test_int [x^7*(2 + 3*x^2)/sqrt(5 + x^4), x, 5, (1/3)*x^4*sqrt(5 + x^4) + (3/8)*x^6*sqrt(5 + x^4) - (5/48)*(32 + 27*x^2)*sqrt(5 + x^4) + (225/16)*asinh(x^2/sqrt(5))]
    @test_int [x^5*(2 + 3*x^2)/sqrt(5 + x^4), x, 4, (x^4*sqrt(5 + x^4))/2 - ((10 - x^2)*sqrt(5 + x^4))/2 - (5*asinh(x^2/sqrt(5)))/2]
    @test_int [x^3*(2 + 3*x^2)/sqrt(5 + x^4), x, 3, ((4 + 3*x^2)*sqrt(5 + x^4))/4 - (15*asinh(x^2/sqrt(5)))/4]
    @test_int [(x^1*(2 + 3*x^2))/sqrt(5 + x^4), x, 3, (3*sqrt(5 + x^4))/2 + asinh(x^2/sqrt(5))]
    @test_int [(2 + 3*x^2)/(x^1*sqrt(5 + x^4)), x, 6, (3*asinh(x^2/sqrt(5)))/2 - atanh(sqrt(5 + x^4)/sqrt(5))/sqrt(5)]
    @test_int [(2 + 3*x^2)/(x^3*sqrt(5 + x^4)), x, 5, -sqrt(5 + x^4)/(5*x^2) - (3*atanh(sqrt(5 + x^4)/sqrt(5)))/(2*sqrt(5))]
    @test_int [(2 + 3*x^2)/(x^5*sqrt(5 + x^4)), x, 6, -(sqrt(5 + x^4)/(10*x^4)) - (3*sqrt(5 + x^4))/(10*x^2) + atanh(sqrt(5 + x^4)/sqrt(5))/(10*sqrt(5))]

    @test_int [x^4*(2 + 3*x^2)/sqrt(5 + x^4), x, 5, (2*x*sqrt(5 + x^4))/3 + (3*x^3*sqrt(5 + x^4))/5 - (9*x*sqrt(5 + x^4))/(sqrt(5) + x^2) + (9*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) - (5^(1/4)*(27 + 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(6*sqrt(5 + x^4))]
    @test_int [x^2*(2 + 3*x^2)/sqrt(5 + x^4), x, 4, x*sqrt(5 + x^4) + (2*x*sqrt(5 + x^4))/(sqrt(5) + x^2) - (2*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + (5^(1/4)*(2 - sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(2*sqrt(5 + x^4))]
    @test_int [x^0*(2 + 3*x^2)/sqrt(5 + x^4), x, 3, (3*x*sqrt(5 + x^4))/(sqrt(5) + x^2) - (3*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/sqrt(5 + x^4) + ((2 + 3*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(2*5^(1/4)*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)/(x^2*sqrt(5 + x^4)), x, 4, (-2*sqrt(5 + x^4))/(5*x) + (2*x*sqrt(5 + x^4))/(5*(sqrt(5) + x^2)) - (2*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(5^(3/4)*sqrt(5 + x^4)) + ((2 + 3*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(2*5^(3/4)*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)/(x^4*sqrt(5 + x^4)), x, 5, (-2*sqrt(5 + x^4))/(15*x^3) - (3*sqrt(5 + x^4))/(5*x) + (3*x*sqrt(5 + x^4))/(5*(sqrt(5) + x^2)) - (3*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(5^(3/4)*sqrt(5 + x^4)) - ((2 - 9*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(30*5^(1/4)*sqrt(5 + x^4))]

    @test_int [x^7*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 4, -((x^4*(2 + 3*x^2))/(2*sqrt(5 + x^4))) + (1/4)*(8 + 9*x^2)*sqrt(5 + x^4) - (45/4)*asinh(x^2/sqrt(5))]
    @test_int [x^5*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 4, -((x^2*(2 + 3*x^2))/(2*sqrt(5 + x^4))) + 3*sqrt(5 + x^4) + asinh(x^2/sqrt(5))]
    @test_int [x^3*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 3, -(2 + 3*x^2)/(2*sqrt(5 + x^4)) + (3*asinh(x^2/sqrt(5)))/2]
    @test_int [(x^1*(2 + 3*x^2))/(5 + x^4)^(3/2), x, 2, -(15 - 2*x^2)/(10*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)/(x^1*(5 + x^4)^(3/2)), x, 6, (2 + 3*x^2)/(10*sqrt(5 + x^4)) - atanh(sqrt(5 + x^4)/sqrt(5))/(5*sqrt(5))]
    @test_int [(2 + 3*x^2)/(x^3*(5 + x^4)^(3/2)), x, 6, (2 + 3*x^2)/(10*x^2*sqrt(5 + x^4)) - (2*sqrt(5 + x^4))/(25*x^2) - (3*atanh(sqrt(5 + x^4)/sqrt(5)))/(10*sqrt(5))]

    @test_int [x^4*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 5, -(x^3*(15 - 2*x^2))/(10*sqrt(5 + x^4)) - (x*sqrt(5 + x^4))/5 + (9*x*sqrt(5 + x^4))/(2*(sqrt(5) + x^2)) - (9*5^(1/4)*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(2*sqrt(5 + x^4)) + ((2 + 9*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(4*5^(1/4)*sqrt(5 + x^4))]
    @test_int [x^2*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 4, -(x*(15 - 2*x^2))/(10*sqrt(5 + x^4)) - (x*sqrt(5 + x^4))/(5*(sqrt(5) + x^2)) + ((sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(5^(3/4)*sqrt(5 + x^4)) - ((2 - 3*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(4*5^(3/4)*sqrt(5 + x^4))]
    @test_int [x^0*(2 + 3*x^2)/(5 + x^4)^(3/2), x, 4, (x*(2 + 3*x^2))/(10*sqrt(5 + x^4)) - (3*x*sqrt(5 + x^4))/(10*(sqrt(5) + x^2)) + (3*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(2*5^(3/4)*sqrt(5 + x^4)) + ((2 - 3*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(20*5^(1/4)*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)/(x^2*(5 + x^4)^(3/2)), x, 5, (2 + 3*x^2)/(10*x*sqrt(5 + x^4)) - (3*sqrt(5 + x^4))/(25*x) + (3*x*sqrt(5 + x^4))/(25*(sqrt(5) + x^2)) - (3*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(5*5^(3/4)*sqrt(5 + x^4)) + (3*(2 + sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(20*5^(3/4)*sqrt(5 + x^4))]
    @test_int [(2 + 3*x^2)/(x^4*(5 + x^4)^(3/2)), x, 6, (2 + 3*x^2)/(10*x^3*sqrt(5 + x^4)) - sqrt(5 + x^4)/(15*x^3) - (9*sqrt(5 + x^4))/(50*x) + (9*x*sqrt(5 + x^4))/(50*(sqrt(5) + x^2)) - (9*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.E(2*atan(x/5^(1/4)), 1/2))/(10*5^(3/4)*sqrt(5 + x^4)) + ((27 - 2*sqrt(5))*(sqrt(5) + x^2)*sqrt((5 + x^4)/(sqrt(5) + x^2)^2)*Elliptic.F(2*atan(x/5^(1/4)), 1/2))/(60*5^(3/4)*sqrt(5 + x^4))]

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

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

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

    @test_int [(f*x)^m*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 3, (d*(f*x)^(1 + m))/(f*(1 + m)) + ((10*d + e)*(f*x)^(3 + m))/(f^3*(3 + m)) + (5*(9*d + 2*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + (15*(8*d + 3*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + (30*(7*d + 4*e)*(f*x)^(9 + m))/(f^9*(9 + m)) + (42*(6*d + 5*e)*(f*x)^(11 + m))/(f^11*(11 + m)) + (42*(5*d + 6*e)*(f*x)^(13 + m))/(f^13*(13 + m)) + (30*(4*d + 7*e)*(f*x)^(15 + m))/(f^15*(15 + m)) + (15*(3*d + 8*e)*(f*x)^(17 + m))/(f^17*(17 + m)) + (5*(2*d + 9*e)*(f*x)^(19 + m))/(f^19*(19 + m)) + ((d + 10*e)*(f*x)^(21 + m))/(f^21*(21 + m)) + (e*(f*x)^(23 + m))/(f^23*(23 + m))]

    @test_int [x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 4, (1/22)*(d - e)*(1 + x^2)^11 - (1/24)*(2*d - 3*e)*(1 + x^2)^12 + (1/26)*(d - 3*e)*(1 + x^2)^13 + (1/28)*e*(1 + x^2)^14]
    @test_int [x^4*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 3, (d*x^5)/5 + (1/7)*(10*d + e)*x^7 + (5/9)*(9*d + 2*e)*x^9 + (15/11)*(8*d + 3*e)*x^11 + (30/13)*(7*d + 4*e)*x^13 + (14/5)*(6*d + 5*e)*x^15 + (42/17)*(5*d + 6*e)*x^17 + (30/19)*(4*d + 7*e)*x^19 + (5/7)*(3*d + 8*e)*x^21 + (5/23)*(2*d + 9*e)*x^23 + (1/25)*(d + 10*e)*x^25 + (e*x^27)/27]
    @test_int [x^3*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 4, (-(1/22))*(d - e)*(1 + x^2)^11 + (1/24)*(d - 2*e)*(1 + x^2)^12 + (1/26)*e*(1 + x^2)^13]
    @test_int [x^2*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 3, (d*x^3)/3 + (1/5)*(10*d + e)*x^5 + (5/7)*(9*d + 2*e)*x^7 + (5/3)*(8*d + 3*e)*x^9 + (30/11)*(7*d + 4*e)*x^11 + (42/13)*(6*d + 5*e)*x^13 + (14/5)*(5*d + 6*e)*x^15 + (30/17)*(4*d + 7*e)*x^17 + (15/19)*(3*d + 8*e)*x^19 + (5/21)*(2*d + 9*e)*x^21 + (1/23)*(d + 10*e)*x^23 + (e*x^25)/25]
    @test_int [x^1*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 4, (1/22)*(d - e)*(1 + x^2)^11 + (1/24)*e*(1 + x^2)^12]
    @test_int [x^0*(d + e*x^2)*(1 + 2*x^2 + x^4)^5, x, 3, d*x + (1/3)*(10*d + e)*x^3 + (9*d + 2*e)*x^5 + (15/7)*(8*d + 3*e)*x^7 + (10/3)*(7*d + 4*e)*x^9 + (42/11)*(6*d + 5*e)*x^11 + (42/13)*(5*d + 6*e)*x^13 + 2*(4*d + 7*e)*x^15 + (15/17)*(3*d + 8*e)*x^17 + (5/19)*(2*d + 9*e)*x^19 + (1/21)*(d + 10*e)*x^21 + (e*x^23)/23]
    @test_int [(d + e*x^2)*(1 + 2*x^2 + x^4)^5/x^1, x, 5, 5*d*x^2 + (45*d*x^4)/4 + 20*d*x^6 + (105*d*x^8)/4 + (126*d*x^10)/5 + (35*d*x^12)/2 + (60*d*x^14)/7 + (45*d*x^16)/16 + (5*d*x^18)/9 + (d*x^20)/20 + (1/22)*e*(1 + x^2)^11 + d*log(x)]
    @test_int [(d + e*x^2)*(1 + 2*x^2 + x^4)^5/x^2, x, 3, -(d/x) + (10*d + e)*x + (5/3)*(9*d + 2*e)*x^3 + 3*(8*d + 3*e)*x^5 + (30/7)*(7*d + 4*e)*x^7 + (14/3)*(6*d + 5*e)*x^9 + (42/11)*(5*d + 6*e)*x^11 + (30/13)*(4*d + 7*e)*x^13 + (3*d + 8*e)*x^15 + (5/17)*(2*d + 9*e)*x^17 + (1/19)*(d + 10*e)*x^19 + (e*x^21)/21]
    @test_int [(d + e*x^2)*(1 + 2*x^2 + x^4)^5/x^3, x, 4, -(d/(2*x^2)) + (5/2)*(9*d + 2*e)*x^2 + (15/4)*(8*d + 3*e)*x^4 + 5*(7*d + 4*e)*x^6 + (21/4)*(6*d + 5*e)*x^8 + (21/5)*(5*d + 6*e)*x^10 + (5/2)*(4*d + 7*e)*x^12 + (15/14)*(3*d + 8*e)*x^14 + (5/16)*(2*d + 9*e)*x^16 + (1/18)*(d + 10*e)*x^18 + (e*x^20)/20 + (10*d + e)*log(x)]

    @test_int [(f*x)^m*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 3, (f*x)^(1 + m)/(f*(1 + m)) + (11*(f*x)^(3 + m))/(f^3*(3 + m)) + (55*(f*x)^(5 + m))/(f^5*(5 + m)) + (165*(f*x)^(7 + m))/(f^7*(7 + m)) + (330*(f*x)^(9 + m))/(f^9*(9 + m)) + (462*(f*x)^(11 + m))/(f^11*(11 + m)) + (462*(f*x)^(13 + m))/(f^13*(13 + m)) + (330*(f*x)^(15 + m))/(f^15*(15 + m)) + (165*(f*x)^(17 + m))/(f^17*(17 + m)) + (55*(f*x)^(19 + m))/(f^19*(19 + m)) + (11*(f*x)^(21 + m))/(f^21*(21 + m)) + (f*x)^(23 + m)/(f^23*(23 + m))]

