text:
@testset "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p" begin
    (a, b, c, d, e, f, m, n, p, x, ) = @variables a b c d e f m n p x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p*with*a=0=#

    @test_int [(a*x^3 + b*x^6)^(5/3), x, 2, -((3*a*(a*x^3 + b*x^6)^(8/3))/(88*b^2*x^8)) + (a*x^3 + b*x^6)^(8/3)/(11*b*x^5)]
    @test_int [(a*x^3 + b*x^6)^(2/3), x, 1, (a*x^3 + b*x^6)^(5/3)/(5*b*x^5)]
    @test_int [1/(a*x^3 + b*x^6)^(2/3), x, 1, -((a*x^3 + b*x^6)^(1/3)/(a*x^2))]
    @test_int [1/(a*x^3 + b*x^6)^(5/3), x, 3, 1/(2*a*x^2*(a*x^3 + b*x^6)^(2/3)) - (3*(a*x^3 + b*x^6)^(1/3))/(4*a^2*x^5) + (9*b*(a*x^3 + b*x^6)^(1/3))/(4*a^3*x^2)]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p*with*b^2-4*a*c=0=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a^2+2*a*b*x^3+b^2*x^6)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a^2+2*a*b*x^2+b^3*x^6)^(p/2)=#

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

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

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

    @test_int [x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*(a + b*x^3)) + (5*a^4*b*x^17*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*(a + b*x^3)) + (a^3*b^2*x^20*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (10*a^2*b^3*x^23*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(23*(a + b*x^3)) + (5*a*b^4*x^26*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(26*(a + b*x^3)) + (b^5*x^29*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(29*(a + b*x^3))]
    @test_int [x^12*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^13*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*(a + b*x^3)) + (5*a^4*b*x^16*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(16*(a + b*x^3)) + (10*a^3*b^2*x^19*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*(a + b*x^3)) + (5*a^2*b^3*x^22*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (a*b^4*x^25*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(5*(a + b*x^3)) + (b^5*x^28*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(28*(a + b*x^3))]
    @test_int [x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 4, -(a^3*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(18*b^4) + (a^2*(a + b*x^3)^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*b^4) - (a*(a + b*x^3)^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*b^4) + ((a + b*x^3)^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(27*b^4)]
    @test_int [x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (5*a^4*b*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*(a + b*x^3)) + (10*a^3*b^2*x^17*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*(a + b*x^3)) + (a^2*b^3*x^20*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (5*a*b^4*x^23*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(23*(a + b*x^3)) + (b^5*x^26*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(26*(a + b*x^3))]
    @test_int [x^9*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^10*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(10*(a + b*x^3)) + (5*a^4*b*x^13*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*(a + b*x^3)) + (5*a^3*b^2*x^16*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3)) + (10*a^2*b^3*x^19*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*(a + b*x^3)) + (5*a*b^4*x^22*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(22*(a + b*x^3)) + (b^5*x^25*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(25*(a + b*x^3))]
    @test_int [x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 4, (a^2*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(18*b^3) - (2*a*(a + b*x^3)^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(21*b^3) + ((a + b*x^3)^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(24*b^3)]
    @test_int [x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3)) + (5*a^4*b*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (5*a^3*b^2*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (10*a^2*b^3*x^17*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*(a + b*x^3)) + (a*b^4*x^20*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3)) + (b^5*x^23*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(23*(a + b*x^3))]
    @test_int [x^6*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (a^4*b*x^10*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (10*a^3*b^2*x^13*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*(a + b*x^3)) + (5*a^2*b^3*x^16*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3)) + (5*a*b^4*x^19*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*(a + b*x^3)) + (b^5*x^22*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(22*(a + b*x^3))]
    @test_int [x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 4, -(a*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(18*b^2) + ((a + b*x^3)^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(21*b^2)]
    @test_int [x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(5*(a + b*x^3)) + (5*a^4*b*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3)) + (10*a^3*b^2*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (5*a^2*b^3*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (5*a*b^4*x^17*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*(a + b*x^3)) + (b^5*x^20*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(20*(a + b*x^3))]
    @test_int [x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3)) + (5*a^4*b*x^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (a^3*b^2*x^10*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (10*a^2*b^3*x^13*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*(a + b*x^3)) + (5*a*b^4*x^16*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(16*(a + b*x^3)) + (b^5*x^19*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*(a + b*x^3))]
    @test_int [x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 2, ((a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(18*b)]
    @test_int [x^1*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (a^4*b*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a^3*b^2*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3)) + (10*a^2*b^3*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (5*a*b^4*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*(a + b*x^3)) + (b^5*x^17*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*(a + b*x^3))]
    @test_int [x^0*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a^4*b*x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(4*(a + b*x^3)^5) + (10*a^3*b^2*x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(7*(a + b*x^3)^5) + (a^2*b^3*x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a*b^4*x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(13*(a + b*x^3)^5) + (b^5*x^16*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(16*(a + b*x^3)^5)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^1, x, 4, (5*a^4*b*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (5*a^3*b^2*x^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (10*a^2*b^3*x^9*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*(a + b*x^3)) + (5*a*b^4*x^12*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(12*(a + b*x^3)) + (b^5*x^15*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(15*(a + b*x^3)) + (a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^2, x, 3, -((a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3))) + (5*a^4*b*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (2*a^3*b^2*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a^2*b^3*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3)) + (5*a*b^4*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3)) + (b^5*x^14*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^3, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^2*(a + b*x^3)) + (5*a^4*b*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a^3*b^2*x^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (10*a^2*b^3*x^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (a*b^4*x^10*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (b^5*x^13*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^4, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^3*(a + b*x^3)) + (10*a^3*b^2*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (5*a^2*b^3*x^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (5*a*b^4*x^9*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*(a + b*x^3)) + (b^5*x^12*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(12*(a + b*x^3)) + (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^5, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^4*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3)) + (5*a^3*b^2*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (2*a^2*b^3*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a*b^4*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3)) + (b^5*x^11*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^6, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(5*x^5*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^2*(a + b*x^3)) + (10*a^3*b^2*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a^2*b^3*x^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (5*a*b^4*x^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3)) + (b^5*x^10*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(10*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^7, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(6*x^6*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^3*(a + b*x^3)) + (10*a^2*b^3*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (5*a*b^4*x^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(6*(a + b*x^3)) + (b^5*x^9*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*(a + b*x^3)) + (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^8, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^4*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3)) + (5*a^2*b^3*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (a*b^4*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (b^5*x^8*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*x^8*(a + b*x^3)) - (a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^5*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^2*(a + b*x^3)) + (10*a^2*b^3*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (5*a*b^4*x^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3)) + (b^5*x^7*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^10, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*x^9*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(6*x^6*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^3*(a + b*x^3)) + (5*a*b^4*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (b^5*x^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(6*(a + b*x^3)) + (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^11, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(10*x^10*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^4*(a + b*x^3)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3)) + (5*a*b^4*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3)) + (b^5*x^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(5*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^12, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*x^11*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*x^8*(a + b*x^3)) - (2*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^5*(a + b*x^3)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^2*(a + b*x^3)) + (5*a*b^4*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3) + (b^5*x^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^13, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(12*x^12*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*x^9*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^6*(a + b*x^3)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^3*(a + b*x^3)) + (b^5*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*(a + b*x^3)) + (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^14, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*x^13*(a + b*x^3)) - (a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^10*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^4*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3)) + (b^5*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^15, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*x^14*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*x^11*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^8*(a + b*x^3)) - (2*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^5*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^2*(a + b*x^3)) + (b^5*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^16, x, 4, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(15*x^15*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(12*x^12*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(9*x^9*(a + b*x^3)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^6*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(3*x^3*(a + b*x^3)) + (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)*log(x))/(a + b*x^3)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(16*x^16*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*x^13*(a + b*x^3)) - (a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^10*(a + b*x^3)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^4*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^18, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*x^17*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(14*x^14*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*x^11*(a + b*x^3)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^8*(a + b*x^3)) - (a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^5*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^2*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^19, x, 2, -((a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(18*a*x^18)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^20, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*x^19*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(16*x^16*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*x^13*(a + b*x^3)) - (a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(x^10*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^4*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^21, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(20*x^20*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*x^17*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^14*(a + b*x^3)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*x^11*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*x^8*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(5*x^5*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^22, x, 4, -(((a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(21*a*x^21)) + (b*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(126*a^2*x^18)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^23, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(22*x^22*(a + b*x^3)) - (5*a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(19*x^19*(a + b*x^3)) - (5*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*x^16*(a + b*x^3)) - (10*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(13*x^13*(a + b*x^3)) - (a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(2*x^10*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^7*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24, x, 3, -(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(23*x^23*(a + b*x^3)) - (a^4*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(4*x^20*(a + b*x^3)) - (10*a^3*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(17*x^17*(a + b*x^3)) - (5*a^2*b^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(7*x^14*(a + b*x^3)) - (5*a*b^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(11*x^11*(a + b*x^3)) - (b^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(8*x^8*(a + b*x^3))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25, x, 5, -((a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(24*a*x^24) + (b*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(84*a^2*x^21) - (b^2*(a + b*x^3)^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(504*a^3*x^18)]

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

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

    @test_int [x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 9, x^2/(9*a*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x^2/(6*b*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(4/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(4/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + ((a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(54*a^(4/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^3/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 9, x/(18*a*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x/(6*b*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(5/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + ((a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(5/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(54*a^(5/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 2, -(1/(6*b*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)))]
    @test_int [x^1/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 9, (2*x^2)/(9*a^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x^2/(6*a*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (2*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(7/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (2*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(7/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + ((a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(27*a^(7/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^0/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 9, (x*(a + b*x^3))/(6*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*x*(a + b*x^3)^2)/(18*a^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) - (5*(a + b*x^3)^3*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(8/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*(a + b*x^3)^3*log(a^(1/3) + b^(1/3)*x))/(27*a^(8/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) - (5*(a + b*x^3)^3*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(54*a^(8/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)), x, 4, 1/(3*a^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(6*a*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + ((a + b*x^3)*log(x))/(a^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*log(a + b*x^3))/(3*a^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)), x, 10, 7/(18*a^2*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(6*a*x*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (14*(a + b*x^3))/(9*a^3*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (14*b^(1/3)*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(10/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (14*b^(1/3)*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(10/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (7*b^(1/3)*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(27*a^(10/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)), x, 10, 4/(9*a^2*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(6*a*x^2*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (10*(a + b*x^3))/(9*a^3*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (20*b^(2/3)*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(11/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (20*b^(2/3)*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(11/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (10*b^(2/3)*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(27*a^(11/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)), x, 4, (-2*b)/(3*a^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - b/(6*a^2*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (a + b*x^3)/(3*a^3*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (3*b*(a + b*x^3)*log(x))/(a^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (b*(a + b*x^3)*log(a + b*x^3))/(a^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]

    @test_int [x^6/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 11, (5*x)/(486*a^2*b^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x^4/(12*b*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x/(27*b^2*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x/(162*a*b^2*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (5*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(8/3)*b^(7/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (5*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(8/3)*b^(7/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (5*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(1458*a^(8/3)*b^(7/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 4, a/(12*b^2*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - 1/(9*b^2*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 11, (7*x^2)/(243*a^3*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x^2/(12*b*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x^2/(54*a*b*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (7*x^2)/(324*a^2*b*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (7*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(10/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (7*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(10/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (7*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(1458*a^(10/3)*b^(5/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^3/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 11, (5*x)/(243*a^3*b*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - x/(12*b*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x/(108*a*b*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x/(81*a^2*b*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (10*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(11/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (10*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(11/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (5*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(729*a^(11/3)*b^(4/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 2, -(1/(12*b*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)))]
    @test_int [x^1/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 11, (35*x^2)/(243*a^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + x^2/(12*a*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (5*x^2)/(54*a^2*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (35*x^2)/(324*a^3*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (35*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(13/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (35*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(13/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (35*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(1458*a^(13/3)*b^(2/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [x^0/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 11, (x*(a + b*x^3))/(12*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^2)/(108*a^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^3)/(81*a^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (55*x*(a + b*x^3)^4)/(243*a^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) - (110*(a + b*x^3)^5*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (110*(a + b*x^3)^5*log(a^(1/3) + b^(1/3)*x))/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) - (55*(a + b*x^3)^5*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))]
    @test_int [1/(x^1*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)), x, 4, 1/(3*a^4*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(12*a*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(9*a^2*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(6*a^3*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + ((a + b*x^3)*log(x))/(a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - ((a + b*x^3)*log(a + b*x^3))/(3*a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)), x, 12, 455/(972*a^4*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(12*a*x*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 13/(108*a^2*x*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 65/(324*a^3*x*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (455*(a + b*x^3))/(243*a^5*x*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (455*b^(1/3)*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(16/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (455*b^(1/3)*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(16/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (455*b^(1/3)*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(1458*a^(16/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)), x, 12, 154/(243*a^4*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 1/(12*a*x^2*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 7/(54*a^2*x^2*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + 77/(324*a^3*x^2*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (385*(a + b*x^3))/(243*a^5*x^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (770*b^(2/3)*(a + b*x^3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(243*sqrt(3)*a^(17/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (770*b^(2/3)*(a + b*x^3)*log(a^(1/3) + b^(1/3)*x))/(729*a^(17/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (385*b^(2/3)*(a + b*x^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(729*a^(17/3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)), x, 4, (-4*b)/(3*a^5*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - b/(12*a^2*(a + b*x^3)^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (2*b)/(9*a^3*(a + b*x^3)^2*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - b/(2*a^4*(a + b*x^3)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (a + b*x^3)/(3*a^5*x^3*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) - (5*b*(a + b*x^3)*log(x))/(a^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6)) + (5*b*(a + b*x^3)*log(a + b*x^3))/(3*a^6*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a^2+2*a*b*x^3+b^2*x^6)^p=#