    @test_int [x^5*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 4, (1/24)*(1 + x^2)^12 - (1/13)*(1 + x^2)^13 + (1/28)*(1 + x^2)^14]
    @test_int [x^4*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 3, x^5/5 + (11*x^7)/7 + (55*x^9)/9 + 15*x^11 + (330*x^13)/13 + (154*x^15)/5 + (462*x^17)/17 + (330*x^19)/19 + (55*x^21)/7 + (55*x^23)/23 + (11*x^25)/25 + x^27/27]
    @test_int [x^3*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 4, (-(1/24))*(1 + x^2)^12 + (1/26)*(1 + x^2)^13]
    @test_int [x^2*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 3, x^3/3 + (11*x^5)/5 + (55*x^7)/7 + (55*x^9)/3 + 30*x^11 + (462*x^13)/13 + (154*x^15)/5 + (330*x^17)/17 + (165*x^19)/19 + (55*x^21)/21 + (11*x^23)/23 + x^25/25]
    @test_int [x^1*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 2, (1/24)*(1 + x^2)^12]
    @test_int [x^0*(1 + x^2)*(1 + 2*x^2 + x^4)^5, x, 3, x + (11*x^3)/3 + 11*x^5 + (165*x^7)/7 + (110*x^9)/3 + 42*x^11 + (462*x^13)/13 + 22*x^15 + (165*x^17)/17 + (55*x^19)/19 + (11*x^21)/21 + x^23/23]
    @test_int [(1 + x^2)*(1 + 2*x^2 + x^4)^5/x^1, x, 4, (11*x^2)/2 + (55*x^4)/4 + (55*x^6)/2 + (165*x^8)/4 + (231*x^10)/5 + (77*x^12)/2 + (165*x^14)/7 + (165*x^16)/16 + (55*x^18)/18 + (11*x^20)/20 + x^22/22 + log(x)]
    @test_int [(1 + x^2)*(1 + 2*x^2 + x^4)^5/x^2, x, 3, -(1/x) + 11*x + (55*x^3)/3 + 33*x^5 + (330*x^7)/7 + (154*x^9)/3 + 42*x^11 + (330*x^13)/13 + 11*x^15 + (55*x^17)/17 + (11*x^19)/19 + x^21/21]
    @test_int [(1 + x^2)*(1 + 2*x^2 + x^4)^5/x^3, x, 4, -(1/(2*x^2)) + (55*x^2)/2 + (165*x^4)/4 + 55*x^6 + (231*x^8)/4 + (231*x^10)/5 + (55*x^12)/2 + (165*x^14)/14 + (55*x^16)/16 + (11*x^18)/18 + x^20/20 + 11*log(x)]

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

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

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

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

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

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

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

    @test_int [(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2), x, 3, (a^5*d*(f*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f*(1 + m)*(a + b*x^2)) + (a^4*(5*b*d + a*e)*(f*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^3*(3 + m)*(a + b*x^2)) + (5*a^3*b*(2*b*d + a*e)*(f*x)^(5 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^5*(5 + m)*(a + b*x^2)) + (10*a^2*b^2*(b*d + a*e)*(f*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^7*(7 + m)*(a + b*x^2)) + (5*a*b^3*(b*d + 2*a*e)*(f*x)^(9 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^9*(9 + m)*(a + b*x^2)) + (b^4*(b*d + 5*a*e)*(f*x)^(11 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^11*(11 + m)*(a + b*x^2)) + (b^5*e*(f*x)^(13 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^13*(13 + m)*(a + b*x^2))]
    @test_int [(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, (a^3*d*(f*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f*(1 + m)*(a + b*x^2)) + (a^2*(3*b*d + a*e)*(f*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^3*(3 + m)*(a + b*x^2)) + (3*a*b*(b*d + a*e)*(f*x)^(5 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^5*(5 + m)*(a + b*x^2)) + (b^2*(b*d + 3*a*e)*(f*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^7*(7 + m)*(a + b*x^2)) + (b^3*e*(f*x)^(9 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^9*(9 + m)*(a + b*x^2))]
    @test_int [(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/2), x, 3, (a*d*(f*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f*(1 + m)*(a + b*x^2)) + ((b*d + a*e)*(f*x)^(3 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^3*(3 + m)*(a + b*x^2)) + (b*e*(f*x)^(5 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(f^5*(5 + m)*(a + b*x^2))]
    @test_int [(f*x)^m*(d + e*x^2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/2), x, 3, (e*(f*x)^(1 + m)*(a + b*x^2))/(b*f*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((b*d - a*e)*(f*x)^(1 + m)*(a + b*x^2)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*b*f*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]
    @test_int [(f*x)^m*(d + e*x^2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2), x, 3, ((b*d - a*e)*(f*x)^(1 + m))/(4*a*b*f*(a + b*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)) + ((b*d*(3 - m) + a*e*(1 + m))*(f*x)^(1 + m)*(a + b*x^2)*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(4*a^3*b*f*(1 + m)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))]

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

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

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

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

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

    @test_int [x^3*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 3, (1/4)*a^3*A*x^4 + (1/6)*a^2*(3*A*b + a*B)*x^6 + (3/8)*a*(a*b*B + A*(b^2 + a*c))*x^8 + (1/10)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^10 + (1/12)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^12 + (3/14)*c*(b^2*B + A*b*c + a*B*c)*x^14 + (1/16)*c^2*(3*b*B + A*c)*x^16 + (1/18)*B*c^3*x^18]
    @test_int [x^2*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 2, (1/3)*a^3*A*x^3 + (1/5)*a^2*(3*A*b + a*B)*x^5 + (3/7)*a*(a*b*B + A*(b^2 + a*c))*x^7 + (1/9)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^9 + (1/11)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^11 + (3/13)*c*(b^2*B + A*b*c + a*B*c)*x^13 + (1/15)*c^2*(3*b*B + A*c)*x^15 + (1/17)*B*c^3*x^17]
    @test_int [x^1*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 3, (1/2)*a^3*A*x^2 + (1/4)*a^2*(3*A*b + a*B)*x^4 + (1/2)*a*(a*b*B + A*(b^2 + a*c))*x^6 + (1/8)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^8 + (1/10)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^10 + (1/4)*c*(b^2*B + A*b*c + a*B*c)*x^12 + (1/14)*c^2*(3*b*B + A*c)*x^14 + (1/16)*B*c^3*x^16]
    @test_int [x^0*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 2, a^3*A*x + (1/3)*a^2*(3*A*b + a*B)*x^3 + (3/5)*a*(a*b*B + A*(b^2 + a*c))*x^5 + (1/7)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7 + (1/9)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^9 + (3/11)*c*(b^2*B + A*b*c + a*B*c)*x^11 + (1/13)*c^2*(3*b*B + A*c)*x^13 + (1/15)*B*c^3*x^15]
    @test_int [1/x^1*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 3, (1/2)*a^2*(3*A*b + a*B)*x^2 + (3/4)*a*(a*b*B + A*(b^2 + a*c))*x^4 + (1/6)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^6 + (1/8)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^8 + (3/10)*c*(b^2*B + A*b*c + a*B*c)*x^10 + (1/12)*c^2*(3*b*B + A*c)*x^12 + (1/14)*B*c^3*x^14 + a^3*A*log(x)]
    @test_int [1/x^2*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 2, -((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + (1/5)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^5 + (1/7)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7 + (1/3)*c*(b^2*B + A*b*c + a*B*c)*x^9 + (1/11)*c^2*(3*b*B + A*c)*x^11 + (1/13)*B*c^3*x^13]
    @test_int [1/x^3*(A + B*x^2)*(a + b*x^2 + c*x^4)^3, x, 3, -((a^3*A)/(2*x^2)) + (3/2)*a*(a*b*B + A*(b^2 + a*c))*x^2 + (1/4)*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^4 + (1/6)*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^6 + (3/8)*c*(b^2*B + A*b*c + a*B*c)*x^8 + (1/10)*c^2*(3*b*B + A*c)*x^10 + (1/12)*B*c^3*x^12 + a^2*(3*A*b + a*B)*log(x)]

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

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

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

    @test_int [x^7*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 7, ((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*log(a + b*x^2 + c*x^4))/(4*c^3)]
    @test_int [x^5*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 6, -((x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + ((b^3*B - 6*a*b*B*c + 4*a*A*c^2)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (B*log(a + b*x^2 + c*x^4))/(4*c^2)]
    @test_int [x^3*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 4, -((a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2)/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) - ((A*b - 2*a*B)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [x^1*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 4, -((A*b - 2*a*B - (b*B - 2*A*c)*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) - ((b*B - 2*A*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [x^(-1)*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 8, -((a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + ((4*a^2*B*c + A*(b^3 - 6*a*b*c))*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*(b^2 - 4*a*c)^(3/2)) + (A*log(x))/a^2 - (A*log(a + b*x^2 + c*x^4))/(4*a^2)]
    @test_int [x^(-3)*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 8, -((2*A*b^2 - a*b*B - 6*a*A*c)/(2*a^2*(b^2 - 4*a*c)*x^2)) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(b^2 - 6*a*c) - 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2))*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*A*b - a*B)*log(x))/a^3 + ((2*A*b - a*B)*log(a + b*x^2 + c*x^4))/(4*a^3)]

    @test_int [x^6*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 6, ((3*b^2*B - A*b*c - 10*a*B*c)*x)/(2*c^2*(b^2 - 4*a*c)) - ((b*B - 2*A*c)*x^3)/(2*c*(b^2 - 4*a*c)) - (x^5*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((3*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 - (3*b^4*B - A*b^3*c - 19*a*b^2*B*c + 8*a*A*b*c^2 + 20*a^2*B*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - ((3*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 + (3*b^4*B - A*b^3*c - 19*a*b^2*B*c + 8*a*A*b*c^2 + 20*a^2*B*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^4*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 5, -(((b*B - 2*A*c)*x)/(2*c*(b^2 - 4*a*c))) - (x^3*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b^2*B + A*b*c - 6*a*B*c - (b^3*B + A*b^2*c - 8*a*b*B*c + 4*a*A*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^2*B + A*b*c - 6*a*B*c + (b^3*B + A*b^2*c - 8*a*b*B*c + 4*a*A*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 4, -((x*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + ((b*B - 2*A*c - (b^2*B - 4*A*b*c + 4*a*B*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b*B - 2*A*c + (b^2*B - 4*A*b*c + 4*a*B*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^0*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 4, (x*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (sqrt(c)*(A*b - 2*a*B + (4*a*b*B + A*(b^2 - 12*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(A*b - 2*a*B - (A*b^2 + 4*a*b*B - 12*a*A*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^(-2)*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 5, -((3*A*b^2 - a*b*B - 10*a*A*c)/(2*a^2*(b^2 - 4*a*c)*x)) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) + (sqrt(c)*(a*B*(b^2 - 12*a*c + b*sqrt(b^2 - 4*a*c)) - A*(3*b^3 - 16*a*b*c + 3*b^2*sqrt(b^2 - 4*a*c) - 10*a*c*sqrt(b^2 - 4*a*c)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(3*A*b^2 - a*b*B - 10*a*A*c + (a*B*(b^2 - 12*a*c) - A*(3*b^3 - 16*a*b*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^(-4)*(A + B*x^2)/(a + b*x^2 + c*x^4)^2, x, 6, -((5*A*b^2 - 3*a*b*B - 14*a*A*c)/(6*a^2*(b^2 - 4*a*c)*x^3)) - (a*B*(3*b^2 - 10*a*c) - A*(5*b^3 - 19*a*b*c))/(2*a^3*(b^2 - 4*a*c)*x) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x^3*(a + b*x^2 + c*x^4)) - (sqrt(c)*(a*B*(3*b^3 - 16*a*b*c + 3*b^2*sqrt(b^2 - 4*a*c) - 10*a*c*sqrt(b^2 - 4*a*c)) - A*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*sqrt(b^2 - 4*a*c) - 19*a*b*c*sqrt(b^2 - 4*a*c)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(a*B*(3*b^3 - 16*a*b*c - 3*b^2*sqrt(b^2 - 4*a*c) + 10*a*c*sqrt(b^2 - 4*a*c)) - A*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - 5*b^3*sqrt(b^2 - 4*a*c) + 19*a*b*c*sqrt(b^2 - 4*a*c)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]

    @test_int [x^11*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 8, ((3*b^4*B - A*b^3*c - 21*a*b^2*B*c + 7*a*A*b*c^2 + 30*a^2*B*c^2)*x^2)/(2*c^3*(b^2 - 4*a*c)^2) - (x^8*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^4*(a*(3*b^3*B - A*b^2*c - 18*a*b*B*c + 16*a*A*c^2) + (3*b^4*B - A*b^3*c - 20*a*b^2*B*c + 10*a*A*b*c^2 + 20*a^2*B*c^2)*x^2))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((3*b^6*B - A*b^5*c - 30*a*b^4*B*c + 10*a*A*b^3*c^2 + 90*a^2*b^2*B*c^2 - 30*a^2*A*b*c^3 - 60*a^3*B*c^3)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^4*(b^2 - 4*a*c)^(5/2)) - ((3*b*B - A*c)*log(a + b*x^2 + c*x^4))/(4*c^4)]
    @test_int [x^9*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 7, -((x^6*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) - (x^2*(2*a*(b^3*B - 7*a*b*B*c + 6*a*A*c^2) + (2*b^4*B - 15*a*b^2*B*c + 6*a*A*b*c^2 + 16*a^2*B*c^2)*x^2))/(4*c^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (B*log(a + b*x^2 + c*x^4))/(4*c^3)]
    @test_int [x^7*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*(A*b - 2*a*B)*x^2*(2*a + b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*a*(A*b - 2*a*B)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^5*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((x^4*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) - (a*(b^2*B - 6*A*b*c + 8*a*B*c) + (b^3*B - 4*A*b^2*c + 2*a*b*B*c + 4*a*A*c^2)*x^2)/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*a*b*B - A*(b^2 + 2*a*c))*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^3*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2)/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + ((b^2*B - 3*A*b*c + 2*a*B*c)*(b + 2*c*x^2))/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((b^2*B - 3*A*b*c + 2*a*B*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^1*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((A*b - 2*a*B - (b*B - 2*A*c)*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) - (3*(b*B - 2*A*c)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*c*(b*B - 2*A*c)*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [x^(-1)*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 9, -((a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (6*a^2*b*B*c + A*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2) + 2*c*(6*a^2*B*c + A*(b^3 - 7*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((12*a^3*B*c^2 - A*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2))*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*(b^2 - 4*a*c)^(5/2)) + (A*log(x))/a^3 - (A*log(a + b*x^2 + c*x^4))/(4*a^3)]
    @test_int [x^(-3)*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 9, (a*b*B*(b^2 - 7*a*c) - 3*A*(b^4 - 7*a*b^2*c + 10*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) - (a*b*B*(b^2 - 10*a*c) - A*(3*b^4 - 20*a*b^2*c + 20*a^2*c^2) + c*(a*B*(b^2 - 16*a*c) - 3*A*(b^3 - 6*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(b^4 - 10*a*b^2*c + 30*a^2*c^2) - 3*A*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3))*atanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^4*(b^2 - 4*a*c)^(5/2)) - ((3*A*b - a*B)*log(x))/a^4 + ((3*A*b - a*B)*log(a + b*x^2 + c*x^4))/(4*a^4)]