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a^2+2*a*b*x^3+b^2*x^6)^(p/2)=#

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a^2+2*a*b*x^2+b^3*x^6)^p*with*m*symbolic=#

    @test_int [(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 3, (a^5*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d*(1 + m)*(a + b*x^3)) + (5*a^4*b*(d*x)^(4 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^4*(4 + m)*(a + b*x^3)) + (10*a^3*b^2*(d*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^7*(7 + m)*(a + b*x^3)) + (10*a^2*b^3*(d*x)^(10 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^10*(10 + m)*(a + b*x^3)) + (5*a*b^4*(d*x)^(13 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^13*(13 + m)*(a + b*x^3)) + (b^5*(d*x)^(16 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^16*(16 + m)*(a + b*x^3))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 3, (a^3*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d*(1 + m)*(a + b*x^3)) + (3*a^2*b*(d*x)^(4 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^4*(4 + m)*(a + b*x^3)) + (3*a*b^2*(d*x)^(7 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^7*(7 + m)*(a + b*x^3)) + (b^3*(d*x)^(10 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^10*(10 + m)*(a + b*x^3))]
    @test_int [(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(1/2), x, 3, (a*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d*(1 + m)*(a + b*x^3)) + (b*(d*x)^(4 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))/(d^4*(4 + m)*(a + b*x^3))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^3 + b^2*x^6)^(1/2), x, 2, ((d*x)^(1 + m)*(a + b*x^3)*HypergeometricFunctions._₂F₁(1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)))/(a*d*(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2), x, 2, ((d*x)^(1 + m)*(a + b*x^3)*HypergeometricFunctions._₂F₁(3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)))/(a^3*d*(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]
    @test_int [(d*x)^m/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2), x, 2, ((d*x)^(1 + m)*(a + b*x^3)*HypergeometricFunctions._₂F₁(5, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)))/(a^5*d*(1 + m)*sqrt(a^2 + 2*a*b*x^3 + b^2*x^6))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a^2+2*a*b*x^2+b^3*x^6)^p*with*p*symbolic=#

    @test_int [(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 2, ((d*x)^(1 + m)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁((1 + m)/3, -2*p, (4 + m)/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^(2*p)*(d*(1 + m)))]

    @test_int [x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 4, -((a^3*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^4*(1 + 2*p))) + (a^2*(a + b*x^3)^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(2*b^4*(1 + p)) - (a*(a + b*x^3)^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(b^4*(3 + 2*p)) + ((a + b*x^3)^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(6*b^4*(2 + p))]
    @test_int [x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 4, (a^2*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^3*(1 + 2*p)) - (a*(a + b*x^3)^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^3*(1 + p)) + ((a + b*x^3)^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^3*(3 + 2*p))]
    @test_int [x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 4, -((a*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^2*(1 + 2*p))) + ((a + b*x^3)^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(6*b^2*(1 + p))]
    @test_int [x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 2, ((1/5)*x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(5/3, -2*p, 8/3, -((b*x^3)/a)))/(1 + (b*x^3)/a)^(2*p)]
    @test_int [x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 2, ((1/4)*x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(4/3, -2*p, 7/3, -((b*x^3)/a)))/(1 + (b*x^3)/a)^(2*p)]
    @test_int [x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 2, ((a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b*(1 + 2*p))]
    @test_int [x^1*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 2, (x^2*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(1, 5/3 + 2*p, 5/3, -((b*x^3)/a)))/(2*a), ((1/2)*x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(2/3, -2*p, 5/3, -((b*x^3)/a)))/(1 + (b*x^3)/a)^(2*p)]
    @test_int [x^0*(a^2 + 2*a*b*x^3 + b^2*x^6)^p, x, 3, (x*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(1, 4/3 + 2*p, 4/3, -((b*x^3)/a)))/a, (x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(1/3, -2*p, 4/3, -((b*x^3)/a)))/(1 + (b*x^3)/a)^(2*p)]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^1, x, 3, -(((a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(1, 1 + 2*p, 2*(1 + p), 1 + (b*x^3)/a))/(3*a*(1 + 2*p)))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^2, x, 2, -(((a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(-(1/3), -2*p, 2/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^(2*p)*x))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^3, x, 2, -(((a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(-(2/3), -2*p, 1/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^(2*p)*(2*x^2)))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^4, x, 3, (b*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(2, 1 + 2*p, 2*(1 + p), 1 + (b*x^3)/a))/(3*a^2*(1 + 2*p))]
    @test_int [(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^5, x, 2, -(((a^2 + 2*a*b*x^3 + b^2*x^6)^p*HypergeometricFunctions._₂F₁(-(4/3), -2*p, -(1/3), -((b*x^3)/a)))/((1 + (b*x^3)/a)^(2*p)*(4*x^4)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^3+c*x^6)^p=#

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

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

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

    @test_int [x^7/(a + b*x^3 + c*x^6), x, 14, x^2/(2*c) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*c^(5/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*c^(5/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*c^(5/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*c^(5/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)) - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*c^(5/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*c^(5/3)*(b + sqrt(b^2 - 4*a*c))^(1/3))]
    @test_int [x^6/(a + b*x^3 + c*x^6), x, 14, x/c + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)) - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3))]
    @test_int [x^4/(a + b*x^3 + c*x^6), x, 13, ((b - sqrt(b^2 - 4*a*c))^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*c^(2/3)*sqrt(b^2 - 4*a*c)) - ((b + sqrt(b^2 - 4*a*c))^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*c^(2/3)*sqrt(b^2 - 4*a*c)) + ((b - sqrt(b^2 - 4*a*c))^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*c^(2/3)*sqrt(b^2 - 4*a*c)) - ((b + sqrt(b^2 - 4*a*c))^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*c^(2/3)*sqrt(b^2 - 4*a*c)) - ((b - sqrt(b^2 - 4*a*c))^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*c^(2/3)*sqrt(b^2 - 4*a*c)) + ((b + sqrt(b^2 - 4*a*c))^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*c^(2/3)*sqrt(b^2 - 4*a*c))]
    @test_int [x^3/(a + b*x^3 + c*x^6), x, 13, ((b - sqrt(b^2 - 4*a*c))^(1/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(1/3)*sqrt(b^2 - 4*a*c)) - ((b + sqrt(b^2 - 4*a*c))^(1/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(1/3)*sqrt(b^2 - 4*a*c)) - ((b - sqrt(b^2 - 4*a*c))^(1/3)*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(1/3)*sqrt(b^2 - 4*a*c)) + ((b + sqrt(b^2 - 4*a*c))^(1/3)*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(1/3)*sqrt(b^2 - 4*a*c)) + ((b - sqrt(b^2 - 4*a*c))^(1/3)*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(1/3)*sqrt(b^2 - 4*a*c)) - ((b + sqrt(b^2 - 4*a*c))^(1/3)*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(1/3)*sqrt(b^2 - 4*a*c))]
    @test_int [x^1/(a + b*x^3 + c*x^6), x, 13, -((2^(1/3)*c^(1/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(3)*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(1/3))) + (2^(1/3)*c^(1/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(3)*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(1/3)) - (2^(1/3)*c^(1/3)*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(1/3)) + (2^(1/3)*c^(1/3)*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(1/3)) + (c^(1/3)*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(3*2^(2/3)*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(1/3)) - (c^(1/3)*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(3*2^(2/3)*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(1/3))]
    @test_int [x^0/(a + b*x^3 + c*x^6), x, 13, -((2^(2/3)*c^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(3)*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3))) + (2^(2/3)*c^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(3)*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3)) + (2^(2/3)*c^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3)) - (2^(2/3)*c^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3)) - (c^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(3*2^(1/3)*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3)) + (c^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(3*2^(1/3)*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3))]
    @test_int [1/(x^2*(a + b*x^3 + c*x^6)), x, 14, -(1/(a*x)) + (c^(1/3)*(1 + b/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*a*(b - sqrt(b^2 - 4*a*c))^(1/3)) + (c^(1/3)*(1 - b/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(2/3)*sqrt(3)*a*(b + sqrt(b^2 - 4*a*c))^(1/3)) + (c^(1/3)*(1 + b/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*a*(b - sqrt(b^2 - 4*a*c))^(1/3)) + (c^(1/3)*(1 - b/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(2/3)*a*(b + sqrt(b^2 - 4*a*c))^(1/3)) - (c^(1/3)*(1 + b/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*a*(b - sqrt(b^2 - 4*a*c))^(1/3)) - (c^(1/3)*(1 - b/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(2/3)*a*(b + sqrt(b^2 - 4*a*c))^(1/3))]
    @test_int [1/(x^3*(a + b*x^3 + c*x^6)), x, 14, -(1/(2*a*x^2)) + (c^(2/3)*(1 + b/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*a*(b - sqrt(b^2 - 4*a*c))^(2/3)) + (c^(2/3)*(1 - b/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*a*(b + sqrt(b^2 - 4*a*c))^(2/3)) - (c^(2/3)*(1 + b/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*a*(b - sqrt(b^2 - 4*a*c))^(2/3)) - (c^(2/3)*(1 - b/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*a*(b + sqrt(b^2 - 4*a*c))^(2/3)) + (c^(2/3)*(1 + b/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*a*(b - sqrt(b^2 - 4*a*c))^(2/3)) + (c^(2/3)*(1 - b/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*a*(b + sqrt(b^2 - 4*a*c))^(2/3))]

    @test_int [x^11/(3 + 4*x^3 + x^6), x, 6, -((4*x^3)/3) + x^6/6 - (1/6)*log(1 + x^3) + (9/2)*log(3 + x^3)]
    @test_int [x^8/(3 + 4*x^3 + x^6), x, 5, x^3/3 + (1/6)*log(1 + x^3) - (3/2)*log(3 + x^3)]
    @test_int [x^5/(3 + 4*x^3 + x^6), x, 4, (-(1/6))*log(1 + x^3) + (1/2)*log(3 + x^3)]
    @test_int [x^2/(3 + 4*x^3 + x^6), x, 4, (-(1/3))*atanh(2 + x^3), (1/6)*log(1 + x^3) - (1/6)*log(3 + x^3)]
    @test_int [1/(x^1*(3 + 4*x^3 + x^6)), x, 6, log(x)/3 - (1/6)*log(1 + x^3) + (1/18)*log(3 + x^3)]
    @test_int [1/(x^4*(3 + 4*x^3 + x^6)), x, 4, -(1/(9*x^3)) - (4*log(x))/9 + (1/6)*log(1 + x^3) - (1/54)*log(3 + x^3)]
    @test_int [1/(x^7*(3 + 4*x^3 + x^6)), x, 4, -(1/(18*x^6)) + 4/(27*x^3) + (13*log(x))/27 - (1/6)*log(1 + x^3) + (1/162)*log(3 + x^3)]

    @test_int [x^10/(3 + 4*x^3 + x^6), x, 15, -2*x^2 + x^5/5 + atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - (9/2)*3^(1/6)*atan((3^(1/3) - 2*x)/3^(5/6)) + (1/6)*log(1 + x) - (3/2)*3^(2/3)*log(3^(1/3) + x) - (1/12)*log(1 - x + x^2) + (3/4)*3^(2/3)*log(3^(2/3) - 3^(1/3)*x + x^2)]
    @test_int [x^9/(3 + 4*x^3 + x^6), x, 15, -4*x + x^4/4 + atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - (3/2)*3^(5/6)*atan((3^(1/3) - 2*x)/3^(5/6)) - (1/6)*log(1 + x) + (3/2)*3^(1/3)*log(3^(1/3) + x) + (1/12)*log(1 - x + x^2) - (3/4)*3^(1/3)*log(3^(2/3) - 3^(1/3)*x + x^2)]
    @test_int [x^7/(3 + 4*x^3 + x^6), x, 14, x^2/2 - atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) + (3/2)*3^(1/6)*atan((3^(1/3) - 2*x)/3^(5/6)) - (1/6)*log(1 + x) + (1/2)*3^(2/3)*log(3^(1/3) + x) + (1/12)*log(1 - x + x^2) - (1/4)*3^(2/3)*log(3^(2/3) - 3^(1/3)*x + x^2)]
    @test_int [x^6/(3 + 4*x^3 + x^6), x, 14, x - atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) + (1/2)*3^(5/6)*atan((3^(1/3) - 2*x)/3^(5/6)) + (1/6)*log(1 + x) - (1/2)*3^(1/3)*log(3^(1/3) + x) - (1/12)*log(1 - x + x^2) + (1/4)*3^(1/3)*log(3^(2/3) - 3^(1/3)*x + x^2)]
    @test_int [x^4/(3 + 4*x^3 + x^6), x, 13, atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - (1/2)*3^(1/6)*atan((3^(1/3) - 2*x)/3^(5/6)) + (1/6)*log(1 + x) - log(3^(1/3) + x)/(2*3^(1/3)) - (1/12)*log(1 - x + x^2) + log(3^(2/3) - 3^(1/3)*x + x^2)/(4*3^(1/3))]
    @test_int [x^3/(3 + 4*x^3 + x^6), x, 13, atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - atan((3^(1/3) - 2*x)/3^(5/6))/(2*3^(1/6)) - (1/6)*log(1 + x) + log(3^(1/3) + x)/(2*3^(2/3)) + (1/12)*log(1 - x + x^2) - log(3^(2/3) - 3^(1/3)*x + x^2)/(4*3^(2/3))]
    @test_int [x^1/(3 + 4*x^3 + x^6), x, 13, -(atan((1 - 2*x)/sqrt(3))/(2*sqrt(3))) + atan((3^(1/3) - 2*x)/3^(5/6))/(2*3^(5/6)) - (1/6)*log(1 + x) + log(3^(1/3) + x)/(6*3^(1/3)) + (1/12)*log(1 - x + x^2) - log(3^(2/3) - 3^(1/3)*x + x^2)/(12*3^(1/3))]
    @test_int [x^0/(3 + 4*x^3 + x^6), x, 13, -(atan((1 - 2*x)/sqrt(3))/(2*sqrt(3))) + atan((3^(1/3) - 2*x)/3^(5/6))/(6*3^(1/6)) + (1/6)*log(1 + x) - log(3^(1/3) + x)/(6*3^(2/3)) - (1/12)*log(1 - x + x^2) + log(3^(2/3) - 3^(1/3)*x + x^2)/(12*3^(2/3))]
    @test_int [1/(x^2*(3 + 4*x^3 + x^6)), x, 14, -(1/(3*x)) + atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - atan((3^(1/3) - 2*x)/3^(5/6))/(6*3^(5/6)) + (1/6)*log(1 + x) - log(3^(1/3) + x)/(18*3^(1/3)) - (1/12)*log(1 - x + x^2) + log(3^(2/3) - 3^(1/3)*x + x^2)/(36*3^(1/3))]
    @test_int [1/(x^3*(3 + 4*x^3 + x^6)), x, 14, -(1/(6*x^2)) + atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) - atan((3^(1/3) - 2*x)/3^(5/6))/(18*3^(1/6)) - (1/6)*log(1 + x) + log(3^(1/3) + x)/(18*3^(2/3)) + (1/12)*log(1 - x + x^2) - log(3^(2/3) - 3^(1/3)*x + x^2)/(36*3^(2/3))]
    @test_int [1/(x^5*(3 + 4*x^3 + x^6)), x, 15, -(1/(12*x^4)) + 4/(9*x) - atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) + atan((3^(1/3) - 2*x)/3^(5/6))/(18*3^(5/6)) - (1/6)*log(1 + x) + log(3^(1/3) + x)/(54*3^(1/3)) + (1/12)*log(1 - x + x^2) - log(3^(2/3) - 3^(1/3)*x + x^2)/(108*3^(1/3))]
    @test_int [1/(x^6*(3 + 4*x^3 + x^6)), x, 15, -(1/(15*x^5)) + 2/(9*x^2) - atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) + atan((3^(1/3) - 2*x)/3^(5/6))/(54*3^(1/6)) + (1/6)*log(1 + x) - log(3^(1/3) + x)/(54*3^(2/3)) - (1/12)*log(1 - x + x^2) + log(3^(2/3) - 3^(1/3)*x + x^2)/(108*3^(2/3))]

    @test_int [x^6/(1 - x^3 + x^6), x, 14, x + ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 - im*sqrt(3))^(2/3)) - ((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 + im*sqrt(3))^(2/3)) + ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 - im*sqrt(3))^(2/3)) + ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 + im*sqrt(3))^(2/3)) - ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 - im*sqrt(3))^(2/3)) - ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 + im*sqrt(3))^(2/3))]
    @test_int [x^5/(1 - x^3 + x^6), x, 5, -(atan((1 - 2*x^3)/sqrt(3))/(3*sqrt(3))) + (1/6)*log(1 - x^3 + x^6)]
    @test_int [x^4/(1 - x^3 + x^6), x, 13, ((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 - im*sqrt(3))^(1/3)) - ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 + im*sqrt(3))^(1/3)) + ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 - im*sqrt(3))^(1/3)) + ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 + im*sqrt(3))^(1/3)) - ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 - im*sqrt(3))^(1/3)) - ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 + im*sqrt(3))^(1/3))]
    @test_int [x^3/(1 - x^3 + x^6), x, 13, -(((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 - im*sqrt(3))^(2/3))) + ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 + im*sqrt(3))^(2/3)) + ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 - im*sqrt(3))^(2/3)) + ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 + im*sqrt(3))^(2/3)) - ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 - im*sqrt(3))^(2/3)) - ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 + im*sqrt(3))^(2/3))]
    @test_int [x^2/(1 - x^3 + x^6), x, 3, -((2*atan((1 - 2*x^3)/sqrt(3)))/(3*sqrt(3)))]
    @test_int [x^1/(1 - x^3 + x^6), x, 13, (im*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*((1/2)*(1 - im*sqrt(3)))^(1/3)) - (im*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*((1/2)*(1 + im*sqrt(3)))^(1/3)) + (im*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(3*sqrt(3)*((1/2)*(1 - im*sqrt(3)))^(1/3)) - (im*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(3*sqrt(3)*((1/2)*(1 + im*sqrt(3)))^(1/3)) - (im*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(2/3)*sqrt(3)*(1 - im*sqrt(3))^(1/3)) + (im*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(2/3)*sqrt(3)*(1 + im*sqrt(3))^(1/3))]
    @test_int [x^0/(1 - x^3 + x^6), x, 13, (-(1/3))*(-1)^(13/18)*atan((1 + 2*(-1)^(1/9)*x)/sqrt(3)) + (1/3)*(-1)^(5/18)*atan((1 - 2*(-1)^(8/9)*x)/sqrt(3)) - ((-1)^(5/18)*(log(2) + 3*log((-1)^(1/9) - x)))/(9*sqrt(3)) + ((-1)^(13/18)*log((-2^(1/3))*((-1)^(8/9) + x)))/(3*sqrt(3)) - ((-1)^(13/18)*log((-2^(2/3))*((-1)^(7/9) + ((-1)^(8/9) - x)*x)))/(6*sqrt(3)) + ((-1)^(5/18)*log(2^(2/3)*((-1)^(2/9) + x*((-1)^(1/9) + x))))/(6*sqrt(3)), -((im*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*((1/2)*(1 - im*sqrt(3)))^(2/3))) + (im*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*((1/2)*(1 + im*sqrt(3)))^(2/3)) + (im*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(3*sqrt(3)*((1/2)*(1 - im*sqrt(3)))^(2/3)) - (im*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(3*sqrt(3)*((1/2)*(1 + im*sqrt(3)))^(2/3)) - (im*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(1/3)*sqrt(3)*(1 - im*sqrt(3))^(2/3)) + (im*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(1/3)*sqrt(3)*(1 + im*sqrt(3))^(2/3))]
    @test_int [1/(x^1*(1 - x^3 + x^6)), x, 7, -(atan((1 - 2*x^3)/sqrt(3))/(3*sqrt(3))) + log(x) - (1/6)*log(1 - x^3 + x^6)]
    @test_int [1/(x^2*(1 - x^3 + x^6)), x, 14, -(1/x) + ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 - im*sqrt(3))^(1/3)) - ((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 + im*sqrt(3))^(1/3)) - ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 - im*sqrt(3))^(1/3)) - ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 + im*sqrt(3))^(1/3)) + ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 - im*sqrt(3))^(1/3)) + ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 + im*sqrt(3))^(1/3))]
    @test_int [1/(x^3*(1 - x^3 + x^6)), x, 14, -(1/(2*x^2)) - ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 - im*sqrt(3))^(2/3)) + ((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(1/3)*(1 + im*sqrt(3))^(2/3)) - ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 - im*sqrt(3))^(2/3)) - ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(1/3)*(1 + im*sqrt(3))^(2/3)) + ((3 - im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 - im*sqrt(3))^(2/3)) + ((3 + im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(1/3)*(1 + im*sqrt(3))^(2/3))]
    @test_int [1/(x^4*(1 - x^3 + x^6)), x, 8, -(1/(3*x^3)) + atan((1 - 2*x^3)/sqrt(3))/(3*sqrt(3)) + log(x) - (1/6)*log(1 - x^3 + x^6)]
    @test_int [1/(x^5*(1 - x^3 + x^6)), x, 16, -(1/(4*x^4)) - 1/x - ((im + sqrt(3))*atan((1 + (2*x)/((1/2)*(1 - im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 - im*sqrt(3))^(1/3)) + ((im - sqrt(3))*atan((1 + (2*x)/((1/2)*(1 + im*sqrt(3)))^(1/3))/sqrt(3)))/(3*2^(2/3)*(1 + im*sqrt(3))^(1/3)) - ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 - im*sqrt(3))^(1/3)) - ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(1/3) - 2^(1/3)*x))/(9*2^(2/3)*(1 + im*sqrt(3))^(1/3)) + ((3 + im*sqrt(3))*log((1 - im*sqrt(3))^(2/3) + (2*(1 - im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 - im*sqrt(3))^(1/3)) + ((3 - im*sqrt(3))*log((1 + im*sqrt(3))^(2/3) + (2*(1 + im*sqrt(3)))^(1/3)*x + 2^(2/3)*x^2))/(18*2^(2/3)*(1 + im*sqrt(3))^(1/3))]

    @test_int [1/(2 + x^3 + x^6), x, 13, (im*atan((1 - (2*x)/((1/2)*(1 - im*sqrt(7)))^(1/3))/sqrt(3)))/(sqrt(21)*((1/2)*(1 - im*sqrt(7)))^(2/3)) - (im*atan((1 - (2*x)/((1/2)*(1 + im*sqrt(7)))^(1/3))/sqrt(3)))/(sqrt(21)*((1/2)*(1 + im*sqrt(7)))^(2/3)) - (im*log((1 - im*sqrt(7))^(1/3) + 2^(1/3)*x))/(3*sqrt(7)*((1/2)*(1 - im*sqrt(7)))^(2/3)) + (im*log((1 + im*sqrt(7))^(1/3) + 2^(1/3)*x))/(3*sqrt(7)*((1/2)*(1 + im*sqrt(7)))^(2/3)) + (im*log((1 - im*sqrt(7))^(2/3) - (2*(1 - im*sqrt(7)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(1/3)*sqrt(7)*(1 - im*sqrt(7))^(2/3)) - (im*log((1 + im*sqrt(7))^(2/3) - (2*(1 + im*sqrt(7)))^(1/3)*x + 2^(2/3)*x^2))/(3*2^(1/3)*sqrt(7)*(1 + im*sqrt(7))^(2/3))]
    @test_int [x^2/(2 + x^3 + x^6), x, 3, (2*atan((1 + 2*x^3)/sqrt(7)))/(3*sqrt(7))]
    @test_int [x^3/(2 + x^3 + x^6), x, 13, -((im*((1/2)*(1 - im*sqrt(7)))^(1/3)*atan((1 - (2*x)/((1/2)*(1 - im*sqrt(7)))^(1/3))/sqrt(3)))/sqrt(21)) + (im*((1/2)*(1 + im*sqrt(7)))^(1/3)*atan((1 - (2*x)/((1/2)*(1 + im*sqrt(7)))^(1/3))/sqrt(3)))/sqrt(21) + ((7 + im*sqrt(7))*log((1 - im*sqrt(7))^(1/3) + 2^(1/3)*x))/(21*2^(1/3)*(1 - im*sqrt(7))^(2/3)) + ((7 - im*sqrt(7))*log((1 + im*sqrt(7))^(1/3) + 2^(1/3)*x))/(21*2^(1/3)*(1 + im*sqrt(7))^(2/3)) - ((7 + im*sqrt(7))*log((1 - im*sqrt(7))^(2/3) - (2*(1 - im*sqrt(7)))^(1/3)*x + 2^(2/3)*x^2))/(42*2^(1/3)*(1 - im*sqrt(7))^(2/3)) - ((7 - im*sqrt(7))*log((1 + im*sqrt(7))^(2/3) - (2*(1 + im*sqrt(7)))^(1/3)*x + 2^(2/3)*x^2))/(42*2^(1/3)*(1 + im*sqrt(7))^(2/3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^3+c*x^6)^(p/2)=#

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

    @test_int [sqrt(a + b*x^3 + c*x^6)*x^14, x, 7, ((21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(1536*c^5) - (b*x^6*(a + b*x^3 + c*x^6)^(3/2))/(20*c^2) + (x^9*(a + b*x^3 + c*x^6)^(3/2))/(18*c) - ((7*b*(15*b^2 - 28*a*c) - 6*c*(21*b^2 - 20*a*c)*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(2880*c^4) - ((b^2 - 4*a*c)*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(3072*c^(11/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^11, x, 6, -((b*(7*b^2 - 12*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(384*c^4)) + (x^6*(a + b*x^3 + c*x^6)^(3/2))/(15*c) + ((35*b^2 - 32*a*c - 42*b*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(720*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(768*c^(9/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^8, x, 6, ((5*b^2 - 4*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(192*c^3) - (5*b*(a + b*x^3 + c*x^6)^(3/2))/(72*c^2) + (x^3*(a + b*x^3 + c*x^6)^(3/2))/(12*c) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(384*c^(7/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^5, x, 5, -((b*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(24*c^2)) + (a + b*x^3 + c*x^6)^(3/2)/(9*c) + (b*(b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(48*c^(5/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^2, x, 4, ((b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(12*c) - ((b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(24*c^(3/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^1, x, 7, (1/3)*sqrt(a + b*x^3 + c*x^6) - (1/3)*sqrt(a)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))) + (b*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(6*sqrt(c))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^4, x, 7, -(sqrt(a + b*x^3 + c*x^6)/(3*x^3)) - (b*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(6*sqrt(a)) + (1/3)*sqrt(c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^7, x, 4, -(((2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(12*a*x^6)) + ((b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(24*a^(3/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^10, x, 5, (b*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(24*a^2*x^6) - (a + b*x^3 + c*x^6)^(3/2)/(9*a*x^9) - (b*(b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(48*a^(5/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^13, x, 6, -(((5*b^2 - 4*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(192*a^3*x^6)) - (a + b*x^3 + c*x^6)^(3/2)/(12*a*x^12) + (5*b*(a + b*x^3 + c*x^6)^(3/2))/(72*a^2*x^9) + ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(384*a^(7/2))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^16, x, 7, (b*(7*b^2 - 12*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(384*a^4*x^6) - (a + b*x^3 + c*x^6)^(3/2)/(15*a*x^15) + (7*b*(a + b*x^3 + c*x^6)^(3/2))/(120*a^2*x^12) - ((35*b^2 - 32*a*c)*(a + b*x^3 + c*x^6)^(3/2))/(720*a^3*x^9) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(768*a^(9/2))]