    @test_int [x^8*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 7, -(((3*b^3*B + A*b^2*c - 24*a*b*B*c + 20*a*A*c^2)*x)/(8*c^2*(b^2 - 4*a*c)^2)) + ((b^2*B + 12*A*b*c - 28*a*B*c)*x^3)/(8*c*(b^2 - 4*a*c)^2) - (x^7*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^5*(7*A*b^2 - 12*a*b*B - 4*a*A*c + (b^2*B + 12*A*b*c - 28*a*B*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*B + A*b^3*c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 - (3*b^5*B + A*b^4*c - 33*a*b^3*B*c - 18*a*A*b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2*A*c^3)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + ((3*b^4*B + A*b^3*c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 + (3*b^5*B + A*b^4*c - 33*a*b^3*B*c - 18*a*A*b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2*A*c^3)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(5/2)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^6*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 6, -(((b^2*B - 12*A*b*c + 20*a*B*c)*x)/(8*c*(b^2 - 4*a*c)^2)) - (x^5*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^3*(5*A*b^2 - 12*a*b*B + 4*a*A*c - (b^2*B - 12*A*b*c + 20*a*B*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 - (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 + (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*c^(3/2)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^4*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((x^3*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*x*(4*a*b*B - A*(b^2 + 4*a*c) + (b^2*B - 4*A*b*c + 4*a*B*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*(b^2*B - 4*A*b*c + 4*a*B*c - (b^3*B - 6*A*b^2*c + 12*a*b*B*c - 8*a*A*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (3*(b^2*B - 4*A*b*c + 4*a*B*c + (b^3*B - 6*A*b^2*c + 12*a*b*B*c - 8*a*A*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, -((x*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) - (x*(a*B*(7*b^2 - 4*a*c) - A*(b^3 + 8*a*b*c) + c*(12*a*b*B - A*(b^2 + 20*a*c))*x^2))/(8*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (sqrt(c)*(6*a*B*(3*b^2 + 4*a*c - 2*b*sqrt(b^2 - 4*a*c)) + A*(b^3 - 52*a*b*c + b^2*sqrt(b^2 - 4*a*c) + 20*a*c*sqrt(b^2 - 4*a*c)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(6*a*B*(3*b^2 + 4*a*c + 2*b*sqrt(b^2 - 4*a*c)) + A*(b^3 - 52*a*b*c - b^2*sqrt(b^2 - 4*a*c) - 20*a*c*sqrt(b^2 - 4*a*c)))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [x^0*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 5, (x*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(a*b*B*(b^2 + 8*a*c) + A*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2) + c*(a*B*(b^2 + 20*a*c) + 3*A*(b^3 - 8*a*b*c))*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (sqrt(c)*(a*B*(b^2 + 20*a*c) + 3*A*(b^3 - 8*a*b*c) + (a*b*B*(b^2 - 52*a*c) + 3*A*(b^4 - 10*a*b^2*c + 56*a^2*c^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(a*B*(b^2 + 20*a*c) + 3*A*(b^3 - 8*a*b*c) - (a*b*B*(b^2 - 52*a*c) + 3*A*(b^4 - 10*a*b^2*c + 56*a^2*c^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c)))]
    #= [x^(-2)*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 6, -((3*(5*A*b^4 - a*b^3*B - 37*a*A*b^2*c + 4*a^2*c*(2*b*B + 15*A*c)))/(8*a^3*(b^2 - 4*a*c)^2*x)) - ((-A)*b^2 + a*b*B + 2*a*A*c + ((-A)*b*c + 2*a*B*c)*x^2)/(4*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)^2) + (5*A*b^4 - a*b^3*B - 35*a*A*b^2*c + 4*a^2*c*(4*b*B + 9*A*c) + c*(5*A*b^3 - a*b^2*B - 32*a*A*b*c + 28*a^2*B*c)*x^2)/(8*a^2*(b^2 - 4*a*c)^2*x*(a + b*x^2 + c*x^4)) - (3*sqrt(c)*(5*A*b^5 - 56*a^3*B*c^2 - 37*a*A*b^2*c*sqrt(b^2 - 4*a*c) - b^4*(a*B - 5*A*sqrt(b^2 - 4*a*c)) - a*b^3*(47*A*c + B*sqrt(b^2 - 4*a*c)) + 2*a^2*c*(5*b^2*B + 4*b*B*sqrt(b^2 - 4*a*c) + 2*A*c*(31*b + 15*sqrt(b^2 - 4*a*c))))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (3*sqrt(c)*(5*A*b^5 - 56*a^3*B*c^2 + 37*a*A*b^2*c*sqrt(b^2 - 4*a*c) - b^4*(a*B + 5*A*sqrt(b^2 - 4*a*c)) - a*b^3*(47*A*c - B*sqrt(b^2 - 4*a*c)) + 2*a^2*c*(5*b^2*B - 4*b*B*sqrt(b^2 - 4*a*c) + 2*A*c*(31*b - 15*sqrt(b^2 - 4*a*c))))*atan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))] =#

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

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

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

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

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

    @test_int [x^5*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 6, (-(1633/256))*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4) + (3/10)*x^4*(3 + 5*x^2 + x^4)^(3/2) + (1/480)*(1837 - 510*x^2)*(3 + 5*x^2 + x^4)^(3/2) + (21229/512)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [x^3*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 5, (259*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4))/128 - ((59 - 18*x^2)*(3 + 5*x^2 + x^4)^(3/2))/48 - (3367*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/256]
    @test_int [x^1*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 5, (-11*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4))/16 + (3 + 5*x^2 + x^4)^(3/2)/2 + (143*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/32]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^1, x, 7, ((23 + 6*x^2)*sqrt(3 + 5*x^2 + x^4))/8 + atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))/16 - sqrt(3)*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^3, x, 7, -((2 - 3*x^2)*sqrt(3 + 5*x^2 + x^4))/(2*x^2) + (19*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/4 - (7*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/sqrt(3)]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^5, x, 7, -(((6 + 23*x^2)*sqrt(3 + 5*x^2 + x^4))/(12*x^4)) + (3/2)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))) - (77*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(24*sqrt(3))]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^7, x, 5, -(((6 + 5*x^2)*sqrt(3 + 5*x^2 + x^4))/(18*x^4)) - (3 + 5*x^2 + x^4)^(3/2)/(9*x^6) + (13*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(36*sqrt(3))]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^9, x, 6, (67*(6 + 5*x^2)*sqrt(3 + 5*x^2 + x^4))/(1728*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(12*x^8) - (11*(3 + 5*x^2 + x^4)^(3/2))/(216*x^6) - (871*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(3456*sqrt(3))]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^11, x, 7, -((161*(6 + 5*x^2)*sqrt(3 + 5*x^2 + x^4))/(5184*x^4)) - (3 + 5*x^2 + x^4)^(3/2)/(15*x^10) - (3 + 5*x^2 + x^4)^(3/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240*x^6) + (2093*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(10368*sqrt(3))]

    @test_int [x^4*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 6, (-1924*x*(5 + sqrt(13) + 2*x^2))/(105*sqrt(3 + 5*x^2 + x^4)) + (13*x*sqrt(3 + 5*x^2 + x^4))/3 - (26*x^3*sqrt(3 + 5*x^2 + x^4))/35 + (x^5*(11 + 7*x^2)*sqrt(3 + 5*x^2 + x^4))/21 + (962*sqrt((2*(5 + sqrt(13)))/3)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(105*sqrt(3 + 5*x^2 + x^4)) - (13*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]
    @test_int [x^2*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 5, (1247*x*(5 + sqrt(13) + 2*x^2))/(210*sqrt(3 + 5*x^2 + x^4)) - (4*x*sqrt(3 + 5*x^2 + x^4))/3 + (x^3*(29 + 15*x^2)*sqrt(3 + 5*x^2 + x^4))/35 - (1247*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(210*sqrt(3 + 5*x^2 + x^4)) + (2*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [x^0*(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4), x, 4, (-23*x*(5 + sqrt(13) + 2*x^2))/(15*sqrt(3 + 5*x^2 + x^4)) + (x*(25 + 9*x^2)*sqrt(3 + 5*x^2 + x^4))/15 + (23*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(15*sqrt(3 + 5*x^2 + x^4)) + (sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^2, x, 4, (9*x*(5 + sqrt(13) + 2*x^2))/(2*sqrt(3 + 5*x^2 + x^4)) - ((2 - x^2)*sqrt(3 + 5*x^2 + x^4))/x - (3*sqrt((3*(5 + sqrt(13)))/2)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(2*sqrt(3 + 5*x^2 + x^4)) + (8*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)*sqrt(3 + 5*x^2 + x^4)/x^4, x, 5, (32*x*(5 + sqrt(13) + 2*x^2))/(9*sqrt(3 + 5*x^2 + x^4)) - (64*sqrt(3 + 5*x^2 + x^4))/(9*x) - ((2 - 9*x^2)*sqrt(3 + 5*x^2 + x^4))/(3*x^3) - (16*sqrt((2*(5 + sqrt(13)))/3)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(9*sqrt(3 + 5*x^2 + x^4)) + (49*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(3*sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]

    @test_int [x^5*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 7, (28379*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4))/2048 - (2183/768)*(5 + 2*x^2)*(3 + 5*x^2 + x^4)^(3/2) + (3/14)*x^4*(3 + 5*x^2 + x^4)^(5/2) + ((3313 - 1070*x^2)*(3 + 5*x^2 + x^4)^(5/2))/1680 - (368927*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/4096]
    @test_int [x^3*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 6, -((4797*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4))/1024) + (123/128)*(5 + 2*x^2)*(3 + 5*x^2 + x^4)^(3/2) - (1/40)*(27 - 10*x^2)*(3 + 5*x^2 + x^4)^(5/2) + (62361*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/2048]
    @test_int [x^1*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 6, (429/256)*(5 + 2*x^2)*sqrt(3 + 5*x^2 + x^4) - (11/32)*(5 + 2*x^2)*(3 + 5*x^2 + x^4)^(3/2) + (3/10)*(3 + 5*x^2 + x^4)^(5/2) - (5577/512)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^1, x, 8, (1/128)*(199 - 74*x^2)*sqrt(3 + 5*x^2 + x^4) + (1/48)*(61 + 18*x^2)*(3 + 5*x^2 + x^4)^(3/2) + (2401/256)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))) - 3*sqrt(3)*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^3, x, 8, (3/16)*(109 + 18*x^2)*sqrt(3 + 5*x^2 + x^4) - ((2 - x^2)*(3 + 5*x^2 + x^4)^(3/2))/(2*x^2) + (609/32)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))) - 12*sqrt(3)*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^5, x, 8, -((3*(28 - 19*x^2)*sqrt(3 + 5*x^2 + x^4))/(8*x^2)) - ((2 - 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/(4*x^4) + (453/16)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))) - (127/8)*sqrt(3)*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^7, x, 8, -(((67 - 32*x^2)*sqrt(3 + 5*x^2 + x^4))/(12*x^2)) - ((2 + 7*x^2)*(3 + 5*x^2 + x^4)^(3/2))/(6*x^6) + (49/4)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))) - (527*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(24*sqrt(3))]

    @test_int [x^4*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 7, (176723*x*(5 + sqrt(13) + 2*x^2))/(4290*sqrt(3 + 5*x^2 + x^4)) - (4210/429)*x*sqrt(3 + 5*x^2 + x^4) + (1251/715)*x^3*sqrt(3 + 5*x^2 + x^4) - (1/429)*x^5*(283 + 272*x^2)*sqrt(3 + 5*x^2 + x^4) + (1/143)*x^5*(71 + 33*x^2)*(3 + 5*x^2 + x^4)^(3/2) - (176723*sqrt((1/6)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(4290*sqrt(3 + 5*x^2 + x^4)) + (2105*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(143*sqrt(3 + 5*x^2 + x^4))]
    @test_int [x^2*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 6, -((49949*x*(5 + sqrt(13) + 2*x^2))/(3465*sqrt(3 + 5*x^2 + x^4))) + (353/99)*x*sqrt(3 + 5*x^2 + x^4) - (x^3*(911 + 890*x^2)*sqrt(3 + 5*x^2 + x^4))/1155 + (1/99)*x^3*(67 + 27*x^2)*(3 + 5*x^2 + x^4)^(3/2) + (49949*sqrt((1/6)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(3465*sqrt(3 + 5*x^2 + x^4)) - (353*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(33*sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]
    @test_int [x^0*(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2), x, 5, (203*x*(5 + sqrt(13) + 2*x^2))/(30*sqrt(3 + 5*x^2 + x^4)) - (1/15)*x*(5 + 12*x^2)*sqrt(3 + 5*x^2 + x^4) + (1/3)*x*(3 + x^2)*(3 + 5*x^2 + x^4)^(3/2) - (203*sqrt((1/6)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(30*sqrt(3 + 5*x^2 + x^4)) + (5*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^2, x, 5, (412*x*(5 + sqrt(13) + 2*x^2))/(35*sqrt(3 + 5*x^2 + x^4)) + (1/35)*x*(655 + 129*x^2)*sqrt(3 + 5*x^2 + x^4) - ((14 - 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/(7*x) - (206*sqrt((2/3)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(35*sqrt(3 + 5*x^2 + x^4)) + (19*sqrt(3/(2*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^4, x, 5, (949*x*(5 + sqrt(13) + 2*x^2))/(30*sqrt(3 + 5*x^2 + x^4)) - (13*(24 - 5*x^2)*sqrt(3 + 5*x^2 + x^4))/(15*x) - ((10 - 9*x^2)*(3 + 5*x^2 + x^4)^(3/2))/(15*x^3) - (949*sqrt((1/6)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(30*sqrt(3 + 5*x^2 + x^4)) + (65*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2)/x^6, x, 6, (361*x*(5 + sqrt(13) + 2*x^2))/(15*sqrt(3 + 5*x^2 + x^4)) - (722*sqrt(3 + 5*x^2 + x^4))/(15*x) - ((40 - 87*x^2)*sqrt(3 + 5*x^2 + x^4))/(5*x^3) - ((2 - 5*x^2)*(3 + 5*x^2 + x^4)^(3/2))/(5*x^5) - (361*sqrt((1/6)*(5 + sqrt(13)))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(15*sqrt(3 + 5*x^2 + x^4)) + (103*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((1/6)*(5 + sqrt(13)))*x), (1/6)*(-13 + 5*sqrt(13))))/(sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]