    @test_int [sqrt(a + b*x^3 + c*x^6)*x^3, x, 2, (x^4*sqrt(a + b*x^3 + c*x^6)*AppellF1(4/3, -(1/2), -(1/2), 7/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(4*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^1, x, 2, (x^2*sqrt(a + b*x^3 + c*x^6)*AppellF1(2/3, -(1/2), -(1/2), 5/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [sqrt(a + b*x^3 + c*x^6)*x^0, x, 2, (x*sqrt(a + b*x^3 + c*x^6)*AppellF1(1/3, -(1/2), -(1/2), 4/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^2, x, 2, -((sqrt(a + b*x^3 + c*x^6)*AppellF1(-(1/3), -(1/2), -(1/2), 2/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(x*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))]
    @test_int [sqrt(a + b*x^3 + c*x^6)/x^3, x, 2, -((sqrt(a + b*x^3 + c*x^6)*AppellF1(-(2/3), -(1/2), -(1/2), 1/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*x^2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))]

    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^14, x, 8, -(((b^2 - 4*a*c)*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(16384*c^6)) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(6144*c^5) - (11*b*x^6*(a + b*x^3 + c*x^6)^(5/2))/(336*c^2) + (x^9*(a + b*x^3 + c*x^6)^(5/2))/(24*c) - ((3*b*(77*b^2 - 124*a*c) - 10*c*(33*b^2 - 28*a*c)*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(13440*c^4) + ((b^2 - 4*a*c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(32768*c^(13/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^11, x, 7, (b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(1024*c^5) - (b*(3*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*c^4) + (x^6*(a + b*x^3 + c*x^6)^(5/2))/(21*c) + ((21*b^2 - 16*a*c - 30*b*c*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(840*c^3) - (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(2048*c^(11/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^8, x, 7, -(((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(1536*c^4)) + ((7*b^2 - 4*a*c)*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*c^3) - (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*c^2) + (x^3*(a + b*x^3 + c*x^6)^(5/2))/(18*c) + ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(3072*c^(9/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^5, x, 6, (b*(b^2 - 4*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(128*c^3) - (b*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*c^2) + (a + b*x^3 + c*x^6)^(5/2)/(15*c) - (b*(b^2 - 4*a*c)^2*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(256*c^(7/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^2, x, 5, -(((b^2 - 4*a*c)*(b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(64*c^2)) + ((b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(24*c) + ((b^2 - 4*a*c)^2*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(128*c^(5/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^1, x, 8, ((b^2 + 8*a*c + 2*b*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(24*c) + (1/9)*(a + b*x^3 + c*x^6)^(3/2) - (1/3)*a^(3/2)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))) - (b*(b^2 - 12*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(48*c^(3/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^4, x, 8, (1/4)*(3*b + 2*c*x^3)*sqrt(a + b*x^3 + c*x^6) - (a + b*x^3 + c*x^6)^(3/2)/(3*x^3) - (1/2)*sqrt(a)*b*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))) + ((b^2 + 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(8*sqrt(c))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^7, x, 8, -(((b - 2*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(4*x^3)) - (a + b*x^3 + c*x^6)^(3/2)/(6*x^6) - ((b^2 + 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(8*sqrt(a)) + (1/2)*b*sqrt(c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^10, x, 8, -(((2*a*b + (b^2 + 8*a*c)*x^3)*sqrt(a + b*x^3 + c*x^6))/(24*a*x^6)) - (a + b*x^3 + c*x^6)^(3/2)/(9*x^9) + (b*(b^2 - 12*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(48*a^(3/2)) + (1/3)*c^(3/2)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^13, x, 5, ((b^2 - 4*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(64*a^2*x^6) - ((2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(24*a*x^12) - ((b^2 - 4*a*c)^2*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(128*a^(5/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^16, x, 6, -((b*(b^2 - 4*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(128*a^3*x^6)) + (b*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*a^2*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(15*a*x^15) + (b*(b^2 - 4*a*c)^2*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(256*a^(7/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^19, x, 7, ((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(1536*a^4*x^6) - ((7*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*a^3*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(18*a*x^18) + (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*a^2*x^15) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(3072*a^(9/2))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^22, x, 8, -((b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(2*a + b*x^3)*sqrt(a + b*x^3 + c*x^6))/(1024*a^5*x^6)) + (b*(3*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*a^4*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(21*a*x^21) + (b*(a + b*x^3 + c*x^6)^(5/2))/(28*a^2*x^18) - ((21*b^2 - 16*a*c)*(a + b*x^3 + c*x^6)^(5/2))/(840*a^3*x^15) + (b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(2048*a^(11/2))]

    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^3, x, 2, (a*x^4*sqrt(a + b*x^3 + c*x^6)*AppellF1(4/3, -(3/2), -(3/2), 7/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(4*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^1, x, 2, (a*x^2*sqrt(a + b*x^3 + c*x^6)*AppellF1(2/3, -(3/2), -(3/2), 5/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)*x^0, x, 2, (a*x*sqrt(a + b*x^3 + c*x^6)*AppellF1(1/3, -(3/2), -(3/2), 4/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^2, x, 2, -((a*sqrt(a + b*x^3 + c*x^6)*AppellF1(-(1/3), -(3/2), -(3/2), 2/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(x*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))]
    @test_int [(a + b*x^3 + c*x^6)^(3/2)/x^3, x, 2, -((a*sqrt(a + b*x^3 + c*x^6)*AppellF1(-(2/3), -(3/2), -(3/2), 1/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*x^2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))]

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

    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^14, x, 6, -((7*b*x^6*sqrt(a + b*x^3 + c*x^6))/(72*c^2)) + (x^9*sqrt(a + b*x^3 + c*x^6))/(12*c) - ((5*b*(21*b^2 - 44*a*c) - 2*c*(35*b^2 - 36*a*c)*x^3)*sqrt(a + b*x^3 + c*x^6))/(576*c^4) + ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(384*c^(9/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^11, x, 5, (x^6*sqrt(a + b*x^3 + c*x^6))/(9*c) + ((15*b^2 - 16*a*c - 10*b*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(72*c^3) - (b*(5*b^2 - 12*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(48*c^(7/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^8, x, 5, -((b*sqrt(a + b*x^3 + c*x^6))/(4*c^2)) + (x^3*sqrt(a + b*x^3 + c*x^6))/(6*c) + ((3*b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(24*c^(5/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^5, x, 4, sqrt(a + b*x^3 + c*x^6)/(3*c) - (b*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(6*c^(3/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^2, x, 3, atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6)))/(3*sqrt(c))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^1, x, 3, -(atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6)))/(3*sqrt(a)))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^4, x, 4, -(sqrt(a + b*x^3 + c*x^6)/(3*a*x^3)) + (b*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(6*a^(3/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^7, x, 5, -(sqrt(a + b*x^3 + c*x^6)/(6*a*x^6)) + (b*sqrt(a + b*x^3 + c*x^6))/(4*a^2*x^3) - ((3*b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(24*a^(5/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^10, x, 6, -(sqrt(a + b*x^3 + c*x^6)/(9*a*x^9)) + (5*b*sqrt(a + b*x^3 + c*x^6))/(36*a^2*x^6) - ((15*b^2 - 16*a*c)*sqrt(a + b*x^3 + c*x^6))/(72*a^3*x^3) + (b*(5*b^2 - 12*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(48*a^(7/2))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^13, x, 7, -(sqrt(a + b*x^3 + c*x^6)/(12*a*x^12)) + (7*b*sqrt(a + b*x^3 + c*x^6))/(72*a^2*x^9) - ((35*b^2 - 36*a*c)*sqrt(a + b*x^3 + c*x^6))/(288*a^3*x^6) + (5*b*(21*b^2 - 44*a*c)*sqrt(a + b*x^3 + c*x^6))/(576*a^4*x^3) - ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(384*a^(9/2))]

    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^3, x, 2, (x^4*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(4/3, 1/2, 1/2, 7/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(4*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^1, x, 2, (x^2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(2/3, 1/2, 1/2, 5/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)*x^0, x, 2, (x*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/3, 1/2, 1/2, 4/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/sqrt(a + b*x^3 + c*x^6)]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^2, x, 2, -((sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/3), 1/2, 1/2, 2/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(x*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [1/sqrt(a + b*x^3 + c*x^6)/x^3, x, 2, -((sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(2/3), 1/2, 1/2, 1/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*x^2*sqrt(a + b*x^3 + c*x^6)))]

    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^14, x, 6, (2*x^9*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6)) - (2*b*x^6*sqrt(a + b*x^3 + c*x^6))/(3*c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 12*a*c)*x^3)*sqrt(a + b*x^3 + c*x^6))/(12*c^3*(b^2 - 4*a*c)) + ((5*b^2 - 4*a*c)*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(8*c^(7/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^11, x, 5, (2*x^6*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6)) + ((3*b^2 - 8*a*c - 2*b*c*x^3)*sqrt(a + b*x^3 + c*x^6))/(3*c^2*(b^2 - 4*a*c)) - (b*atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6))))/(2*c^(5/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^8, x, 5, (2*x^3*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6)) - (2*b*sqrt(a + b*x^3 + c*x^6))/(3*c*(b^2 - 4*a*c)) + atanh((b + 2*c*x^3)/(2*sqrt(c)*sqrt(a + b*x^3 + c*x^6)))/(3*c^(3/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^5, x, 2, (2*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^2, x, 2, -((2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^1, x, 5, (2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*sqrt(a + b*x^3 + c*x^6)) - atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6)))/(3*a^(3/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^4, x, 5, (2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^3*sqrt(a + b*x^3 + c*x^6)) - ((3*b^2 - 8*a*c)*sqrt(a + b*x^3 + c*x^6))/(3*a^2*(b^2 - 4*a*c)*x^3) + (b*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(2*a^(5/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^7, x, 6, (2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^6*sqrt(a + b*x^3 + c*x^6)) - ((5*b^2 - 12*a*c)*sqrt(a + b*x^3 + c*x^6))/(6*a^2*(b^2 - 4*a*c)*x^6) + (b*(15*b^2 - 52*a*c)*sqrt(a + b*x^3 + c*x^6))/(12*a^3*(b^2 - 4*a*c)*x^3) - ((5*b^2 - 4*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(8*a^(7/2))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^10, x, 7, (2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^9*sqrt(a + b*x^3 + c*x^6)) - ((7*b^2 - 16*a*c)*sqrt(a + b*x^3 + c*x^6))/(9*a^2*(b^2 - 4*a*c)*x^9) + (b*(35*b^2 - 116*a*c)*sqrt(a + b*x^3 + c*x^6))/(36*a^3*(b^2 - 4*a*c)*x^6) - ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2)*sqrt(a + b*x^3 + c*x^6))/(72*a^4*(b^2 - 4*a*c)*x^3) + (5*b*(7*b^2 - 12*a*c)*atanh((2*a + b*x^3)/(2*sqrt(a)*sqrt(a + b*x^3 + c*x^6))))/(48*a^(9/2))]

    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^3, x, 2, (x^4*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(4/3, 3/2, 3/2, 7/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(4*a*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^1, x, 2, (x^2*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(2/3, 3/2, 3/2, 5/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*a*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)*x^0, x, 2, (x*sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/3, 3/2, 3/2, 4/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(a*sqrt(a + b*x^3 + c*x^6))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^2, x, 2, -((sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(1/3), 3/2, 3/2, 2/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(a*x*sqrt(a + b*x^3 + c*x^6)))]
    @test_int [1/(a + b*x^3 + c*x^6)^(3/2)/x^3, x, 2, -((sqrt(1 + (2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(-(2/3), 3/2, 3/2, 1/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*a*x^2*sqrt(a + b*x^3 + c*x^6)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p*with*m*symbolic=#

    @test_int [(d*x)^m*(a + b*x^3 + c*x^6)^2, x, 2, (a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(4 + m))/(d^4*(4 + m)) + ((b^2 + 2*a*c)*(d*x)^(7 + m))/(d^7*(7 + m)) + (2*b*c*(d*x)^(10 + m))/(d^10*(10 + m)) + (c^2*(d*x)^(13 + m))/(d^13*(13 + m))]
    @test_int [(d*x)^m*(a + b*x^3 + c*x^6)^1, x, 2, (a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(4 + m))/(d^4*(4 + m)) + (c*(d*x)^(7 + m))/(d^7*(7 + m))]
    @test_int [(d*x)^m/(a + b*x^3 + c*x^6)^1, x, 3, (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/3, (4 + m)/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)) - (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/3, (4 + m)/3, -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m))]
    @test_int [(d*x)^m/(a + b*x^3 + c*x^6)^2, x, 4, ((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*d*(a + b*x^3 + c*x^6)) + (c*(b^2*(2 - m) + b*sqrt(b^2 - 4*a*c)*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/3, (4 + m)/3, -((2*c*x^3)/(b - sqrt(b^2 - 4*a*c)))))/(3*a*(b^2 - 4*a*c)^(3/2)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)) - (c*(b^2*(2 - m) - b*sqrt(b^2 - 4*a*c)*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/3, (4 + m)/3, -((2*c*x^3)/(b + sqrt(b^2 - 4*a*c)))))/(3*a*(b^2 - 4*a*c)^(3/2)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^3+c*x^6)^p*with*p*symbolic=#

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

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

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

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^4+c*x^8)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^4+c*x^8)^p*with*b^2-4*a*c=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^4+c*x^8)^p=#

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

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

    @test_int [x^m/(1 + 2*x^4 + x^8), x, 2, (x^(1 + m)*HypergeometricFunctions._₂F₁(2, (1 + m)/4, (5 + m)/4, -x^4))/(1 + m)]

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

    @test_int [x^8/(1 + 2*x^4 + x^8), x, 12, (5*x)/4 - x^5/(4*(1 + x^4)) + (5*atan(1 - sqrt(2)*x))/(8*sqrt(2)) - (5*atan(1 + sqrt(2)*x))/(8*sqrt(2)) + (5*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) - (5*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [x^6/(1 + 2*x^4 + x^8), x, 11, -(x^3/(4*(1 + x^4))) - (3*atan(1 - sqrt(2)*x))/(8*sqrt(2)) + (3*atan(1 + sqrt(2)*x))/(8*sqrt(2)) + (3*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) - (3*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [x^4/(1 + 2*x^4 + x^8), x, 11, -(x/(4*(1 + x^4))) - atan(1 - sqrt(2)*x)/(8*sqrt(2)) + atan(1 + sqrt(2)*x)/(8*sqrt(2)) - log(1 - sqrt(2)*x + x^2)/(16*sqrt(2)) + log(1 + sqrt(2)*x + x^2)/(16*sqrt(2))]
    @test_int [x^2/(1 + 2*x^4 + x^8), x, 11, x^3/(4*(1 + x^4)) - atan(1 - sqrt(2)*x)/(8*sqrt(2)) + atan(1 + sqrt(2)*x)/(8*sqrt(2)) + log(1 - sqrt(2)*x + x^2)/(16*sqrt(2)) - log(1 + sqrt(2)*x + x^2)/(16*sqrt(2))]
    @test_int [1/(1 + 2*x^4 + x^8), x, 11, x/(4*(1 + x^4)) - (3*atan(1 - sqrt(2)*x))/(8*sqrt(2)) + (3*atan(1 + sqrt(2)*x))/(8*sqrt(2)) - (3*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) + (3*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [1/(x^2*(1 + 2*x^4 + x^8)), x, 12, -(5/(4*x)) + 1/(4*x*(1 + x^4)) + (5*atan(1 - sqrt(2)*x))/(8*sqrt(2)) - (5*atan(1 + sqrt(2)*x))/(8*sqrt(2)) - (5*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) + (5*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [1/(x^4*(1 + 2*x^4 + x^8)), x, 12, -(7/(12*x^3)) + 1/(4*x^3*(1 + x^4)) + (7*atan(1 - sqrt(2)*x))/(8*sqrt(2)) - (7*atan(1 + sqrt(2)*x))/(8*sqrt(2)) + (7*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) - (7*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [1/(x^6*(1 + 2*x^4 + x^8)), x, 13, -(9/(20*x^5)) + 9/(4*x) + 1/(4*x^5*(1 + x^4)) - (9*atan(1 - sqrt(2)*x))/(8*sqrt(2)) + (9*atan(1 + sqrt(2)*x))/(8*sqrt(2)) + (9*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) - (9*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]
    @test_int [1/(x^8*(1 + 2*x^4 + x^8)), x, 13, -(11/(28*x^7)) + 11/(12*x^3) + 1/(4*x^7*(1 + x^4)) - (11*atan(1 - sqrt(2)*x))/(8*sqrt(2)) + (11*atan(1 + sqrt(2)*x))/(8*sqrt(2)) - (11*log(1 - sqrt(2)*x + x^2))/(16*sqrt(2)) + (11*log(1 + sqrt(2)*x + x^2))/(16*sqrt(2))]

    @test_int [x^m/(1 - 2*x^4 + x^8), x, 2, (x^(1 + m)*HypergeometricFunctions._₂F₁(2, (1 + m)/4, (5 + m)/4, x^4))/(1 + m)]

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

    @test_int [x^8/(1 - 2*x^4 + x^8), x, 6, (5*x)/4 + x^5/(4*(1 - x^4)) - (5*atan(x))/8 - (5*atanh(x))/8]
    @test_int [x^6/(1 - 2*x^4 + x^8), x, 5, x^3/(4*(1 - x^4)) + (3*atan(x))/8 - (3*atanh(x))/8]
    @test_int [x^4/(1 - 2*x^4 + x^8), x, 5, x/(4*(1 - x^4)) - atan(x)/8 - atanh(x)/8]
    @test_int [x^2/(1 - 2*x^4 + x^8), x, 5, x^3/(4*(1 - x^4)) - atan(x)/8 + atanh(x)/8]
    @test_int [1/(1 - 2*x^4 + x^8), x, 5, x/(4*(1 - x^4)) + (3*atan(x))/8 + (3*atanh(x))/8]
    @test_int [1/(x^2*(1 - 2*x^4 + x^8)), x, 6, -(5/(4*x)) + 1/(4*x*(1 - x^4)) - (5*atan(x))/8 + (5*atanh(x))/8]
    @test_int [1/(x^4*(1 - 2*x^4 + x^8)), x, 6, -(7/(12*x^3)) + 1/(4*x^3*(1 - x^4)) + (7*atan(x))/8 + (7*atanh(x))/8]
    @test_int [1/(x^6*(1 - 2*x^4 + x^8)), x, 7, -(9/(20*x^5)) - 9/(4*x) + 1/(4*x^5*(1 - x^4)) - (9*atan(x))/8 + (9*atanh(x))/8]
    @test_int [1/(x^8*(1 - 2*x^4 + x^8)), x, 7, -(11/(28*x^7)) - 11/(12*x^3) + 1/(4*x^7*(1 - x^4)) + (11*atan(x))/8 + (11*atanh(x))/8]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^4+c*x^8)^(p/2)=#

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^4+c*x^8)^p=#

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^4+c*x^8)^(p/2)=#

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^4+c*x^8)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^4+c*x^8)^p=#

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

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

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

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

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

    @test_int [x^m/(1 + x^4 + x^8), x, 3, (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, -((2*x^4)/(1 - im*sqrt(3)))))/(sqrt(3)*(im + sqrt(3))*(1 + m)) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, -((2*x^4)/(1 + im*sqrt(3)))))/(sqrt(3)*(im - sqrt(3))*(1 + m))]

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

    @test_int [x^8/(1 + x^4 + x^8), x, 20, x + atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) + 2*x) + (1/8)*log(1 - x + x^2) - (1/8)*log(1 + x + x^2) + log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]
    @test_int [x^6/(1 + x^4 + x^8), x, 9, -(atan((1 - 2*x)/sqrt(3))/(2*sqrt(3))) + atan((1 + 2*x)/sqrt(3))/(2*sqrt(3)) + log(1 - sqrt(3)*x + x^2)/(4*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(4*sqrt(3))]
    @test_int [x^4/(1 + x^4 + x^8), x, 19, atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) + 2*x) - (1/8)*log(1 - x + x^2) + (1/8)*log(1 + x + x^2) + log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]
    @test_int [x^2/(1 + x^4 + x^8), x, 19, atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) + 2*x) + (1/8)*log(1 - x + x^2) - (1/8)*log(1 + x + x^2) - log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) + log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]
    @test_int [x^0/(1 + x^4 + x^8), x, 9, -(atan((1 - 2*x)/sqrt(3))/(2*sqrt(3))) + atan((1 + 2*x)/sqrt(3))/(2*sqrt(3)) - log(1 - sqrt(3)*x + x^2)/(4*sqrt(3)) + log(1 + sqrt(3)*x + x^2)/(4*sqrt(3))]
    @test_int [1/(x^2*(1 + x^4 + x^8)), x, 20, -(1/x) + atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) + 2*x) - (1/8)*log(1 - x + x^2) + (1/8)*log(1 + x + x^2) - log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) + log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]
    @test_int [1/(x^4*(1 + x^4 + x^8)), x, 20, -(1/(3*x^3)) + atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) + 2*x) + (1/8)*log(1 - x + x^2) - (1/8)*log(1 + x + x^2) + log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]
    @test_int [1/(x^6*(1 + x^4 + x^8)), x, 12, -(1/(5*x^5)) + 1/x - atan((1 - 2*x)/sqrt(3))/(2*sqrt(3)) + atan((1 + 2*x)/sqrt(3))/(2*sqrt(3)) + log(1 - sqrt(3)*x + x^2)/(4*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(4*sqrt(3))]
    @test_int [1/(x^8*(1 + x^4 + x^8)), x, 22, -(1/(7*x^7)) + 1/(3*x^3) + atan((1 - 2*x)/sqrt(3))/(4*sqrt(3)) - (1/4)*atan(sqrt(3) - 2*x) - atan((1 + 2*x)/sqrt(3))/(4*sqrt(3)) + (1/4)*atan(sqrt(3) + 2*x) - (1/8)*log(1 - x + x^2) + (1/8)*log(1 + x + x^2) + log(1 - sqrt(3)*x + x^2)/(8*sqrt(3)) - log(1 + sqrt(3)*x + x^2)/(8*sqrt(3))]

    @test_int [x^m/(1 - x^4 + x^8), x, 3, (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 - im*sqrt(3))))/(sqrt(3)*(im + sqrt(3))*(1 + m)) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 + im*sqrt(3))))/(sqrt(3)*(im - sqrt(3))*(1 + m))]

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

    @test_int [x^8/(1 - x^4 + x^8), x, 20, x + atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) - atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) - atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) + atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) - (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 - sqrt(2 - sqrt(3))*x + x^2) + (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 + sqrt(2 - sqrt(3))*x + x^2) + (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 - sqrt(2 + sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 + sqrt(2 + sqrt(3))*x + x^2)]
    @test_int [x^6/(1 - x^4 + x^8), x, 19, -(atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6))) - atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) + atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6)) + atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) + log(1 - sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) - log(1 + sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) + log(1 - sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6)) - log(1 + sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6))]
    @test_int [x^4/(1 - x^4 + x^8), x, 19, atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) - atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) - atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) + atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) - log(1 - sqrt(2 - sqrt(3))*x + x^2)/(8*sqrt(3*(2 - sqrt(3)))) + log(1 + sqrt(2 - sqrt(3))*x + x^2)/(8*sqrt(3*(2 - sqrt(3)))) + log(1 - sqrt(2 + sqrt(3))*x + x^2)/(8*sqrt(3*(2 + sqrt(3)))) - log(1 + sqrt(2 + sqrt(3))*x + x^2)/(8*sqrt(3*(2 + sqrt(3))))]
    @test_int [x^2/(1 - x^4 + x^8), x, 19, (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3))) - (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3))) - (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3))) + (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3))) + log(1 - sqrt(2 - sqrt(3))*x + x^2)/(8*sqrt(3*(2 - sqrt(3)))) - log(1 + sqrt(2 - sqrt(3))*x + x^2)/(8*sqrt(3*(2 - sqrt(3)))) - log(1 - sqrt(2 + sqrt(3))*x + x^2)/(8*sqrt(3*(2 + sqrt(3)))) + log(1 + sqrt(2 + sqrt(3))*x + x^2)/(8*sqrt(3*(2 + sqrt(3))))]
    @test_int [x^0/(1 - x^4 + x^8), x, 19, -(atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6))) - atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) + atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6)) + atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) - log(1 - sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) + log(1 + sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) - log(1 - sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6)) + log(1 + sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6))]
    @test_int [1/(x^2*(1 - x^4 + x^8)), x, 22, -(1/x) + atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) - atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) - atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(4*sqrt(3*(2 - sqrt(3)))) + atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(4*sqrt(3*(2 + sqrt(3)))) + (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 - sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 + sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 - sqrt(2 + sqrt(3))*x + x^2) + (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 + sqrt(2 + sqrt(3))*x + x^2)]
    @test_int [1/(x^4*(1 - x^4 + x^8)), x, 20, -(1/(3*x^3)) - (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3))) + (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3))) + (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3))) - (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3))) + (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 - sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 + sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 - sqrt(2 + sqrt(3))*x + x^2) + (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 + sqrt(2 + sqrt(3))*x + x^2)]
    @test_int [1/(x^6*(1 - x^4 + x^8)), x, 22, -(1/(5*x^5)) - 1/x + atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6)) + atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) - atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3)))/(2*sqrt(6)) - atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3)))/(2*sqrt(6)) - log(1 - sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) + log(1 + sqrt(2 - sqrt(3))*x + x^2)/(4*sqrt(6)) - log(1 - sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6)) + log(1 + sqrt(2 + sqrt(3))*x + x^2)/(4*sqrt(6))]
    @test_int [1/(x^8*(1 - x^4 + x^8)), x, 22, -(1/(7*x^7)) - 1/(3*x^3) - (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 - sqrt(3)) - 2*x)/sqrt(2 + sqrt(3))) + (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 + sqrt(3)) - 2*x)/sqrt(2 - sqrt(3))) + (1/4)*sqrt((1/3)*(2 - sqrt(3)))*atan((sqrt(2 - sqrt(3)) + 2*x)/sqrt(2 + sqrt(3))) - (1/4)*sqrt((1/3)*(2 + sqrt(3)))*atan((sqrt(2 + sqrt(3)) + 2*x)/sqrt(2 - sqrt(3))) + (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 - sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 + sqrt(3)))*log(1 + sqrt(2 - sqrt(3))*x + x^2) - (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 - sqrt(2 + sqrt(3))*x + x^2) + (1/8)*sqrt((1/3)*(2 - sqrt(3)))*log(1 + sqrt(2 + sqrt(3))*x + x^2)]