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

    @test_int [x^5*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 5, (B*x^4*sqrt(a + b*x^2 + c*x^4))/(6*c) + ((15*b^2*B - 18*A*b*c - 16*a*B*c - 2*c*(5*b*B - 6*A*c)*x^2)*sqrt(a + b*x^2 + c*x^4))/(48*c^3) - ((5*b^3*B - 6*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(7/2))]
    @test_int [x^3*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 4, -(((3*b*B - 4*A*c - 2*B*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*c^2)) + ((3*b^2*B - 4*A*b*c - 4*a*B*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*c^(5/2))]
    @test_int [x^1*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 4, (B*sqrt(a + b*x^2 + c*x^4))/(2*c) - ((b*B - 2*A*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*c^(3/2))]
    @test_int [(A + B*x^2)/(x^1*sqrt(a + b*x^2 + c*x^4)), x, 6, -((A*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt(a))) + (B*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(2*sqrt(c))]
    @test_int [(A + B*x^2)/(x^3*sqrt(a + b*x^2 + c*x^4)), x, 4, -((A*sqrt(a + b*x^2 + c*x^4))/(2*a*x^2)) + ((A*b - 2*a*B)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*a^(3/2))]
    @test_int [(A + B*x^2)/(x^5*sqrt(a + b*x^2 + c*x^4)), x, 5, -((A*sqrt(a + b*x^2 + c*x^4))/(4*a*x^4)) + ((3*A*b - 4*a*B)*sqrt(a + b*x^2 + c*x^4))/(8*a^2*x^2) - ((3*A*b^2 - 4*a*b*B - 4*a*A*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(16*a^(5/2))]
    @test_int [(A + B*x^2)/(x^7*sqrt(a + b*x^2 + c*x^4)), x, 6, -((A*sqrt(a + b*x^2 + c*x^4))/(6*a*x^6)) + ((5*A*b - 6*a*B)*sqrt(a + b*x^2 + c*x^4))/(24*a^2*x^4) - ((15*A*b^2 - 18*a*b*B - 16*a*A*c)*sqrt(a + b*x^2 + c*x^4))/(48*a^3*x^2) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(32*a^(7/2))]

    @test_int [x^4*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 5, -(((4*b*B - 5*A*c)*x*sqrt(a + b*x^2 + c*x^4))/(15*c^2)) + (B*x^3*sqrt(a + b*x^2 + c*x^4))/(5*c) + ((8*b^2*B - 10*A*b*c - 9*a*B*c)*x*sqrt(a + b*x^2 + c*x^4))/(15*c^(5/2)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*(8*b^2*B - 10*A*b*c - 9*a*B*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(15*c^(11/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(8*b^2*B - 10*A*b*c - 9*a*B*c + sqrt(a)*sqrt(c)*(4*b*B - 5*A*c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(30*c^(11/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^2*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 4, (B*x*sqrt(a + b*x^2 + c*x^4))/(3*c) - ((2*b*B - 3*A*c)*x*sqrt(a + b*x^2 + c*x^4))/(3*c^(3/2)*(sqrt(a) + sqrt(c)*x^2)) + (a^(1/4)*(2*b*B - 3*A*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*c^(7/4)*sqrt(a + b*x^2 + c*x^4)) - (a^(1/4)*(2*b*B + sqrt(a)*B*sqrt(c) - 3*A*c)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*c^(7/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [x^0*(A + B*x^2)/sqrt(a + b*x^2 + c*x^4), x, 3, (B*x*sqrt(a + b*x^2 + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a^(1/4)*B*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(c^(3/4)*sqrt(a + b*x^2 + c*x^4)) + (a^(1/4)*(B + (A*sqrt(c))/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)))))/(2*c^(3/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^2*sqrt(a + b*x^2 + c*x^4)), x, 4, -((A*sqrt(a + b*x^2 + c*x^4))/(a*x)) + (A*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(a*(sqrt(a) + sqrt(c)*x^2)) - (A*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(a^(3/4)*sqrt(a + b*x^2 + c*x^4)) + ((sqrt(a)*B + A*sqrt(c))*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(2*a^(3/4)*c^(1/4)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^4*sqrt(a + b*x^2 + c*x^4)), x, 5, -((A*sqrt(a + b*x^2 + c*x^4))/(3*a*x^3)) + ((2*A*b - 3*a*B)*sqrt(a + b*x^2 + c*x^4))/(3*a^2*x) - ((2*A*b - 3*a*B)*sqrt(c)*x*sqrt(a + b*x^2 + c*x^4))/(3*a^2*(sqrt(a) + sqrt(c)*x^2)) + ((2*A*b - 3*a*B)*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(3*a^(7/4)*sqrt(a + b*x^2 + c*x^4)) - ((2*A*b - 3*a*B + sqrt(a)*A*sqrt(c))*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + b*x^2 + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), (1/4)*(2 - b/(sqrt(a)*sqrt(c)))))/(6*a^(7/4)*sqrt(a + b*x^2 + c*x^4))]

    @test_int [x^7*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 6, (-(89/48))*x^4*sqrt(3 + 5*x^2 + x^4) + (3/8)*x^6*sqrt(3 + 5*x^2 + x^4) - (1/384)*(24243 - 3802*x^2)*sqrt(3 + 5*x^2 + x^4) + (32801/256)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [x^5*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 5, (1/2)*x^4*sqrt(3 + 5*x^2 + x^4) + (3/16)*(89 - 14*x^2)*sqrt(3 + 5*x^2 + x^4) - (1083/32)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [x^3*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 4, -((37 - 6*x^2)*sqrt(3 + 5*x^2 + x^4))/8 + (149*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/16]
    @test_int [x^1*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 4, (3*sqrt(3 + 5*x^2 + x^4))/2 - (11*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/4]
    @test_int [(2 + 3*x^2)/(x^1*sqrt(3 + 5*x^2 + x^4)), x, 6, (3*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/2 - atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))/sqrt(3)]
    @test_int [(2 + 3*x^2)/(x^3*sqrt(3 + 5*x^2 + x^4)), x, 4, -sqrt(3 + 5*x^2 + x^4)/(3*x^2) - (2*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(3*sqrt(3))]
    @test_int [(2 + 3*x^2)/(x^5*sqrt(3 + 5*x^2 + x^4)), x, 5, -(sqrt(3 + 5*x^2 + x^4)/(6*x^4)) - sqrt(3 + 5*x^2 + x^4)/(12*x^2) + (1/8)*sqrt(3)*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [(2 + 3*x^2)/(x^7*sqrt(3 + 5*x^2 + x^4)), x, 6, -(sqrt(3 + 5*x^2 + x^4)/(9*x^6)) - sqrt(3 + 5*x^2 + x^4)/(54*x^4) + (13*sqrt(3 + 5*x^2 + x^4))/(108*x^2) - (61*atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4))))/(216*sqrt(3))]

    @test_int [x^4*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 5, (419*x*(5 + sqrt(13) + 2*x^2))/(30*sqrt(3 + 5*x^2 + x^4)) - (10*x*sqrt(3 + 5*x^2 + x^4))/3 + (3*x^3*sqrt(3 + 5*x^2 + x^4))/5 - (419*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(30*sqrt(3 + 5*x^2 + x^4)) + (5*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [x^2*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 4, (-4*x*(5 + sqrt(13) + 2*x^2))/sqrt(3 + 5*x^2 + x^4) + x*sqrt(3 + 5*x^2 + x^4) + (2*sqrt((2*(5 + sqrt(13)))/3)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4) - (sqrt(3/(2*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [x^0*(2 + 3*x^2)/sqrt(3 + 5*x^2 + x^4), x, 3, (3*x*(5 + sqrt(13) + 2*x^2))/(2*sqrt(3 + 5*x^2 + x^4)) - (sqrt((3*(5 + sqrt(13)))/2)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(2*sqrt(3 + 5*x^2 + x^4)) + (sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)/(x^2*sqrt(3 + 5*x^2 + x^4)), x, 4, (x*(5 + sqrt(13) + 2*x^2))/(3*sqrt(3 + 5*x^2 + x^4)) - (2*sqrt(3 + 5*x^2 + x^4))/(3*x) - (sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(3*sqrt(3 + 5*x^2 + x^4)) + (sqrt(3/(2*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/sqrt(3 + 5*x^2 + x^4)]
    @test_int [(2 + 3*x^2)/(x^4*sqrt(3 + 5*x^2 + x^4)), x, 5, (7*x*(5 + sqrt(13) + 2*x^2))/(54*sqrt(3 + 5*x^2 + x^4)) - (2*sqrt(3 + 5*x^2 + x^4))/(9*x^3) - (7*sqrt(3 + 5*x^2 + x^4))/(27*x) - (7*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(54*sqrt(3 + 5*x^2 + x^4)) - (sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(9*sqrt(3 + 5*x^2 + x^4))]

    @test_int [x^5*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 5, -((x^2*(33 + 47*x^2))/(13*sqrt(3 + 5*x^2 + x^4))) + (133/26)*sqrt(3 + 5*x^2 + x^4) - (41/4)*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4)))]
    @test_int [x^3*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 4, -(33 + 47*x^2)/(13*sqrt(3 + 5*x^2 + x^4)) + (3*atanh((5 + 2*x^2)/(2*sqrt(3 + 5*x^2 + x^4))))/2]
    @test_int [x^1*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 2, (8 + 11*x^2)/(13*sqrt(3 + 5*x^2 + x^4))]
    @test_int [(2 + 3*x^2)/(x^1*(3 + 5*x^2 + x^4)^(3/2)), x, 5, -(7 + 8*x^2)/(39*sqrt(3 + 5*x^2 + x^4)) - atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))/(3*sqrt(3))]
    @test_int [(2 + 3*x^2)/(x^3*(3 + 5*x^2 + x^4)^(3/2)), x, 5, -(7 + 8*x^2)/(39*x^2*sqrt(3 + 5*x^2 + x^4)) - (2*sqrt(3 + 5*x^2 + x^4))/(39*x^2) + atanh((6 + 5*x^2)/(2*sqrt(3)*sqrt(3 + 5*x^2 + x^4)))/(3*sqrt(3))]

    @test_int [x^4*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 5, (43*x*(5 + sqrt(13) + 2*x^2))/(13*sqrt(3 + 5*x^2 + x^4)) + (x^3*(8 + 11*x^2))/(13*sqrt(3 + 5*x^2 + x^4)) - (11*x*sqrt(3 + 5*x^2 + x^4))/13 - (43*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(13*sqrt(3 + 5*x^2 + x^4)) + (11*sqrt(3/(2*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(13*sqrt(3 + 5*x^2 + x^4))]
    @test_int [x^2*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 4, (-11*x*(5 + sqrt(13) + 2*x^2))/(26*sqrt(3 + 5*x^2 + x^4)) + (x*(8 + 11*x^2))/(13*sqrt(3 + 5*x^2 + x^4)) + (11*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(26*sqrt(3 + 5*x^2 + x^4)) - (4*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(13*sqrt(3 + 5*x^2 + x^4))]
    @test_int [x^0*(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2), x, 4, (4*x*(5 + sqrt(13) + 2*x^2))/(39*sqrt(3 + 5*x^2 + x^4)) - (x*(7 + 8*x^2))/(39*sqrt(3 + 5*x^2 + x^4)) - (2*sqrt((2*(5 + sqrt(13)))/3)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(39*sqrt(3 + 5*x^2 + x^4)) + (11*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(13*sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]
    @test_int [(2 + 3*x^2)/(x^2*(3 + 5*x^2 + x^4)^(3/2)), x, 5, (19*x*(5 + sqrt(13) + 2*x^2))/(234*sqrt(3 + 5*x^2 + x^4)) - (7 + 8*x^2)/(39*x*sqrt(3 + 5*x^2 + x^4)) - (19*sqrt(3 + 5*x^2 + x^4))/(117*x) - (19*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(234*sqrt(3 + 5*x^2 + x^4)) - (4*sqrt(2/(3*(5 + sqrt(13))))*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(39*sqrt(3 + 5*x^2 + x^4))]
    @test_int [(2 + 3*x^2)/(x^4*(3 + 5*x^2 + x^4)^(3/2)), x, 6, (-133*x*(5 + sqrt(13) + 2*x^2))/(1053*sqrt(3 + 5*x^2 + x^4)) - (7 + 8*x^2)/(39*x^3*sqrt(3 + 5*x^2 + x^4)) - (5*sqrt(3 + 5*x^2 + x^4))/(351*x^3) + (266*sqrt(3 + 5*x^2 + x^4))/(1053*x) + (133*sqrt((5 + sqrt(13))/6)*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.E(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(1053*sqrt(3 + 5*x^2 + x^4)) - (5*sqrt((6 + (5 - sqrt(13))*x^2)/(6 + (5 + sqrt(13))*x^2))*(6 + (5 + sqrt(13))*x^2)*Elliptic.F(atan(sqrt((5 + sqrt(13))/6)*x), (-13 + 5*sqrt(13))/6))/(351*sqrt(6*(5 + sqrt(13)))*sqrt(3 + 5*x^2 + x^4))]

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

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

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

    @test_int [(d + e*x^2)*sqrt(a + b*x^2 + c*x^4)*(f*x)^(3/2), x, 6, (2*d*(f*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(1/2), -(1/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f*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)))) + (2*e*(f*x)^(9/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(9/4, -(1/2), -(1/2), 13/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(9*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*sqrt(a + b*x^2 + c*x^4)*(f*x)^(1/2), x, 6, (2*d*(f*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(1/2), -(1/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f*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)))) + (2*e*(f*x)^(7/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(7/4, -(1/2), -(1/2), 11/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(7*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*sqrt(a + b*x^2 + c*x^4)/(f*x)^(1/2), x, 6, (2*d*sqrt(f*x)*sqrt(a + b*x^2 + c*x^4)*AppellF1(1/4, -(1/2), -(1/2), 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*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)))) + (2*e*(f*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(1/2), -(1/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*sqrt(a + b*x^2 + c*x^4)/(f*x)^(3/2), x, 6, -((2*d*sqrt(a + b*x^2 + c*x^4)*AppellF1(-(1/4), -(1/2), -(1/2), 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*sqrt(f*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))) + (2*e*(f*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(1/2), -(1/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]

    @test_int [(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)*(f*x)^(3/2), x, 6, (2*a*d*(f*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(3/2), -(3/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f*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)))) + (2*a*e*(f*x)^(9/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(9/4, -(3/2), -(3/2), 13/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(9*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)*(f*x)^(1/2), x, 6, (2*a*d*(f*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(3/2), -(3/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f*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)))) + (2*a*e*(f*x)^(7/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(7/4, -(3/2), -(3/2), 11/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(7*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(1/2), x, 6, (2*a*d*sqrt(f*x)*sqrt(a + b*x^2 + c*x^4)*AppellF1(1/4, -(3/2), -(3/2), 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*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)))) + (2*a*e*(f*x)^(5/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(5/4, -(3/2), -(3/2), 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(3/2), x, 6, -((2*a*d*sqrt(a + b*x^2 + c*x^4)*AppellF1(-(1/4), -(3/2), -(3/2), 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*sqrt(f*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))) + (2*a*e*(f*x)^(3/2)*sqrt(a + b*x^2 + c*x^4)*AppellF1(3/4, -(3/2), -(3/2), 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f^3*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]

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

    @test_int [(d + e*x^2)/sqrt(a + b*x^2 + c*x^4)*(f*x)^(3/2), x, 6, (2*d*(f*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 1/2, 1/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(9/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(9/4, 1/2, 1/2, 13/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(9*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/sqrt(a + b*x^2 + c*x^4)*(f*x)^(1/2), x, 6, (2*d*(f*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 1/2, 1/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(7/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(7/4, 1/2, 1/2, 11/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(7*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/sqrt(a + b*x^2 + c*x^4)/(f*x)^(1/2), x, 6, (2*d*sqrt(f*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/4, 1/2, 1/2, 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 1/2, 1/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/sqrt(a + b*x^2 + c*x^4)/(f*x)^(3/2), x, 6, -((2*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/4), 1/2, 1/2, 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*sqrt(f*x)*sqrt(a + b*x^2 + c*x^4))) + (2*e*(f*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 1/2, 1/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*f^3*sqrt(a + b*x^2 + c*x^4))]

    @test_int [(d + e*x^2)/(a + b*x^2 + c*x^4)^(3/2)*(f*x)^(3/2), x, 6, (2*d*(f*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 3/2, 3/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*a*f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(9/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(9/4, 3/2, 3/2, 13/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(9*a*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/(a + b*x^2 + c*x^4)^(3/2)*(f*x)^(1/2), x, 6, (2*d*(f*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 3/2, 3/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*a*f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(7/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(7/4, 3/2, 3/2, 11/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(7*a*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(1/2), x, 6, (2*d*sqrt(f*x)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/4, 3/2, 3/2, 5/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*f*sqrt(a + b*x^2 + c*x^4)) + (2*e*(f*x)^(5/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(5/4, 3/2, 3/2, 9/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(5*a*f^3*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(d + e*x^2)/(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(3/2), x, 6, -((2*d*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/4), 3/2, 3/2, 3/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*f*sqrt(f*x)*sqrt(a + b*x^2 + c*x^4))) + (2*e*(f*x)^(3/2)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(3/4, 3/2, 3/2, 7/4, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(3*a*f^3*sqrt(a + b*x^2 + c*x^4))]

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

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

    @test_int [(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2), x, 6, (a*d*(f*x)^(1 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((1 + m)/2, -(3/2), -(3/2), (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))) + (a*e*(f*x)^(3 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((3 + m)/2, -(3/2), -(3/2), (5 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f^3*(3 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^(1/2), x, 6, (d*(f*x)^(1 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((1 + m)/2, -(1/2), -(1/2), (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))) + (e*(f*x)^(3 + m)*sqrt(a + b*x^2 + c*x^4)*AppellF1((3 + m)/2, -(1/2), -(1/2), (5 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f^3*(3 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(f*x)^m*(d + e*x^2)/(a + b*x^2 + c*x^4)^(1/2), x, 6, (d*(f*x)^(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((1 + m)/2, 1/2, 1/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f*(1 + m)*sqrt(a + b*x^2 + c*x^4)) + (e*(f*x)^(3 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((3 + m)/2, 1/2, 1/2, (5 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(f^3*(3 + m)*sqrt(a + b*x^2 + c*x^4))]
    @test_int [(f*x)^m*(d + e*x^2)/(a + b*x^2 + c*x^4)^(3/2), x, 6, (d*(f*x)^(1 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((1 + m)/2, 3/2, 3/2, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*f*(1 + m)*sqrt(a + b*x^2 + c*x^4)) + (e*(f*x)^(3 + m)*sqrt(1 + (2*c*x^2)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))*AppellF1((3 + m)/2, 3/2, 3/2, (5 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c)))))/(a*f^3*(3 + m)*sqrt(a + b*x^2 + c*x^4))]

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

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

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

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

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

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

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

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

    @test_int [x^8/((d + e*x^2)*(a + c*x^4)^2), x, 24, (d*x)/(4*c*(c*d^2 + a*e^2)) - (x^3*(a*e + c*d*x^2))/(4*c*(c*d^2 + a*e^2)*(a + c*x^4)) + (d^(7/2)*atan((sqrt(e)*x)/sqrt(d)))/(sqrt(e)*(c*d^2 + a*e^2)^2) + (a^(1/4)*d^2*(sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*c^(3/4)*(c*d^2 + a*e^2)^2) + (a^(1/4)*(sqrt(c)*d - 3*sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*c^(7/4)*(c*d^2 + a*e^2)) - (a^(1/4)*d^2*(sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*c^(3/4)*(c*d^2 + a*e^2)^2) - (a^(1/4)*(sqrt(c)*d - 3*sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*c^(7/4)*(c*d^2 + a*e^2)) + (a^(1/4)*d^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)*c^(3/4)*(c*d^2 + a*e^2)^2) + (a^(1/4)*(sqrt(c)*d + 3*sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*c^(7/4)*(c*d^2 + a*e^2)) - (a^(1/4)*d^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)*c^(3/4)*(c*d^2 + a*e^2)^2) - (a^(1/4)*(sqrt(c)*d + 3*sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*c^(7/4)*(c*d^2 + a*e^2))]
    @test_int [x^6/((d + e*x^2)*(a + c*x^4)^2), x, 23, -(x*(a*e + c*d*x^2))/(4*c*(c*d^2 + a*e^2)*(a + c*x^4)) - (d^(5/2)*sqrt(e)*atan((sqrt(e)*x)/sqrt(d)))/(c*d^2 + a*e^2)^2 - (d^2*(sqrt(c)*d + sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(1/4)*c^(1/4)*(c*d^2 + a*e^2)^2) + ((sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(1/4)*c^(5/4)*(c*d^2 + a*e^2)) + (d^2*(sqrt(c)*d + sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(1/4)*c^(1/4)*(c*d^2 + a*e^2)^2) - ((sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(1/4)*c^(5/4)*(c*d^2 + a*e^2)) + (d^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^(1/4)*c^(1/4)*(c*d^2 + a*e^2)^2) - ((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^(1/4)*c^(5/4)*(c*d^2 + a*e^2)) - (d^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^(1/4)*c^(1/4)*(c*d^2 + a*e^2)^2) + ((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^(1/4)*c^(5/4)*(c*d^2 + a*e^2))]
    @test_int [x^4/((d + e*x^2)*(a + c*x^4)^2), x, 23, -(x*(d - e*x^2))/(4*(c*d^2 + a*e^2)*(a + c*x^4)) + (d^(3/2)*e^(3/2)*atan((sqrt(e)*x)/sqrt(d)))/(c*d^2 + a*e^2)^2 - (c^(1/4)*d^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) + ((3*sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(3/4)*c^(3/4)*(c*d^2 + a*e^2)) + (c^(1/4)*d^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) - ((3*sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(3/4)*c^(3/4)*(c*d^2 + a*e^2)) - (c^(1/4)*d^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) + ((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^(3/4)*c^(3/4)*(c*d^2 + a*e^2)) + (c^(1/4)*d^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) - ((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^(3/4)*c^(3/4)*(c*d^2 + a*e^2))]
    @test_int [x^2/((d + e*x^2)*(a + c*x^4)^2), x, 23, (x*(a*e + c*d*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) - (sqrt(d)*e^(5/2)*atan((sqrt(e)*x)/sqrt(d)))/(c*d^2 + a*e^2)^2 + (c^(1/4)*d*e*(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) - ((sqrt(c)*d + 3*sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)) - (c^(1/4)*d*e*(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) + ((sqrt(c)*d + 3*sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)) + (c^(1/4)*d*e*(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) + ((sqrt(c)*d - 3*sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)) - (c^(1/4)*d*e*(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) - ((sqrt(c)*d - 3*sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2))]
    @test_int [1/((d + e*x^2)*(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/(x^2*(d + e*x^2)*(a + c*x^4)^2), x, 22, -(1/(a^2*d*x)) - (c*x*(a*e + c*d*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (e^(9/2)*atan((sqrt(e)*x)/sqrt(d)))/(d^(3/2)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(sqrt(c)*d + 3*sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)) + (c^(3/4)*(a^(3/2)*e^3 + sqrt(c)*d*(c*d^2 + 2*a*e^2))*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*(sqrt(c)*d + 3*sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)) - (c^(3/4)*(a^(3/2)*e^3 + sqrt(c)*d*(c*d^2 + 2*a*e^2))*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*(sqrt(c)*d - 3*sqrt(a)*e)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)) + (c^(3/4)*(a^(3/2)*e^3 - sqrt(c)*d*(c*d^2 + 2*a*e^2))*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*(sqrt(c)*d - 3*sqrt(a)*e)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(16*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)) - (c^(3/4)*(a^(3/2)*e^3 - sqrt(c)*d*(c*d^2 + 2*a*e^2))*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(9/4)*(c*d^2 + a*e^2)^2)]
    @test_int [1/(x^4*(d + e*x^2)*(a + c*x^4)^2), x, 22, -1/(3*a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(11/2)*atan((sqrt(e)*x)/sqrt(d)))/(d^(5/2)*(c*d^2 + a*e^2)^2) + (c^(5/4)*(3*sqrt(c)*d - sqrt(a)*e)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + 2*a*e^2)*atan(1 - (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*sqrt(c)*d - sqrt(a)*e)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(sqrt(c)*d - sqrt(a)*e)*(c*d^2 + 2*a*e^2)*atan(1 + (sqrt(2)*c^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)^2) + (c^(5/4)*(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^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(sqrt(c)*d + sqrt(a)*e)*(c*d^2 + 2*a*e^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(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^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(sqrt(c)*d + sqrt(a)*e)*(c*d^2 + 2*a*e^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(c)*x^2))/(4*sqrt(2)*a^(11/4)*(c*d^2 + a*e^2)^2)]

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

    @test_int [x^2/((1 + x^2)*sqrt(-1 + x^4)), x, 7, -((x*(1 - x^2))/(2*sqrt(-1 + x^4))) - (sqrt(1 - x^2)*sqrt(1 + x^2)*Elliptic.E(asin(x), -1))/(2*sqrt(-1 + x^4)) + (sqrt(-1 + x^2)*sqrt(1 + x^2)*Elliptic.F(asin((sqrt(2)*x)/sqrt(-1 + x^2)), 1/2))/(sqrt(2)*sqrt(-1 + x^4))]
    @test_int [x^2/((1 - x^2)*sqrt(-1 + x^4)), x, 4, (x*(1 + x^2))/(2*sqrt(-1 + x^4)) - (sqrt(1 - x^2)*sqrt(1 + x^2)*Elliptic.E(asin(x), -1))/(2*sqrt(-1 + x^4))]
    @test_int [x^2/((1 + x^2)*sqrt(-1 - x^4)), x, 4, -(atanh((sqrt(2)*x)/sqrt(-1 - x^4))/(2*sqrt(2))) + ((1 + x^2)*sqrt((1 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/2))/(4*sqrt(-1 - x^4))]
    @test_int [x^2/((1 - x^2)*sqrt(-1 - x^4)), x, 4, atan((sqrt(2)*x)/sqrt(-1 - x^4))/(2*sqrt(2)) - ((1 + x^2)*sqrt((1 + x^4)/(1 + x^2)^2)*Elliptic.F(2*atan(x), 1/2))/(4*sqrt(-1 - x^4))]

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

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

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

    @test_int [x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 6, -((c*(b*c - 2*a*d)*x*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(16*d^2*(a + b*x^2))) - ((b*c - 2*a*d)*x^3*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*d*(a + b*x^2)) + (b*x^3*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(6*d*(a + b*x^2)) + (c^2*(b*c - 2*a*d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(16*d^(5/2)*(a + b*x^2))]
    @test_int [x^1*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 4, -(((b*c - a*d)*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d^2*(a + b*x^2))) + (b*(c + d*x^2)^(5/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(5*d^2*(a + b*x^2))]
    @test_int [x^0*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 5, -(((b*c - 4*a*d)*x*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(8*d*(a + b*x^2))) + (b*x*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(4*d*(a + b*x^2)) - (c*(b*c - 4*a*d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(3/2)*(a + b*x^2))]
    @test_int [1/x^1*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 6, (a*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(a + b*x^2) + (b*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(3*d*(a + b*x^2)) - (a*sqrt(c)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(a + b*x^2)]
    @test_int [1/x^2*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 5, ((b*c + 2*a*d)*x*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*c*(a + b*x^2)) - (a*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(c*x*(a + b*x^2)) + ((b*c + 2*a*d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*sqrt(d)*(a + b*x^2))]
    @test_int [1/x^3*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*sqrt(c + d*x^2), x, 6, ((2*b*c + a*d)*sqrt(c + d*x^2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*c*(a + b*x^2)) - (a*(c + d*x^2)^(3/2)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4))/(2*c*x^2*(a + b*x^2)) - ((2*b*c + a*d)*sqrt(a^2 + 2*a*b*x^2 + b^2*x^4)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*sqrt(c)*(a + b*x^2))]