    @test_int [x^m/(1 + 3*x^4 + x^8), x, 3, (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, -((2*x^4)/(3 - sqrt(5)))))/(sqrt(5)*(3 - sqrt(5))*(1 + m)) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, -((2*x^4)/(3 + sqrt(5)))))/(sqrt(5)*(3 + sqrt(5))*(1 + m))]

    @test_int [x^11/(1 + 3*x^4 + x^8), x, 5, x^4/4 - (1/40)*(15 - 7*sqrt(5))*log(3 - sqrt(5) + 2*x^4) - (1/40)*(15 + 7*sqrt(5))*log(3 + sqrt(5) + 2*x^4)]
    @test_int [x^9/(1 + 3*x^4 + x^8), x, 5, x^2/2 - (1/2)*sqrt((1/5)*(9 + 4*sqrt(5)))*atan(sqrt(2/(3 + sqrt(5)))*x^2) + (1/2)*sqrt((1/5)*(9 - 4*sqrt(5)))*atan(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [x^7/(1 + 3*x^4 + x^8), x, 4, (1/40)*(5 - 3*sqrt(5))*log(3 - sqrt(5) + 2*x^4) + (1/40)*(5 + 3*sqrt(5))*log(3 + sqrt(5) + 2*x^4)]
    @test_int [x^5/(1 + 3*x^4 + x^8), x, 4, (1/2)*sqrt((1/10)*(3 + sqrt(5)))*atan(sqrt(2/(3 + sqrt(5)))*x^2) - (1/2)*sqrt((1/10)*(3 - sqrt(5)))*atan(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [x^3/(1 + 3*x^4 + x^8), x, 3, -(atanh((3 + 2*x^4)/sqrt(5))/(2*sqrt(5)))]
    @test_int [x^1/(1 + 3*x^4 + x^8), x, 4, -(atan(sqrt(2/(3 + sqrt(5)))*x^2)/sqrt(10*(3 + sqrt(5)))) + (1/2)*sqrt((1/10)*(3 + sqrt(5)))*atan(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [1/(x^1*(1 + 3*x^4 + x^8)), x, 6, log(x) - (1/40)*(5 + 3*sqrt(5))*log(3 - sqrt(5) + 2*x^4) - (1/40)*(5 - 3*sqrt(5))*log(3 + sqrt(5) + 2*x^4)]
    @test_int [1/(x^3*(1 + 3*x^4 + x^8)), x, 5, -(1/(2*x^2)) + (1/2)*sqrt((1/5)*(9 - 4*sqrt(5)))*atan(sqrt(2/(3 + sqrt(5)))*x^2) - ((3 + sqrt(5))^(3/2)*atan(sqrt((1/2)*(3 + sqrt(5)))*x^2))/(4*sqrt(10))]
    @test_int [1/(x^5*(1 + 3*x^4 + x^8)), x, 7, -(1/(4*x^4)) - 3*log(x) + (1/40)*(15 + 7*sqrt(5))*log(3 - sqrt(5) + 2*x^4) + (1/40)*(15 - 7*sqrt(5))*log(3 + sqrt(5) + 2*x^4)]
    @test_int [1/(x^7*(1 + 3*x^4 + x^8)), x, 6, -(1/(6*x^6)) + 3/(2*x^2) - (1/2)*sqrt((1/10)*(123 - 55*sqrt(5)))*atan(sqrt(2/(3 + sqrt(5)))*x^2) + (1/2)*sqrt((1/10)*(123 + 55*sqrt(5)))*atan(sqrt((1/2)*(3 + sqrt(5)))*x^2)]

    @test_int [x^8/(1 + 3*x^4 + x^8), x, 20, x - ((123 - 55*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((123 - 55*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((123 + 55*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((123 + 55*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((123 - 55*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) + ((123 - 55*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) + ((123 + 55*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) - ((123 + 55*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5))]
    @test_int [x^6/(1 + 3*x^4 + x^8), x, 19, If(13<11, ((3 - sqrt(5))^(3/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) - ((3 - sqrt(5))^(3/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) - ((3 + sqrt(5))^(3/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) + ((3 + sqrt(5))^(3/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) - ((3 - sqrt(5))^(3/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)) + ((3 - sqrt(5))^(3/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)) + ((3 + sqrt(5))^(3/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)) - ((3 + sqrt(5))^(3/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)), ((9 - 4*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*sqrt(10)) - ((9 - 4*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*sqrt(10)) - ((3 + sqrt(5))^(3/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) + ((3 + sqrt(5))^(3/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(4*2^(1/4)*sqrt(5)) - ((9 - 4*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*sqrt(10)) + ((9 - 4*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*sqrt(10)) + ((3 + sqrt(5))^(3/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)) - ((3 + sqrt(5))^(3/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(1/4)*sqrt(5)))]
    @test_int [x^4/(1 + 3*x^4 + x^8), x, 19, ((3 - sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((3 - sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((3 + sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((3 + sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((3 - sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) - ((3 - sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) - ((3 + sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) + ((3 + sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5))]
    @test_int [x^2/(1 + 3*x^4 + x^8), x, 19, -(atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4))/(2*sqrt(5)*(2*(3 - sqrt(5)))^(1/4))) + atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4))/(2*sqrt(5)*(2*(3 - sqrt(5)))^(1/4)) + atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(2*sqrt(5)*(2*(3 + sqrt(5)))^(1/4)) - atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(2*sqrt(5)*(2*(3 + sqrt(5)))^(1/4)) + log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2)/(4*sqrt(5)*(2*(3 - sqrt(5)))^(1/4)) - log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2)/(4*sqrt(5)*(2*(3 - sqrt(5)))^(1/4)) - log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(4*sqrt(5)*(2*(3 + sqrt(5)))^(1/4)) + log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(4*sqrt(5)*(2*(3 + sqrt(5)))^(1/4))]
    @test_int [x^0/(1 + 3*x^4 + x^8), x, 19, If(13<11, (-(1/20))*sqrt(20 + 10*sqrt(5))*atan(1 - sqrt(1 + sqrt(5))*x) + (1/20)*sqrt(-20 + 10*sqrt(5))*atan(1 - sqrt(-1 + sqrt(5))*x) + (1/20)*sqrt(20 + 10*sqrt(5))*atan(1 + sqrt(1 + sqrt(5))*x) - (1/20)*sqrt(-20 + 10*sqrt(5))*atan(1 + sqrt(-1 + sqrt(5))*x) - (1/40)*sqrt(20 + 10*sqrt(5))*log(-1 + sqrt(5) - 2*sqrt(-1 + sqrt(5))*x + 2*x^2) + (1/40)*sqrt(20 + 10*sqrt(5))*log(-1 + sqrt(5) + 2*sqrt(-1 + sqrt(5))*x + 2*x^2) + (1/40)*sqrt(-20 + 10*sqrt(5))*log(1 + sqrt(5) - 2*sqrt(1 + sqrt(5))*x + 2*x^2) - (1/40)*sqrt(-20 + 10*sqrt(5))*log(1 + sqrt(5) + 2*sqrt(1 + sqrt(5))*x + 2*x^2), -(((9 + 4*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*sqrt(10))) + ((9 + 4*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*sqrt(10)) + atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - ((9 + 4*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*sqrt(10)) + ((9 + 4*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*sqrt(10)) + log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 + sqrt(5)))^(3/4))), -(atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4))/(sqrt(5)*(2*(3 - sqrt(5)))^(3/4))) + atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4))/(sqrt(5)*(2*(3 - sqrt(5)))^(3/4)) + atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4))/(sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 - sqrt(5)))^(3/4)) + log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 - sqrt(5)))^(3/4)) + log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 + sqrt(5)))^(3/4)) - log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2)/(2*sqrt(5)*(2*(3 + sqrt(5)))^(3/4))]
    @test_int [1/(x^2*(1 + 3*x^4 + x^8)), x, 20, If(13<11, -(1/x) + ((246 + 110*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*sqrt(5)) - ((246 + 110*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*sqrt(5)) - (1/20)*(6150 - 2750*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)) + (1/20)*(6150 - 2750*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)) - ((246 + 110*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*sqrt(5)) + ((246 + 110*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*sqrt(5)) + (1/40)*(6150 - 2750*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2) - (1/40)*(6150 - 2750*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2), -(1/x) + ((3 + sqrt(5))^(5/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*2^(3/4)*sqrt(5)) - ((3 + sqrt(5))^(5/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(4*2^(3/4)*sqrt(5)) - (1/20)*(6150 - 2750*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)) + (1/20)*(6150 - 2750*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)) - ((3 + sqrt(5))^(5/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(3/4)*sqrt(5)) + ((3 + sqrt(5))^(5/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(8*2^(3/4)*sqrt(5)) + (1/40)*(6150 - 2750*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2) - (1/40)*(6150 - 2750*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))]
    @test_int [1/(x^4*(1 + 3*x^4 + x^8)), x, 20, -(1/(3*x^3)) + ((843 + 377*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((843 + 377*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 - sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) - ((843 - 377*sqrt(5))^(1/4)*atan(1 - (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((843 - 377*sqrt(5))^(1/4)*atan(1 + (2^(3/4)*x)/(3 + sqrt(5))^(1/4)))/(2*2^(3/4)*sqrt(5)) + ((843 + 377*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) - 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) - ((843 + 377*sqrt(5))^(1/4)*log(sqrt(2*(3 - sqrt(5))) + 2*(2*(3 - sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) - ((843 - 377*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) - 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5)) + ((843 - 377*sqrt(5))^(1/4)*log(sqrt(2*(3 + sqrt(5))) + 2*(2*(3 + sqrt(5)))^(1/4)*x + 2*x^2))/(4*2^(3/4)*sqrt(5))]

    @test_int [x^m/(1 - 3*x^4 + x^8), x, 3, (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, (2*x^4)/(3 - sqrt(5))))/(sqrt(5)*(3 - sqrt(5))*(1 + m)) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/4, (5 + m)/4, (2*x^4)/(3 + sqrt(5))))/(sqrt(5)*(3 + sqrt(5))*(1 + m))]

    @test_int [x^11/(1 - 3*x^4 + x^8), x, 5, x^4/4 + (1/40)*(15 - 7*sqrt(5))*log(3 - sqrt(5) - 2*x^4) + (1/40)*(15 + 7*sqrt(5))*log(3 + sqrt(5) - 2*x^4)]
    @test_int [x^9/(1 - 3*x^4 + x^8), x, 5, x^2/2 - (1/2)*sqrt((1/5)*(9 + 4*sqrt(5)))*atanh(sqrt(2/(3 + sqrt(5)))*x^2) + (1/2)*sqrt((1/5)*(9 - 4*sqrt(5)))*atanh(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [x^7/(1 - 3*x^4 + x^8), x, 4, (1/40)*(5 - 3*sqrt(5))*log(3 - sqrt(5) - 2*x^4) + (1/40)*(5 + 3*sqrt(5))*log(3 + sqrt(5) - 2*x^4)]
    @test_int [x^5/(1 - 3*x^4 + x^8), x, 4, (-(1/2))*sqrt((1/10)*(3 + sqrt(5)))*atanh(sqrt(2/(3 + sqrt(5)))*x^2) + (1/2)*sqrt((1/10)*(3 - sqrt(5)))*atanh(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [x^3/(1 - 3*x^4 + x^8), x, 3, atanh((3 - 2*x^4)/sqrt(5))/(2*sqrt(5))]
    @test_int [x^1/(1 - 3*x^4 + x^8), x, 4, -(atanh(sqrt(2/(3 + sqrt(5)))*x^2)/sqrt(10*(3 + sqrt(5)))) + (1/2)*sqrt((1/10)*(3 + sqrt(5)))*atanh(sqrt((1/2)*(3 + sqrt(5)))*x^2)]
    @test_int [1/(x^1*(1 - 3*x^4 + x^8)), x, 6, log(x) - (1/40)*(5 + 3*sqrt(5))*log(3 - sqrt(5) - 2*x^4) - (1/40)*(5 - 3*sqrt(5))*log(3 + sqrt(5) - 2*x^4)]
    @test_int [1/(x^3*(1 - 3*x^4 + x^8)), x, 5, -(1/(2*x^2)) - (1/2)*sqrt((1/5)*(9 - 4*sqrt(5)))*atanh(sqrt(2/(3 + sqrt(5)))*x^2) + ((3 + sqrt(5))^(3/2)*atanh(sqrt((1/2)*(3 + sqrt(5)))*x^2))/(4*sqrt(10))]
    @test_int [1/(x^5*(1 - 3*x^4 + x^8)), x, 7, -(1/(4*x^4)) + 3*log(x) - (1/40)*(15 + 7*sqrt(5))*log(3 - sqrt(5) - 2*x^4) - (1/40)*(15 - 7*sqrt(5))*log(3 + sqrt(5) - 2*x^4)]
    @test_int [1/(x^7*(1 - 3*x^4 + x^8)), x, 6, -(1/(6*x^6)) - 3/(2*x^2) - (1/2)*sqrt((1/10)*(123 - 55*sqrt(5)))*atanh(sqrt(2/(3 + sqrt(5)))*x^2) + (1/2)*sqrt((1/10)*(123 + 55*sqrt(5)))*atanh(sqrt((1/2)*(3 + sqrt(5)))*x^2)]