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

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

    #= ::Subsubsection::Closed:: =#
    #=q>0=#

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

    #= ::Subsubsection::Closed:: =#
    #=q<0=#

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

    @test_int [x^6*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 5, ((6*c*d^2 - e*(3*b*d - a*e))*x)/e^5 - ((3*c*d - b*e)*x^3)/(3*e^4) + (c*x^5)/(5*e^3) - (d^2*(c*d^2 - b*d*e + a*e^2)*x)/(4*e^5*(d + e*x^2)^2) + (d*(17*c*d^2 - e*(13*b*d - 9*a*e))*x)/(8*e^5*(d + e*x^2)) - (sqrt(d)*(63*c*d^2 - 35*b*d*e + 15*a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(8*e^(11/2))]
    @test_int [x^4*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 5, -(((3*c*d - b*e)*x)/e^4) + (c*x^3)/(3*e^3) + (d*(c*d^2 - b*d*e + a*e^2)*x)/(4*e^4*(d + e*x^2)^2) - ((13*c*d^2 - e*(9*b*d - 5*a*e))*x)/(8*e^4*(d + e*x^2)) + ((35*c*d^2 - 3*e*(5*b*d - a*e))*atan((sqrt(e)*x)/sqrt(d)))/(8*sqrt(d)*e^(9/2))]
    @test_int [x^2*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 4, (c*x)/e^3 - ((c*d^2 - b*d*e + a*e^2)*x)/(4*e^3*(d + e*x^2)^2) + ((9*c*d^2 - e*(5*b*d - a*e))*x)/(8*d*e^3*(d + e*x^2)) - ((15*c*d^2 - e*(3*b*d + a*e))*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(3/2)*e^(7/2))]
    @test_int [x^0*(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 [1/x^2*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 4, -(a/(d^3*x)) - ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^2*e*(d + e*x^2)^2) + ((c*d^2 + e*(3*b*d - 7*a*e))*x)/(8*d^3*e*(d + e*x^2)) + ((c*d^2 + 3*e*(b*d - 5*a*e))*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(7/2)*e^(3/2))]
    @test_int [1/x^4*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 5, -(a/(3*d^3*x^3)) - (b*d - 3*a*e)/(d^4*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^3*(d + e*x^2)^2) + ((3*c*d^2 - e*(7*b*d - 11*a*e))*x)/(8*d^4*(d + e*x^2)) + ((3*c*d^2 - 15*b*d*e + 35*a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(9/2)*sqrt(e))]
    @test_int [1/x^6*(a + b*x^2 + c*x^4)/(d + e*x^2)^3, x, 5, -(a/(5*d^3*x^5)) - (b*d - 3*a*e)/(3*d^4*x^3) - (c*d^2 - 3*b*d*e + 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a*e^2)*x)/(4*d^4*(d + e*x^2)^2) - (e*(7*c*d^2 - e*(11*b*d - 15*a*e))*x)/(8*d^5*(d + e*x^2)) - (sqrt(e)*(15*c*d^2 - 35*b*d*e + 63*a*e^2)*atan((sqrt(e)*x)/sqrt(d)))/(8*d^(11/2))]

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

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

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

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

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

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

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

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

    @test_int [1/(sqrt(f*x)*(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 19, (c^(3/4)*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atan((2^(1/4)*c^(1/4)*sqrt(f*x))/((-b - sqrt(b^2 - 4*a*c))^(1/4)*sqrt(f))))/(2^(1/4)*sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) - (c^(3/4)*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atan((2^(1/4)*c^(1/4)*sqrt(f*x))/((-b + sqrt(b^2 - 4*a*c))^(1/4)*sqrt(f))))/(2^(1/4)*sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) - (e^(7/4)*atan(1 - (sqrt(2)*e^(1/4)*sqrt(f*x))/(d^(1/4)*sqrt(f))))/(sqrt(2)*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) + (e^(7/4)*atan(1 + (sqrt(2)*e^(1/4)*sqrt(f*x))/(d^(1/4)*sqrt(f))))/(sqrt(2)*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) + (c^(3/4)*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((2^(1/4)*c^(1/4)*sqrt(f*x))/((-b - sqrt(b^2 - 4*a*c))^(1/4)*sqrt(f))))/(2^(1/4)*sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) - (c^(3/4)*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((2^(1/4)*c^(1/4)*sqrt(f*x))/((-b + sqrt(b^2 - 4*a*c))^(1/4)*sqrt(f))))/(2^(1/4)*sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) - (e^(7/4)*log(sqrt(d)*sqrt(f) + sqrt(e)*sqrt(f)*x - sqrt(2)*d^(1/4)*e^(1/4)*sqrt(f*x)))/(2*sqrt(2)*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f)) + (e^(7/4)*log(sqrt(d)*sqrt(f) + sqrt(e)*sqrt(f)*x + sqrt(2)*d^(1/4)*e^(1/4)*sqrt(f*x)))/(2*sqrt(2)*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*sqrt(f))]

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

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

    @test_int [x^5*sqrt(a + b*x^2 + c*x^4)/(d + e*x^2), x, 8, (((2*c*d - b*e)*(4*c*d + b*e) - 2*c*e*(2*c*d + b*e)*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c^2*e^3) + (a + b*x^2 + c*x^4)^(3/2)/(6*c*e) - ((16*c^3*d^3 - b^3*e^3 - 2*b*c*e^2*(b*d - 2*a*e) - 8*c^2*d*e*(b*d - a*e))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(5/2)*e^4) + (d^2*sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^4)]
    @test_int [x^3*sqrt(a + b*x^2 + c*x^4)/(d + e*x^2), x, 7, -((4*c*d - b*e - 2*c*e*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*c*e^2) + ((8*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - a*e))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*c^(3/2)*e^3) - (d*sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^3)]
    @test_int [x^1*sqrt(a + b*x^2 + c*x^4)/(d + e*x^2), x, 7, sqrt(a + b*x^2 + c*x^4)/(2*e) - ((2*c*d - b*e)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(c)*e^2) + (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^2)]
    @test_int [sqrt(a + b*x^2 + c*x^4)/(x^1*(d + e*x^2)), x, 9, -(sqrt(a)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*d) + (sqrt(c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(2*e) - (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d*e)]
    @test_int [sqrt(a + b*x^2 + c*x^4)/(x^3*(d + e*x^2)), x, 21, -sqrt(a + b*x^2 + c*x^4)/(2*d*x^2) - (b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(a)*d) + (sqrt(a)*e*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*d^2) + (sqrt(c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(2*d) - (b*e*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(c)*d^2) - ((2*c*d - b*e)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(c)*d^2) + (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d^2)]

    @test_int [x^4*sqrt(1 + 2*x^2 + 2*x^4)/(3 + 2*x^2), x, 17, (-(1/60))*x*(13 - 6*x^2)*sqrt(1 + 2*x^2 + 2*x^4) + (109*x*sqrt(1 + 2*x^2 + 2*x^4))/(60*sqrt(2)*(1 + sqrt(2)*x^2)) + (3/16)*sqrt(15)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (109*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(60*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((-70 + 263*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(60*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + (15*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(16*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (-(1/4))*x*sqrt(1 + 2*x^2 + 2*x^4) + (1/30)*x*(1 + 3*x^2)*sqrt(1 + 2*x^2 + 2*x^4) + (109*x*sqrt(1 + 2*x^2 + 2*x^4))/(60*sqrt(2)*(1 + sqrt(2)*x^2)) + (3/16)*sqrt(15)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (109*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(60*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (139*(1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(240*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(4*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (45*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(112*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (15*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(224*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^2*sqrt(1 + 2*x^2 + 2*x^4)/(3 + 2*x^2), x, 13, (1/6)*x*sqrt(1 + 2*x^2 + 2*x^4) - (7*x*sqrt(1 + 2*x^2 + 2*x^4))/(6*sqrt(2)*(1 + sqrt(2)*x^2)) - (1/8)*sqrt(15)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (7*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(6*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((-4 + 17*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(6*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - (5*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (1/6)*x*sqrt(1 + 2*x^2 + 2*x^4) - (7*x*sqrt(1 + 2*x^2 + 2*x^4))/(6*sqrt(2)*(1 + sqrt(2)*x^2)) - (1/8)*sqrt(15)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (7*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(6*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (3*(1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((1 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(6*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (15*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(56*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(112*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^0*sqrt(1 + 2*x^2 + 2*x^4)/(3 + 2*x^2), x, 7, (x*sqrt(1 + 2*x^2 + 2*x^4))/(sqrt(2)*(1 + sqrt(2)*x^2)) + (1/4)*sqrt(5/3)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (2^(3/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/((-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(12*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (x*sqrt(1 + 2*x^2 + 2*x^4))/(sqrt(2)*(1 + sqrt(2)*x^2)) + (1/4)*sqrt(5/3)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(4*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(28*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (5*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(168*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [sqrt(1 + 2*x^2 + 2*x^4)/(x^2*(3 + 2*x^2)), x, 8, -(sqrt(1 + 2*x^2 + 2*x^4)/(3*x)) + (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(3*(1 + sqrt(2)*x^2)) - (1/6)*sqrt(5/3)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(21*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(252*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [sqrt(1 + 2*x^2 + 2*x^4)/(x^4*(3 + 2*x^2)), x, 7, -(sqrt(1 + 2*x^2 + 2*x^4)/(9*x^3)) + (1/9)*sqrt(5/3)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(9*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(63*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (5*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(378*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [sqrt(1 + 2*x^2 + 2*x^4)/(x^6*(3 + 2*x^2)), x, 13, -(sqrt(1 + 2*x^2 + 2*x^4)/(15*x^5)) + (4*sqrt(1 + 2*x^2 + 2*x^4))/(135*x^3) - (4*sqrt(1 + 2*x^2 + 2*x^4))/(45*x) + (4*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(45*(1 + sqrt(2)*x^2)) - (2/27)*sqrt(5/3)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (4*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(45*sqrt(1 + 2*x^2 + 2*x^4)) + (5*2^(1/4)*(5 - 3*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(189*sqrt(1 + 2*x^2 + 2*x^4)) - (2^(1/4)*(19 - 2*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(135*sqrt(1 + 2*x^2 + 2*x^4)) + (5*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(567*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]

    @test_int [x^5*(a + b*x^2 + c*x^4)^(3/2)/(d + e*x^2), x, 9, ((128*c^4*d^4 + 3*b^4*e^4 - 32*c^3*d^2*e*(5*b*d - 4*a*e) + 8*b*c^2*d*e^2*(2*b*d - 3*a*e) + 6*b^2*c*e^3*(b*d - 2*a*e) - 2*c*e*(32*c^3*d^3 - 3*b^3*e^3 - 8*c^2*d*e*(2*b*d - 3*a*e) - 6*b*c*e^2*(b*d - 2*a*e))*x^2)*sqrt(a + b*x^2 + c*x^4))/(256*c^3*e^5) + ((16*c^2*d^2 - 6*b*c*d*e - 3*b^2*e^2 - 6*c*e*(2*c*d + b*e)*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(96*c^2*e^3) + (a + b*x^2 + c*x^4)^(5/2)/(10*c*e) - ((256*c^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a*e) - 384*c^4*d^3*e*(b*d - a*e) + 96*c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(512*c^(7/2)*e^6) + (d^2*(c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^6)]
    @test_int [x^3*(a + b*x^2 + c*x^4)^(3/2)/(d + e*x^2), x, 8, -((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 3*a*e) - 2*c*e*(16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*x^2)*sqrt(a + b*x^2 + c*x^4))/(128*c^2*e^4) - ((8*c*d - 3*b*e - 6*c*e*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(48*c*e^2) + ((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(256*c^(5/2)*e^5) - (d*(c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^5)]
    @test_int [x^1*(a + b*x^2 + c*x^4)^(3/2)/(d + e*x^2), x, 8, ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c*e^3) + (a + b*x^2 + c*x^4)^(3/2)/(6*e) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(3/2)*e^4) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e^4)]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/(x^1*(d + e*x^2)), x, 14, (a*sqrt(a + b*x^2 + c*x^4))/(2*d) - ((4*c*d^2 - e*(5*b*d - 4*a*e) - 2*c*d*e*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*d*e^2) - (a^(3/2)*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*d) + (a*b*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(4*sqrt(c)*d) + ((8*c^2*d^3 + b*e^2*(3*b*d - 4*a*e) - 12*c*d*e*(b*d - a*e))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*sqrt(c)*d*e^3) - ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d*e^3)]
    @test_int [(a + b*x^2 + c*x^4)^(3/2)/(x^3*(d + e*x^2)), x, 24, (3*(3*b + 2*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(8*d) - (e*(b^2 + 8*a*c + 2*b*c*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c*d^2) + ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x^2)*sqrt(a + b*x^2 + c*x^4))/(16*c*d^2*e) - (a + b*x^2 + c*x^4)^(3/2)/(2*d*x^2) - (3*sqrt(a)*b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*d) + (a^(3/2)*e*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*d^2) + (3*(b^2 + 4*a*c)*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(16*sqrt(c)*d) + (b*(b^2 - 12*a*c)*e*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(3/2)*d^2) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4))))/(32*c^(3/2)*d^2*e^2) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d^2*e^2)]

    @test_int [x^2*(1 + 2*x^2 + 2*x^4)^(3/2)/(3 - 2*x^2), x, 19, (-(213/140))*x*sqrt(1 + 2*x^2 + 2*x^4) - (27/70)*x^3*sqrt(1 + 2*x^2 + 2*x^4) - (2211*x*sqrt(1 + 2*x^2 + 2*x^4))/(140*sqrt(2)*(1 + sqrt(2)*x^2)) - (1/14)*x*(1 + 2*x^2 + 2*x^4)^(3/2) + (17/16)*sqrt(51)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (2211*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(140*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (3*(514 + 2717*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(140*2^(3/4)*(2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(16*2^(3/4)*(2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (-(3/35))*x*(2 + x^2)*sqrt(1 + 2*x^2 + 2*x^4) - (3/20)*x*(9 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4) - (309*x*sqrt(1 + 2*x^2 + 2*x^4))/(20*sqrt(2)*(1 + sqrt(2)*x^2)) - (6*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(35*(1 + sqrt(2)*x^2)) - (1/14)*x*(1 + 2*x^2 + 2*x^4)^(3/2) + (17/16)*sqrt(51)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (309*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (6*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(35*sqrt(1 + 2*x^2 + 2*x^4)) + (867*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(112*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (51*(5 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(16*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (3*(3 + 2*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(70*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (3*(9 + 8*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(11 - 6*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(224*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^0*(1 + 2*x^2 + 2*x^4)^(3/2)/(3 - 2*x^2), x, 12, (-(1/10))*x*(9 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4) - (103*x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) + (17/8)*sqrt(17/3)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (103*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((66 + 383*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*(2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(24*2^(3/4)*(2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (-(1/10))*x*(9 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4) - (103*x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) + (17/8)*sqrt(17/3)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (103*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (289*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(56*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (17*(5 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((9 + 8*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(11 - 6*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(336*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [(1 + 2*x^2 + 2*x^4)^(3/2)/(x^2*(3 - 2*x^2)), x, 13, -(((1 + x^2)*sqrt(1 + 2*x^2 + 2*x^4))/(3*x)) - (17*x*sqrt(1 + 2*x^2 + 2*x^4))/(3*sqrt(2)*(1 + sqrt(2)*x^2)) + (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(3*(1 + sqrt(2)*x^2)) + (17/12)*sqrt(17/3)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (17*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*sqrt(1 + 2*x^2 + 2*x^4)) + ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (289*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(84*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (17*(5 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(12*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(11 - 6*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(504*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [(1 + 2*x^2 + 2*x^4)^(3/2)/(x^4*(3 - 2*x^2)), x, 13, -((2*sqrt(1 + 2*x^2 + 2*x^4))/x) - ((1 - 8*x^2)*sqrt(1 + 2*x^2 + 2*x^4))/(9*x^3) + (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(9*(1 + sqrt(2)*x^2)) + (17/18)*sqrt(17/3)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(9*sqrt(1 + 2*x^2 + 2*x^4)) + (289*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(126*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (17*(5 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(18*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (2^(1/4)*(9 + 5*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(9*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(11 - 6*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(756*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)), x, 15, (74*sqrt(1 + 2*x^2 + 2*x^4))/(135*x^3) - (262*sqrt(1 + 2*x^2 + 2*x^4))/(135*x) - ((3 + 40*x^2)*sqrt(1 + 2*x^2 + 2*x^4))/(45*x^5) + (262*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(135*(1 + sqrt(2)*x^2)) + (17/27)*sqrt(17/3)*atanh((sqrt(17/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - (262*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(135*sqrt(1 + 2*x^2 + 2*x^4)) + (85*2^(3/4)*(3 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(189*sqrt(1 + 2*x^2 + 2*x^4)) + (2^(3/4)*(37 + 23*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(135*sqrt(1 + 2*x^2 + 2*x^4)) - (289*(11 - 6*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 + 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(1134*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]

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

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

    @test_int [x^4/((3 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4)), x, 4, (x*sqrt(1 + 2*x^2 + 2*x^4))/(2*sqrt(2)*(1 + sqrt(2)*x^2)) - (3*sqrt(3/10)*(3 - sqrt(2))*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)))/(4*(2 - 3*sqrt(2))) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(2*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((1 - 3*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(2*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + (3*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^2/((3 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4)), x, 3, (-(1/4))*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(14*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(56*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^0/((3 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4)), x, 3, atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4))/(2*sqrt(15)) + ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(14*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(84*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [1/(x^2*(3 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4)), x, 6, -(sqrt(1 + 2*x^2 + 2*x^4)/(3*x)) + (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(3*(1 + sqrt(2)*x^2)) - atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4))/(3*sqrt(15)) - (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*sqrt(1 + 2*x^2 + 2*x^4)) + ((5 - 3*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(21*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(126*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [1/(x^4*(3 + 2*x^2)*sqrt(1 + 2*x^2 + 2*x^4)), x, 7, -(sqrt(1 + 2*x^2 + 2*x^4)/(9*x^3)) + (2*sqrt(1 + 2*x^2 + 2*x^4))/(3*x) - (2*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(3*(1 + sqrt(2)*x^2)) + (2*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)))/(9*sqrt(15)) + (2*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + 19*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(63*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(189*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]

    @test_int [x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 7, (a*(b^2*d - 2*a*c*d - a*b*e) + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*x^2)/(c*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4)) + atanh((b + 2*c*x^2)/(2*sqrt(c)*sqrt(a + b*x^2 + c*x^4)))/(2*c^(3/2)*e) - (d^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*e*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 5, -((a*(b*d - 2*a*e) + (b^2*d - 2*a*c*d - a*b*e)*x^2)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))) + (d^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [x^3/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 5, (a*(2*c*d - b*e) + c*(b*d - 2*a*e)*x^2)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4)) - (d*e*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [x^1/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 5, -((b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))) + (e^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [1/(x^1*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 11, (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*d*sqrt(a + b*x^2 + c*x^4)) + (e*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c)*d*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4)) - atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4)))/(2*a^(3/2)*d) - (e^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x, 15, -((e*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*d^2*sqrt(a + b*x^2 + c*x^4))) + (b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*d*x^2*sqrt(a + b*x^2 + c*x^4)) - (e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c)*d^2*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4)) - ((3*b^2 - 8*a*c)*sqrt(a + b*x^2 + c*x^4))/(2*a^2*(b^2 - 4*a*c)*d*x^2) + (3*b*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(4*a^(5/2)*d) + (e*atanh((2*a + b*x^2)/(2*sqrt(a)*sqrt(a + b*x^2 + c*x^4))))/(2*a^(3/2)*d^2) + (e^4*atanh((b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x^2 + c*x^4))))/(2*d^2*(c*d^2 - b*d*e + a*e^2)^(3/2))]

    @test_int [x^8/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 10, (x^3*(1 - 2*x^2))/(20*sqrt(1 + 2*x^2 + 2*x^4)) + (1/20)*x*sqrt(1 + 2*x^2 + 2*x^4) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) + (27/80)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((-2 + 7*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + (27*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(80*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (x^3*(1 - 2*x^2))/(20*sqrt(1 + 2*x^2 + 2*x^4)) + (1/20)*x*sqrt(1 + 2*x^2 + 2*x^4) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) - (27*sqrt(3/10)*(3 - sqrt(2))*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)))/(40*(2 - 3*sqrt(2))) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (9*(1 - 3*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - ((7 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(40*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (27*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(80*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^6/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 8, (x*(1 - 2*x^2))/(20*sqrt(1 + 2*x^2 + 2*x^4)) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) - (9/40)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((2^(1/4) + 2^(3/4))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(8*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - (9*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(40*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (x*(1 - 2*x^2))/(20*sqrt(1 + 2*x^2 + 2*x^4)) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) - (9/40)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(40*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (9*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(140*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (9*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(560*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^4/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 8, -((x*(2 + x^2))/(10*sqrt(1 + 2*x^2 + 2*x^4))) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) + (3/20)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((2 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(4*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + (3*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), -((x*(2 + x^2))/(10*sqrt(1 + 2*x^2 + 2*x^4))) + (x*sqrt(1 + 2*x^2 + 2*x^4))/(10*sqrt(2)*(1 + sqrt(2)*x^2)) + (3/20)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + (9*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(140*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (3*(3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(280*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^2/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 8, (x*(3 + 4*x^2))/(10*sqrt(1 + 2*x^2 + 2*x^4)) - (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(5*(1 + sqrt(2)*x^2)) - (1/10)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(5*sqrt(1 + 2*x^2 + 2*x^4)) - ((2^(1/4) + 2^(3/4))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(4*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), (x*(3 + 4*x^2))/(10*sqrt(1 + 2*x^2 + 2*x^4)) - (sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(5*(1 + sqrt(2)*x^2)) - (1/10)*sqrt(3/5)*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)) + (2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(5*sqrt(1 + 2*x^2 + 2*x^4)) - (3*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(70*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 + 2*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(20*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(140*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [x^0/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 8, -((x*(1 + 3*x^2))/(5*sqrt(1 + 2*x^2 + 2*x^4))) + (3*x*sqrt(1 + 2*x^2 + 2*x^4))/(5*sqrt(2)*(1 + sqrt(2)*x^2)) + atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4))/(5*sqrt(15)) - (3*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(5*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((2 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(2*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(15*2^(3/4)*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), -((x*(1 + 3*x^2))/(5*sqrt(1 + 2*x^2 + 2*x^4))) + (3*x*sqrt(1 + 2*x^2 + 2*x^4))/(5*sqrt(2)*(1 + sqrt(2)*x^2)) + atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4))/(5*sqrt(15)) - (3*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(5*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(35*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + 2*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(10*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(210*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]
    @test_int [1/(x^2*(3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)), x, 15, -(x/(3*sqrt(1 + 2*x^2 + 2*x^4))) + (2*x*(1 + 3*x^2))/(15*sqrt(1 + 2*x^2 + 2*x^4)) - sqrt(1 + 2*x^2 + 2*x^4)/(3*x) + (2*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(15*(1 + sqrt(2)*x^2)) - (2*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)))/(15*sqrt(15)) - (2*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(15*sqrt(1 + 2*x^2 + 2*x^4)) + ((-7 + 3*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(3*2^(3/4)*(-2 + 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)) - (2^(1/4)*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(45*(2 - 3*sqrt(2))*sqrt(1 + 2*x^2 + 2*x^4)), -(x/(3*sqrt(1 + 2*x^2 + 2*x^4))) + (2*x*(1 + 3*x^2))/(15*sqrt(1 + 2*x^2 + 2*x^4)) - sqrt(1 + 2*x^2 + 2*x^4)/(3*x) + (2*sqrt(2)*x*sqrt(1 + 2*x^2 + 2*x^4))/(15*(1 + sqrt(2)*x^2)) - (2*atan((sqrt(5/3)*x)/sqrt(1 + 2*x^2 + 2*x^4)))/(15*sqrt(15)) - (2*2^(1/4)*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.E(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(15*sqrt(1 + 2*x^2 + 2*x^4)) - ((1 - sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(6*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4)) - (2^(3/4)*(3 + sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(105*sqrt(1 + 2*x^2 + 2*x^4)) - ((3 + 2*sqrt(2))*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.F(2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(15*2^(3/4)*sqrt(1 + 2*x^2 + 2*x^4)) + ((3 + sqrt(2))^2*(1 + sqrt(2)*x^2)*sqrt((1 + 2*x^2 + 2*x^4)/(1 + sqrt(2)*x^2)^2)*Elliptic.Pi((1/24)*(12 - 11*sqrt(2)), 2*atan(2^(1/4)*x), (1/4)*(2 - sqrt(2))))/(315*2^(1/4)*sqrt(1 + 2*x^2 + 2*x^4))]

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

    #= ::Subsubsection::Closed:: =#
    #=q>0=#

    @test_int [x^7*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 7, ((b^2 - a*c)*sqrt(d + e*x^2))/c^3 - ((c*d + b*e)*(d + e*x^2)^(3/2))/(3*c^2*e^2) + (d + e*x^2)^(5/2)/(5*c*e^2) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(7/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(7/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^5*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 7, -((b*sqrt(d + e*x^2))/c^2) + (d + e*x^2)^(3/2)/(3*c*e) + ((b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(5/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + ((b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(5/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^3*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 6, sqrt(d + e*x^2)/c + ((b*c*d - b^2*e + 2*a*c*e - sqrt(b^2 - 4*a*c)*(c*d - b*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - ((b*c*d - b^2*e + 2*a*c*e + sqrt(b^2 - 4*a*c)*(c*d - b*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^1*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 5, -((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c))) + (sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*sqrt(c)*sqrt(b^2 - 4*a*c))]
    @test_int [sqrt(d + e*x^2)/(x^1*(a + b*x^2 + c*x^4)), x, 8, -((sqrt(d)*atanh(sqrt(d + e*x^2)/sqrt(d)))/a) + (sqrt(c)*(b*d + sqrt(b^2 - 4*a*c)*d - 2*a*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(c)*(b*d - sqrt(b^2 - 4*a*c)*d - 2*a*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [sqrt(d + e*x^2)/(x^3*(a + b*x^2 + c*x^4)), x, 10, If(13>=8, -(sqrt(d + e*x^2)/(2*a*x^2)) + (e*atanh(sqrt(d + e*x^2)/sqrt(d)))/(2*a*sqrt(d)) + ((b*d - a*e)*atanh(sqrt(d + e*x^2)/sqrt(d)))/(a^2*sqrt(d)) - (sqrt(c)*(b^2*d - 2*a*c*d - a*b*e + sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(c)*(b^2*d - b*(sqrt(b^2 - 4*a*c)*d + a*e) - a*(2*c*d - sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)), -(sqrt(d + e*x^2)/(2*a*x^2)) + (e*atanh(sqrt(d + e*x^2)/sqrt(d)))/(2*a*sqrt(d)) + ((b*d - a*e)*atanh(sqrt(d + e*x^2)/sqrt(d)))/(a^2*sqrt(d)) - (sqrt(c)*(b^2*d - 2*a*c*d - a*b*e + sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(c)*(b^2*d - 2*a*c*d - a*b*e - sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [sqrt(d + e*x^2)/(x^5*(a + b*x^2 + c*x^4)), x, 13, -(sqrt(d + e*x^2)/(4*a*x^4)) + (3*e*sqrt(d + e*x^2))/(8*a*d*x^2) + ((b*d - a*e)*sqrt(d + e*x^2))/(2*a^2*d*x^2) - (3*e^2*atanh(sqrt(d + e*x^2)/sqrt(d)))/(8*a*d^(3/2)) - (e*(b*d - a*e)*atanh(sqrt(d + e*x^2)/sqrt(d)))/(2*a^2*d^(3/2)) - ((b^2*d - a*c*d - a*b*e)*atanh(sqrt(d + e*x^2)/sqrt(d)))/(a^3*sqrt(d)) + (sqrt(c)*(b^3*d - a*c*(sqrt(b^2 - 4*a*c)*d - 2*a*e) + b^2*(sqrt(b^2 - 4*a*c)*d - a*e) - a*b*(3*c*d + sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(c)*(b^3*d - b^2*(sqrt(b^2 - 4*a*c)*d + a*e) + a*c*(sqrt(b^2 - 4*a*c)*d + 2*a*e) - a*b*(3*c*d - sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x^2))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]