    @test_int [x^8/(1 - 3*x^4 + x^8), x, 8, x - (((1/2)*(123 + 55*sqrt(5)))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + ((984 - 440*sqrt(5))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5)) - (((1/2)*(123 + 55*sqrt(5)))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + ((984 - 440*sqrt(5))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5))]
    @test_int [x^6/(1 - 3*x^4 + x^8), x, 7, ((3 + sqrt(5))^(3/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*2^(3/4)*sqrt(5)) - ((144 - 64*sqrt(5))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5)) - ((3 + sqrt(5))^(3/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*2^(3/4)*sqrt(5)) + ((144 - 64*sqrt(5))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5))]
    @test_int [x^4/(1 - 3*x^4 + x^8), x, 7, -((((1/2)*(3 + sqrt(5)))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5))) + (((1/2)*(3 - sqrt(5)))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - (((1/2)*(3 + sqrt(5)))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + (((1/2)*(3 - sqrt(5)))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5))]
    @test_int [x^2/(1 - 3*x^4 + x^8), x, 7, (1/20)*sqrt(-10 + 10*sqrt(5))*atan((1/2)*sqrt(-2 + 2*sqrt(5))*x) - (1/20)*sqrt(10 + 10*sqrt(5))*atan((1/2)*sqrt(2 + 2*sqrt(5))*x) - (1/20)*sqrt(-10 + 10*sqrt(5))*atanh((1/2)*sqrt(-2 + 2*sqrt(5))*x) + (1/20)*sqrt(10 + 10*sqrt(5))*atanh((1/2)*sqrt(2 + 2*sqrt(5))*x), atan((2/(3 + sqrt(5)))^(1/4)*x)/(2^(3/4)*sqrt(5)*(3 + sqrt(5))^(1/4)) - (((1/2)*(3 + sqrt(5)))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - atanh((2/(3 + sqrt(5)))^(1/4)*x)/(2^(3/4)*sqrt(5)*(3 + sqrt(5))^(1/4)) + (((1/2)*(3 + sqrt(5)))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5))]
    @test_int [x^0/(1 - 3*x^4 + x^8), x, 7, If(13<11, -(atan((2/(3 + sqrt(5)))^(1/4)*x)/(2^(1/4)*sqrt(5)*(3 + sqrt(5))^(3/4))) + ((9 + 4*sqrt(5))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - atanh((2/(3 + sqrt(5)))^(1/4)*x)/(2^(1/4)*sqrt(5)*(3 + sqrt(5))^(3/4)) + ((9 + 4*sqrt(5))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)), -(atan((2/(3 + sqrt(5)))^(1/4)*x)/(2^(1/4)*sqrt(5)*(3 + sqrt(5))^(3/4))) + ((3 + sqrt(5))^(3/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*2^(3/4)*sqrt(5)) - atanh((2/(3 + sqrt(5)))^(1/4)*x)/(2^(1/4)*sqrt(5)*(3 + sqrt(5))^(3/4)) + ((3 + sqrt(5))^(3/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*2^(3/4)*sqrt(5)))]
    @test_int [1/(x^2*(1 - 3*x^4 + x^8)), x, 8, -(1/x) + ((984 - 440*sqrt(5))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5)) - ((3 + sqrt(5))^(5/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*2^(1/4)*sqrt(5)) - ((984 - 440*sqrt(5))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(4*sqrt(5)) + ((3 + sqrt(5))^(5/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*2^(1/4)*sqrt(5))]
    @test_int [1/(x^4*(1 - 3*x^4 + x^8)), x, 8, If(13<11, -(1/(3*x^3)) - (((1/2)*(843 - 377*sqrt(5)))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + (((1/2)*(843 + 377*sqrt(5)))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - (((1/2)*(843 - 377*sqrt(5)))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + (((1/2)*(843 + 377*sqrt(5)))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)), -(1/(3*x^3)) - (((1/2)*(843 - 377*sqrt(5)))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + ((3 + sqrt(5))^(7/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*2^(3/4)*sqrt(5)) - (((1/2)*(843 - 377*sqrt(5)))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + ((3 + sqrt(5))^(7/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(4*2^(3/4)*sqrt(5)))]
    @test_int [1/(x^6*(1 - 3*x^4 + x^8)), x, 9, -(1/(5*x^5)) - 3/x + ((2889 - 1292*sqrt(5))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - ((2889 + 1292*sqrt(5))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - ((2889 - 1292*sqrt(5))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + ((2889 + 1292*sqrt(5))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5))]
    @test_int [1/(x^8*(1 - 3*x^4 + x^8)), x, 9, -(1/(7*x^7)) - 1/x^3 - (((1/2)*(39603 - 17711*sqrt(5)))^(1/4)*atan((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + (((1/2)*(39603 + 17711*sqrt(5)))^(1/4)*atan(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) - (((1/2)*(39603 - 17711*sqrt(5)))^(1/4)*atanh((2/(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5)) + (((1/2)*(39603 + 17711*sqrt(5)))^(1/4)*atanh(((1/2)*(3 + sqrt(5)))^(1/4)*x))/(2*sqrt(5))]

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

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^4+c*x^8)^(p/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^4+c*x^8)^p=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a+b*x^4+c*x^8)^(p/2)=#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^5+c*x^10)^p=#

    #= ::Section:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^5+c*x^10)^p*with*b^2-4*a*c=0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^5+c*x^10)^p=#

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

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

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

    @test_int [1/(x + x^6 + x^11), x, 8, -atan((1 + 2*x^5)/sqrt(3))/(5*sqrt(3)) + log(x) - log(1 + x^5 + x^10)/10]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^n+c*x^(2*n))^p*with*integer*n<0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^1+c/x^2)^p=#

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

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

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

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

    @test_int [x^1/(a/x^2 + b/x + c)^2, x, 8, -((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (b*x^3)/(c*(b^2 - 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*log(a + b*x + c*x^2))/(2*c^4)]
    @test_int [x^0/(a/x^2 + b/x + c)^2, x, 8, (2*(b^2 - 3*a*c)*x)/(c^2*(b^2 - 4*a*c)) - (b*x^2)/(c*(b^2 - 4*a*c)) + (x^3*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^3*(b^2 - 4*a*c)^(3/2)) - (b*log(a + b*x + c*x^2))/c^3]
    @test_int [1/((a/x^2 + b/x + c)^2*x^1), x, 7, -((b*x)/(c*(b^2 - 4*a*c))) + (x^2*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^2*(b^2 - 4*a*c)^(3/2)) + log(a + b*x + c*x^2)/(2*c^2)]
    @test_int [1/((a/x^2 + b/x + c)^2*x^2), x, 4, (b + (2*a)/x)/((b^2 - 4*a*c)*(c + a/x^2 + b/x)) - (4*a*atanh((b + (2*a)/x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [1/((a/x^2 + b/x + c)^2*x^3), x, 4, (2*a + b*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*b*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [1/((a/x^2 + b/x + c)^2*x^4), x, 4, -((b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)]
    @test_int [1/((a/x^2 + b/x + c)^2*x^5), x, 8, (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(3/2)) + log(x)/a^2 - log(a + b*x + c*x^2)/(2*a^2)]
    @test_int [1/((a/x^2 + b/x + c)^2*x^6), x, 8, (-2*(b^2 - 3*a*c))/(a^2*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) - (2*b*log(x))/a^3 + (b*log(a + b*x + c*x^2))/a^3]
    @test_int [1/((a/x^2 + b/x + c)^2*x^7), x, 8, -((3*b^2 - 8*a*c)/(2*a^2*(b^2 - 4*a*c)*x^2)) + (b*(3*b^2 - 11*a*c))/(a^3*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*log(x))/a^4 - ((3*b^2 - 2*a*c)*log(a + b*x + c*x^2))/(2*a^4)]

    @test_int [x^0/(a/x^2 + b/x + c)^3, x, 9, (3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*log(a + b*x + c*x^2))/(2*c^4)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^1), x, 8, -((b*(b^2 - 7*a*c)*x)/(c^2*(b^2 - 4*a*c)^2)) + (x^4*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^2*(a*(b^2 - 16*a*c) + b*(b^2 - 10*a*c)*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^3*(b^2 - 4*a*c)^(5/2)) + log(a + b*x + c*x^2)/(2*c^3)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^2), x, 5, (b + (2*a)/x)/(2*(b^2 - 4*a*c)*(c + a/x^2 + b/x)^2) - (3*a*(b + (2*a)/x))/((b^2 - 4*a*c)^2*(c + a/x^2 + b/x)) + (12*a^2*atanh((b + (2*a)/x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^3), x, 5, -(x^3*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*b*x*(2*a + b*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*a*b*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^4), x, 5, (x*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*a*b + (b^2 + 2*a*c)*x)/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*(b^2 + 2*a*c)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^5), x, 5, (2*a + b*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*b*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*b*c*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^6), x, 5, -(b + 2*c*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*c*(b + 2*c*x))/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (12*c^2*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^7), x, 9, (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(5/2)) + log(x)/a^3 - log(a + b*x + c*x^2)/(2*a^3)]
    @test_int [1/((a/x^2 + b/x + c)^3*x^8), x, 9, -((3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(a^3*(b^2 - 4*a*c)^2*x)) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x*(a + b*x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x + c*x^2))/(2*a^4)]

    @test_int [x^2/(2/x^2 + 13/x + 15), x, 6, (139*x)/3375 - (13*x^2)/450 + x^3/45 - (16/567)*log(2 + 3*x) + log(1 + 5*x)/4375]
    @test_int [x^1/(2/x^2 + 13/x + 15), x, 6, -((13*x)/225) + x^2/30 + (8/189)*log(2 + 3*x) - (1/875)*log(1 + 5*x)]
    @test_int [x^0/(2/x^2 + 13/x + 15), x, 5, x/15 - (4/63)*log(2 + 3*x) + (1/175)*log(1 + 5*x)]
    @test_int [1/(x^1*(2/x^2 + 13/x + 15)), x, 4, (2/21)*log(2 + 3*x) - (1/35)*log(1 + 5*x)]
    @test_int [1/(x^2*(2/x^2 + 13/x + 15)), x, 4, (1/7)*log(5 + 1/x) - (1/7)*log(3 + 2/x)]
    @test_int [1/(x^3*(2/x^2 + 13/x + 15)), x, 6, log(x)/2 + (3/14)*log(2 + 3*x) - (5/7)*log(1 + 5*x)]
    @test_int [1/(x^4*(2/x^2 + 13/x + 15)), x, 4, -(1/(2*x)) - (13*log(x))/4 - (9/28)*log(2 + 3*x) + (25/7)*log(1 + 5*x)]
    @test_int [1/(x^5*(2/x^2 + 13/x + 15)), x, 4, -(1/(4*x^2)) + 13/(4*x) + (139*log(x))/8 + (27/56)*log(2 + 3*x) - (125/7)*log(1 + 5*x)]
    @test_int [1/(x^6*(2/x^2 + 13/x + 15)), x, 4, -(1/(6*x^3)) + 13/(8*x^2) - 139/(8*x) - (1417*log(x))/16 - (81/112)*log(2 + 3*x) + (625/7)*log(1 + 5*x)]

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

    @test_int [(a + b/x + c/x^2)^(5/2), x, 9, (-(5/24))*(a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x) - (5*sqrt(a + c/x^2 + b/x)*(b*(b^2 + 44*a*c) + (2*c*(b^2 + 12*a*c))/x))/(64*c) + (a + c/x^2 + b/x)^(5/2)*x + (5/2)*a^(3/2)*b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))) + (5*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*atanh((b + (2*c)/x)/(2*sqrt(c)*sqrt(a + c/x^2 + b/x))))/(128*c^(3/2))]
    @test_int [(a + b/x + c/x^2)^(3/2), x, 8, (-(3/4))*sqrt(a + c/x^2 + b/x)*(3*b + (2*c)/x) + (a + c/x^2 + b/x)^(3/2)*x + (3/2)*sqrt(a)*b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))) - (3*(b^2 + 4*a*c)*atanh((b + (2*c)/x)/(2*sqrt(c)*sqrt(a + c/x^2 + b/x))))/(8*sqrt(c))]
    @test_int [(a + b/x + c/x^2)^(1/2), x, 7, sqrt(a + c/x^2 + b/x)*x + (b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))))/(2*sqrt(a)) - sqrt(c)*atanh((b + (2*c)/x)/(2*sqrt(c)*sqrt(a + c/x^2 + b/x)))]
    @test_int [1/(a + b/x + c/x^2)^(1/2), x, 4, (sqrt(a + c/x^2 + b/x)*x)/a - (b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))))/(2*a^(3/2))]
    @test_int [1/(a + b/x + c/x^2)^(3/2), x, 5, ((3*b^2 - 8*a*c)*sqrt(a + c/x^2 + b/x)*x)/(a^2*(b^2 - 4*a*c)) - (2*(b^2 - 2*a*c + (b*c)/x)*x)/(a*(b^2 - 4*a*c)*sqrt(a + c/x^2 + b/x)) - (3*b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))))/(2*a^(5/2))]
    @test_int [1/(a + b/x + c/x^2)^(5/2), x, 6, ((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*sqrt(a + c/x^2 + b/x)*x)/(3*a^3*(b^2 - 4*a*c)^2) - (2*(b^2 - 2*a*c + (b*c)/x)*x)/(3*a*(b^2 - 4*a*c)*(a + c/x^2 + b/x)^(3/2)) - (2*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2 + (b*c*(5*b^2 - 28*a*c))/x)*x)/(3*a^2*(b^2 - 4*a*c)^2*sqrt(a + c/x^2 + b/x)) - (5*b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))))/(2*a^(7/2))]