    @test_int [x^4*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 10, (x*sqrt(d + e*x^2))/(2*c) - ((b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - ((b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) + ((c*d - 2*b*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^2*sqrt(e))]
    @test_int [x^2*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 9, ((c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + ((c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) + (sqrt(e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/c]
    @test_int [x^0*sqrt(d + e*x^2)/(a + b*x^2 + c*x^4), x, 11, (sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [sqrt(d + e*x^2)/(x^2*(a + b*x^2 + c*x^4)), x, 8, -(sqrt(d + e*x^2)/(a*x)) - (c*(d + (b*d - 2*a*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (c*(d - (b*d - 2*a*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [sqrt(d + e*x^2)/(x^4*(a + b*x^2 + c*x^4)), x, 12, -(sqrt(d + e*x^2)/(3*a*x^3)) + (2*e*sqrt(d + e*x^2))/(3*a*d*x) + ((b*d - a*e)*sqrt(d + e*x^2))/(a^2*d*x) + (c*(b*d - a*e + (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (c*(b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [sqrt(d + e*x^2)/(x^6*(a + b*x^2 + c*x^4)), x, 15, -(sqrt(d + e*x^2)/(5*a*x^5)) + (4*e*sqrt(d + e*x^2))/(15*a*d*x^3) + ((b*d - a*e)*sqrt(d + e*x^2))/(3*a^2*d*x^3) - (8*e^2*sqrt(d + e*x^2))/(15*a*d^2*x) - (2*e*(b*d - a*e)*sqrt(d + e*x^2))/(3*a^2*d^2*x) - ((b^2*d - a*c*d - a*b*e)*sqrt(d + e*x^2))/(a^3*d*x) - (c*(b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^3*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (c*(b^2*d - a*c*d - a*b*e - (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^3*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]

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

    @test_int [x^4*(d + e*x^2)^(3/2)/(a + b*x^2 + c*x^4), x, 17, ((3*c*d - 4*b*e)*x*sqrt(d + e*x^2))/(8*c^2) + (x*(d + e*x^2)^(3/2))/(4*c) - (sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*c^3*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*c^3*sqrt(b + sqrt(b^2 - 4*a*c))) + (d*(3*c*d - 4*b*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(8*c^2*sqrt(e)) - (sqrt(e)*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^3) - (sqrt(e)*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^3)]
    @test_int [x^2*(d + e*x^2)^(3/2)/(a + b*x^2 + c*x^4), x, 16, (e*x*sqrt(d + e*x^2))/(2*c) + (sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*c^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*c^2*sqrt(b + sqrt(b^2 - 4*a*c))) + (d*sqrt(e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c) + (sqrt(e)*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^2) + (sqrt(e)*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^2)]
    @test_int [x^0*(d + e*x^2)^(3/2)/(a + b*x^2 + c*x^4), x, 13, ((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*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - ((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*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) + (sqrt(e)*(3*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c*sqrt(b^2 - 4*a*c)) - (sqrt(e)*(3*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c*sqrt(b^2 - 4*a*c))]
    @test_int [(d + e*x^2)^(3/2)/(x^2*(a + b*x^2 + c*x^4)), x, -16, -((d*sqrt(d + e*x^2))/(a*x)) - ((2*c*d - (b - sqrt(b^2 - 4*a*c))*e)^(3/2)*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(3/2)) + ((2*c*d - (b + sqrt(b^2 - 4*a*c))*e)^(3/2)*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(3/2))]
    @test_int [(d + e*x^2)^(3/2)/(x^4*(a + b*x^2 + c*x^4)), x, 19, ((b*d - a*e)*sqrt(d + e*x^2))/(a^2*x) - (d + e*x^2)^(3/2)/(3*a*x^3) + (sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(b*d - a*e + (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*a^2*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(2*a^2*sqrt(b + sqrt(b^2 - 4*a*c))) - (sqrt(e)*(b*d - a*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/a^2 + (sqrt(e)*(b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*a^2) + (sqrt(e)*(b*d - a*e + (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*a^2)]

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

    @test_int [x^4*sqrt(1 - x^2)/(a + b*x^2 + c*x^4), x, 9, (x*sqrt(1 - x^2))/(2*c) + ((2*b + c)*asin(x))/(2*c^2) - ((b^2 - a*c + b*c - (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c - sqrt(b^2 - 4*a*c))*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(c^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(b + 2*c - sqrt(b^2 - 4*a*c))) - ((b^2 - a*c + b*c + (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c + sqrt(b^2 - 4*a*c))*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(c^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(b + 2*c + sqrt(b^2 - 4*a*c)))]
    @test_int [x^2*sqrt(1 - x^2)/(a + b*x^2 + c*x^4), x, 8, -(asin(x)/c) + ((b + c - (b^2 - 2*a*c + b*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c - sqrt(b^2 - 4*a*c))*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(c*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(b + 2*c - sqrt(b^2 - 4*a*c))) + ((b + c + (b^2 - 2*a*c + b*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c + sqrt(b^2 - 4*a*c))*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(c*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(b + 2*c + sqrt(b^2 - 4*a*c)))]
    @test_int [x^0*sqrt(1 - x^2)/(a + b*x^2 + c*x^4), x, 9, (sqrt(b + 2*c - sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c - sqrt(b^2 - 4*a*c))*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(b + 2*c + sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c + sqrt(b^2 - 4*a*c))*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))]
    @test_int [sqrt(1 - x^2)/(x^2*(a + b*x^2 + c*x^4)), x, 8, -(sqrt(1 - x^2)/(a*x)) - (c*(1 + (2*a + b)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c - sqrt(b^2 - 4*a*c))*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(a*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(b + 2*c - sqrt(b^2 - 4*a*c))) - (c*(1 - (2*a + b)/sqrt(b^2 - 4*a*c))*atan((sqrt(b + 2*c + sqrt(b^2 - 4*a*c))*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(1 - x^2))))/(a*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(b + 2*c + sqrt(b^2 - 4*a*c)))]

    @test_int [x^2*sqrt(1 - x^2)/(-1 + x^2 + x^4), x, 8, -asin(x) + sqrt((1/5)*(2 + sqrt(5)))*atan((sqrt((1/2)*(1 + sqrt(5)))*x)/sqrt(1 - x^2)) - sqrt((1/5)*(-2 + sqrt(5)))*atanh((sqrt((1/2)*(-1 + sqrt(5)))*x)/sqrt(1 - x^2))]

    #= ::Subsubsection::Closed:: =#
    #=q<0=#

    @test_int [x^8/(sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 17, -((3*d*x*sqrt(d + e*x^2))/(8*c*e^2)) - (b*x*sqrt(d + e*x^2))/(2*c^2*e) + (x^3*sqrt(d + e*x^2))/(4*c*e) - ((b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^3*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - ((b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^3*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) + (3*d^2*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(8*c*e^(5/2)) + (b*d*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c^2*e^(3/2)) + ((b^2 - a*c)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(c^3*sqrt(e))]
    @test_int [x^6/(sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 13, (x*sqrt(d + e*x^2))/(2*c*e) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) - (d*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(2*c*e^(3/2)) - (b*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(c^2*sqrt(e))]
    @test_int [x^4/(sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 10, -(((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)) + atanh((sqrt(e)*x)/sqrt(d + e*x^2))/(c*sqrt(e))]
    @test_int [x^2/(sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 6, -((sqrt(b - sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))) + (sqrt(b + sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^0/(sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 5, (2*c*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (2*c*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [1/(x^2*sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 9, -(sqrt(d + e*x^2)/(a*d*x)) - (c*(1 + b/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (c*(1 - b/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [1/(x^4*sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 11, -(sqrt(d + e*x^2)/(3*a*d*x^3)) + (b*sqrt(d + e*x^2))/(a^2*d*x) + (2*e*sqrt(d + e*x^2))/(3*a*d^2*x) + (c*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (c*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [1/(x^6*sqrt(d + e*x^2)*(a + b*x^2 + c*x^4)), x, 14, -(sqrt(d + e*x^2)/(5*a*d*x^5)) + (b*sqrt(d + e*x^2))/(3*a^2*d*x^3) + (4*e*sqrt(d + e*x^2))/(15*a*d^2*x^3) - ((b^2 - a*c)*sqrt(d + e*x^2))/(a^3*d*x) - (2*b*e*sqrt(d + e*x^2))/(3*a^2*d^2*x) - (8*e^2*sqrt(d + e*x^2))/(15*a*d^3*x) - (c*(b^2 - a*c + (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^3*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (c*(b^2 - a*c - (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^3*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]

    @test_int [x^6/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 14, -((d^2*x)/(e*(c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2))) + (2*(b^2 - a*c - (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b - sqrt(b^2 - 4*a*c))*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)^(3/2)) + (2*(b^2 - a*c + (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b + sqrt(b^2 - 4*a*c))*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)^(3/2)) + atanh((sqrt(e)*x)/sqrt(d + e*x^2))/(c*e^(3/2)), -((d^2*x)/(e*(c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2))) + ((b^2*d - a*c*d - a*b*e - (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) + ((b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(c*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) + (d^2*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(e^(3/2)*(c*d^2 - b*d*e + a*e^2)) - ((b*d - a*e)*atanh((sqrt(e)*x)/sqrt(d + e*x^2)))/(c*sqrt(e)*(c*d^2 - b*d*e + a*e^2))]
    @test_int [x^4/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 8, (d*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2)) - ((b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) - ((b*d - a*e + (b^2*d - 2*a*c*d - a*b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))]
    @test_int [x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 8, -((e*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2))) + (c*(d - (b*d - 2*a*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) + (c*(d + (b*d - 2*a*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))]
    @test_int [x^0/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 8, (e^2*x)/(d*(c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2)) - (c*(e - (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) - (c*(e + (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))]
    @test_int [1/(x^2*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 12, (-d - 2*e*x^2)/(a*d^2*x*sqrt(d + e*x^2)) + (e*(c*d - b*e)*x)/(a*d*(c*d^2 + e*((-b)*d + a*e))*sqrt(d + e*x^2)) - (2*c^2*(1 + b/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b - sqrt(b^2 - 4*a*c))*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)^(3/2)) - (2*c^2*(1 - b/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b + sqrt(b^2 - 4*a*c))*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)^(3/2)), -(e^2/(d*(c*d^2 - b*d*e + a*e^2)*x*sqrt(d + e*x^2))) - (2*e^3*x)/(d^2*(c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2)) - ((c*d - b*e)*sqrt(d + e*x^2))/(a*d*(c*d^2 - b*d*e + a*e^2)*x) - (c*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) - (c*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))]
    @test_int [1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x, 15, -(1/(3*a*d*x^3*sqrt(d + e*x^2))) + (3*b*d + 4*a*e)/(3*a^2*d^2*x*sqrt(d + e*x^2)) + (2*e*(3*b*d + 4*a*e)*x)/(3*a^2*d^3*sqrt(d + e*x^2)) - (e*(b*c*d - b^2*e + a*c*e)*x)/(a^2*d*(c*d^2 + e*((-b)*d + a*e))*sqrt(d + e*x^2)) + (2*c^2*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b - sqrt(b^2 - 4*a*c))*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)^(3/2)) + (2*c^2*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b + sqrt(b^2 - 4*a*c))*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)^(3/2)), -(e^2/(3*d*(c*d^2 - b*d*e + a*e^2)*x^3*sqrt(d + e*x^2))) + (4*e^3)/(3*d^2*(c*d^2 - b*d*e + a*e^2)*x*sqrt(d + e*x^2)) + (8*e^4*x)/(3*d^3*(c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x^2)) - ((c*d - b*e)*sqrt(d + e*x^2))/(3*a*d*(c*d^2 - b*d*e + a*e^2)*x^3) + (2*e*(c*d - b*e)*sqrt(d + e*x^2))/(3*a*d^2*(c*d^2 - b*d*e + a*e^2)*x) + ((b*c*d - b^2*e + a*c*e)*sqrt(d + e*x^2))/(a^2*d*(c*d^2 - b*d*e + a*e^2)*x) + (c*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b - sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) + (c*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atan((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*x)/(sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(d + e*x^2))))/(a^2*sqrt(b + sqrt(b^2 - 4*a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))]

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

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

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

    @test_int [(f*x)^m*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x, 6, (2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1((1 + m)/2, 1, -q, (3 + m)/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*f*(1 + m))) - (2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1((1 + m)/2, 1, -q, (3 + m)/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))*f*(1 + m)))]

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

    @test_int [x^6*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x, 12, ((b^2 - a*c - (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(c^2*(b - sqrt(b^2 - 4*a*c)))) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(c^2*(b + sqrt(b^2 - 4*a*c)))) - (b*x*(d + e*x^2)^q*HypergeometricFunctions._₂F₁(1/2, -q, 3/2, -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*c^2) + (x^3*(d + e*x^2)^q*HypergeometricFunctions._₂F₁(3/2, -q, 5/2, -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(3*c))]
    @test_int [x^4*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x, 10, -(((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(c*(b - sqrt(b^2 - 4*a*c))))) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(c*(b + sqrt(b^2 - 4*a*c)))) + (x*(d + e*x^2)^q*HypergeometricFunctions._₂F₁(1/2, -q, 3/2, -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*c)]
    @test_int [x^2*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x, 6, -((x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*sqrt(b^2 - 4*a*c))) + (x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*sqrt(b^2 - 4*a*c))]
    @test_int [x^0*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x, 5, -((2*c*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c)))) - (2*c*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c)))]
    @test_int [(d + e*x^2)^q/(x^2*(a + b*x^2 + c*x^4)), x, 10, -((c*(1 + b/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a*(b - sqrt(b^2 - 4*a*c))))) - (c*(1 - b/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a*(b + sqrt(b^2 - 4*a*c)))) - ((d + e*x^2)^q*HypergeometricFunctions._₂F₁(-(1/2), -q, 1/2, -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a*x))]
    @test_int [(d + e*x^2)^q/(x^4*(a + b*x^2 + c*x^4)), x, 12, (c*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b - sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a^2*(b - sqrt(b^2 - 4*a*c)))) + (c*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*x*(d + e*x^2)^q*AppellF1(1/2, 1, -q, 3/2, -((2*c*x^2)/(b + sqrt(b^2 - 4*a*c))), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a^2*(b + sqrt(b^2 - 4*a*c)))) - ((d + e*x^2)^q*HypergeometricFunctions._₂F₁(-(3/2), -q, -(1/2), -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(3*a*x^3)) + (b*(d + e*x^2)^q*HypergeometricFunctions._₂F₁(-(1/2), -q, 1/2, -((e*x^2)/d)))/((1 + (e*x^2)/d)^q*(a^2*x))]

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

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

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

    #= ::Subsubsection::Closed:: =#
    #=p<0 & q>0=#

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

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.jl