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

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

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

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

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^3+c/x^6)^p=#

    @test_int [1/(a/x^6 + b/x^3 + c), x, 15, x/c + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*c^(1/3)*x)/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(2^(1/3)*sqrt(3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)) - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x))/(3*2^(1/3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x + 2^(2/3)*c^(2/3)*x^2))/(6*2^(1/3)*c^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^4+c/x^8)^p=#

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

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^n+c*x^(2*n))^p*with*fraction*n>0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/2)+c*x^(2/2))^p=#

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

    @test_int [sqrt(a + b*sqrt(x) + c*x)/x, x, 7, 2*sqrt(a + b*sqrt(x) + c*x) - 2*sqrt(a)*atanh((2*a + b*sqrt(x))/(2*sqrt(a)*sqrt(a + b*sqrt(x) + c*x))) + (b*atanh((b + 2*c*sqrt(x))/(2*sqrt(c)*sqrt(a + b*sqrt(x) + c*x))))/sqrt(c)]

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

    @test_int [(b^2/(4*c) + b*sqrt(x) + c*x)^2, x, 4, -((b*(b + 2*c*sqrt(x))^5)/(160*c^4)) + (b + 2*c*sqrt(x))^6/(192*c^4)]

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/3)+c*x^(2/3))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/3)+c*x^(2/3))^(p/2)*with*b^2-4*a*c=0=#

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/3)+c*x^(2/3))^p*with*b^2-4*a*c=0*and*p*symbolic=#

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

    @test_int [x^2*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p, x, 4, (3*a^9*(1 + (b*x^(1/3))/a)*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(1 + 2*p)) - (12*a^9*(1 + (b*x^(1/3))/a)^2*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(1 + p)) + (84*a^9*(1 + (b*x^(1/3))/a)^3*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(3 + 2*p)) - (84*a^9*(1 + (b*x^(1/3))/a)^4*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(2 + p)) + (210*a^9*(1 + (b*x^(1/3))/a)^5*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(5 + 2*p)) - (84*a^9*(1 + (b*x^(1/3))/a)^6*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(3 + p)) + (84*a^9*(1 + (b*x^(1/3))/a)^7*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(7 + 2*p)) - (12*a^9*(1 + (b*x^(1/3))/a)^8*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(4 + p)) + (3*a^9*(1 + (b*x^(1/3))/a)^9*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^9*(9 + 2*p))]
    @test_int [x^1*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p, x, 4, -((3*a^6*(1 + (b*x^(1/3))/a)*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^6*(1 + 2*p))) + (15*a^6*(1 + (b*x^(1/3))/a)^2*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(2*b^6*(1 + p)) - (30*a^6*(1 + (b*x^(1/3))/a)^3*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^6*(3 + 2*p)) + (15*a^6*(1 + (b*x^(1/3))/a)^4*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^6*(2 + p)) - (15*a^6*(1 + (b*x^(1/3))/a)^5*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^6*(5 + 2*p)) + (3*a^6*(1 + (b*x^(1/3))/a)^6*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(2*b^6*(3 + p))]
    @test_int [x^0*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p, x, 4, (3*a^2*(a + b*x^(1/3))*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^3*(1 + 2*p)) - (3*a*(a + b*x^(1/3))^2*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^3*(1 + p)) + (3*(a + b*x^(1/3))^3*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(b^3*(3 + 2*p))]
    @test_int [1/x^1*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p, x, 3, -((3*(1 + (b*x^(1/3))/a)*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*HypergeometricFunctions._₂F₁(1, 1 + 2*p, 2*(1 + p), 1 + (b*x^(1/3))/a))/(1 + 2*p))]
    @test_int [1/x^2*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p, x, 3, (3*b^3*(1 + (b*x^(1/3))/a)*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*HypergeometricFunctions._₂F₁(4, 1 + 2*p, 2*(1 + p), 1 + (b*x^(1/3))/a))/(a^3*(1 + 2*p))]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/4)+c*x^(2/4))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/4)+c*sqrt(x))^(p/2)*with*b^2-4*a*c=0=#

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/6)+c*x^(2/6))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^(1/6)+c*x^(1/3))^(p/2)*with*b^2-4*a*c=0=#

    @test_int [1/(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(5/2), x, 4, -((60*a^2)/(b^6*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)))) + (3*a^5)/(2*b^6*(a + b*x^(1/6))^3*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))) - (10*a^4)/(b^6*(a + b*x^(1/6))^2*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))) + (30*a^3)/(b^6*(a + b*x^(1/6))*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))) + (6*(a + b*x^(1/6))*x^(1/6))/(b^5*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))) - (30*a*(a + b*x^(1/6))*log(a + b*x^(1/6)))/(b^6*sqrt(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^n+c*x^(2*n))^p*with*fraction*n<0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/2)+c/x^(2/2))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/2)+c/x^(2/2))*with*b^2-4*a*c=0=#

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/3)+c/x^(2/3))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/3)+c/x^(2/3))^(p/2)*with*b^2-4*a*c=0=#

    @test_int [(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(7/2), x, 5, -((3*b^7*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/(4*(a + b/x^(1/3))*x^(4/3))) - (7*a*b^6*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/((a + b/x^(1/3))*x) - (63*a^2*b^5*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/(2*(a + b/x^(1/3))*x^(2/3)) - (105*a^3*b^4*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/((a + b/x^(1/3))*x^(1/3)) + (63*a^5*b^2*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(1/3))/(a + b/x^(1/3)) + (21*a^6*b*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(2/3))/(2*(a + b/x^(1/3))) + (a^7*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x)/(a + b/x^(1/3)) + (105*a^4*b^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*log(x^(1/3)))/(a + b/x^(1/3))]
    @test_int [(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(5/2), x, 5, -((3*b^5*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/(2*(a + b/x^(1/3))*x^(2/3))) - (15*a*b^4*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))/((a + b/x^(1/3))*x^(1/3)) + (30*a^3*b^2*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(1/3))/(a + b/x^(1/3)) + (15*a^4*b*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(2/3))/(2*(a + b/x^(1/3))) + (a^5*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x)/(a + b/x^(1/3)) + (30*a^2*b^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*log(x^(1/3)))/(a + b/x^(1/3))]
    @test_int [(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(3/2), x, 5, (9*a*b^2*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(1/3))/(a + b/x^(1/3)) + (9*a^2*b*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(2/3))/(2*(a + b/x^(1/3))) + (a^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x)/(a + b/x^(1/3)) + (3*b^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*log(x^(1/3)))/(a + b/x^(1/3))]
    @test_int [(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(1/2), x, 4, (3*b*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x^(2/3))/(2*(a + b/x^(1/3))) + (a*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*x)/(a + b/x^(1/3))]
    @test_int [1/(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(1/2), x, 5, (3*b^2*(a + b/x^(1/3))*x^(1/3))/(a^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (3*b*(a + b/x^(1/3))*x^(2/3))/(2*a^2*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) + ((a + b/x^(1/3))*x)/(a*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (3*b^3*(a + b/x^(1/3))*log(b + a*x^(1/3)))/(a^4*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))]
    @test_int [1/(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(3/2), x, 5, (3*b^5*(a + b/x^(1/3)))/(2*a^6*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))^2) - (15*b^4*(a + b/x^(1/3)))/(a^6*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))) + (18*b^2*(a + b/x^(1/3))*x^(1/3))/(a^5*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (9*b*(a + b/x^(1/3))*x^(2/3))/(2*a^4*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) + ((a + b/x^(1/3))*x)/(a^3*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (30*b^3*(a + b/x^(1/3))*log(b + a*x^(1/3)))/(a^6*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))]
    @test_int [1/(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(5/2), x, 5, (3*b^7*(a + b/x^(1/3)))/(4*a^8*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))^4) - (7*b^6*(a + b/x^(1/3)))/(a^8*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))^3) + (63*b^5*(a + b/x^(1/3)))/(2*a^8*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))^2) - (105*b^4*(a + b/x^(1/3)))/(a^8*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))*(b + a*x^(1/3))) + (45*b^2*(a + b/x^(1/3))*x^(1/3))/(a^7*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (15*b*(a + b/x^(1/3))*x^(2/3))/(2*a^6*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) + ((a + b/x^(1/3))*x)/(a^5*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))) - (105*b^3*(a + b/x^(1/3))*log(b + a*x^(1/3)))/(a^8*sqrt(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/4)+c/x^(2/4))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/4)+c/sqrt(x))^(p/2)*with*b^2-4*a*c=0=#

    @test_int [(a^2 + 2*a*b/x^(1/4) + b^2/sqrt(x))^(5/2), x, 5, -((4*b^5*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4)))/((a + b/x^(1/4))*x^(1/4))) + (40*a^2*b^3*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4))*x^(1/4))/(a + b/x^(1/4)) + (20*a^3*b^2*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4))*sqrt(x))/(a + b/x^(1/4)) + (20*a^4*b*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4))*x^(3/4))/(3*(a + b/x^(1/4))) + (a^5*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4))*x)/(a + b/x^(1/4)) + (20*a*b^4*sqrt(a^2 + b^2/sqrt(x) + (2*a*b)/x^(1/4))*log(x^(1/4)))/(a + b/x^(1/4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/5)+c/x^(2/5))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/5)+c/x^(2/5))^(p/2)*with*b^2-4*a*c=0=#

    @test_int [(a^2 + 2*a*b/x^(1/5) + b^2/x^(2/5))^(5/2), x, 5, (25*a*b^4*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*x^(1/5))/(a + b/x^(1/5)) + (25*a^2*b^3*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*x^(2/5))/(a + b/x^(1/5)) + (50*a^3*b^2*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*x^(3/5))/(3*(a + b/x^(1/5))) + (25*a^4*b*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*x^(4/5))/(4*(a + b/x^(1/5))) + (a^5*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*x)/(a + b/x^(1/5)) + (5*b^5*sqrt(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))*log(x^(1/5)))/(a + b/x^(1/5))]
    @test_int [1/(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))^(5/2), x, 4, (20*a)/(b^5*sqrt(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))) - (5*a^4)/(4*b^5*(a + b*x^(1/5))^3*sqrt(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))) + (20*a^3)/(3*b^5*(a + b*x^(1/5))^2*sqrt(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))) - (15*a^2)/(b^5*(a + b*x^(1/5))*sqrt(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))) + (5*(a + b*x^(1/5))*log(a + b*x^(1/5)))/(b^5*sqrt(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/6)+c/x^(2/6))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b/x^(1/6)+c/x^(1/3))^(p/2)*with*b^2-4*a*c=0=#

    @test_int [(a^2 + 2*a*b/x^(1/6) + b^2/x^(1/3))^(7/2), x, 5, -((6*b^7*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)))/((a + b/x^(1/6))*x^(1/6))) + (126*a^2*b^5*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*x^(1/6))/(a + b/x^(1/6)) + (105*a^3*b^4*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*x^(1/3))/(a + b/x^(1/6)) + (70*a^4*b^3*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*sqrt(x))/(a + b/x^(1/6)) + (63*a^5*b^2*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*x^(2/3))/(2*(a + b/x^(1/6))) + (42*a^6*b*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*x^(5/6))/(5*(a + b/x^(1/6))) + (a^7*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*x)/(a + b/x^(1/6)) + (42*a*b^6*sqrt(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))*log(x^(1/6)))/(a + b/x^(1/6))]

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

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

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

    @test_int [x^(n/4 - 1)/(b*x^n + c*x^(2*n)), x, 12, -(4/(x^((3*n)/4)*(3*b*n))) + (sqrt(2)*c^(3/4)*atan(1 - (sqrt(2)*c^(1/4)*x^(n/4))/b^(1/4)))/(b^(7/4)*n) - (sqrt(2)*c^(3/4)*atan(1 + (sqrt(2)*c^(1/4)*x^(n/4))/b^(1/4)))/(b^(7/4)*n) + (c^(3/4)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*x^(n/4) + sqrt(c)*x^(n/2)))/(sqrt(2)*b^(7/4)*n) - (c^(3/4)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*x^(n/4) + sqrt(c)*x^(n/2)))/(sqrt(2)*b^(7/4)*n)]
    @test_int [x^(n/3 - 1)/(b*x^n + c*x^(2*n)), x, 9, -(3/(x^((2*n)/3)*(2*b*n))) + (sqrt(3)*c^(2/3)*atan((b^(1/3) - 2*c^(1/3)*x^(n/3))/(sqrt(3)*b^(1/3))))/(b^(5/3)*n) - (c^(2/3)*log(b^(1/3) + c^(1/3)*x^(n/3)))/(b^(5/3)*n) + (c^(2/3)*log(b^(2/3) - b^(1/3)*c^(1/3)*x^(n/3) + c^(2/3)*x^((2*n)/3)))/(2*b^(5/3)*n)]
    @test_int [x^(n/2 - 1)/(b*x^n + c*x^(2*n)), x, 5, -(2/(x^(n/2)*(b*n))) + (2*sqrt(c)*atan(sqrt(b)/(x^(n/2)*sqrt(c))))/(b^(3/2)*n)]
    @test_int [x^(-n/2 - 1)/(b*x^n + c*x^(2*n)), x, 6, -(2/(x^((3*n)/2)*(3*b*n))) + (2*c)/(x^(n/2)*(b^2*n)) - (2*c^(3/2)*atan(sqrt(b)/(x^(n/2)*sqrt(c))))/(b^(5/2)*n)]
    @test_int [x^(-n/3 - 1)/(b*x^n + c*x^(2*n)), x, 11, If(13>=8, -(3/(x^((4*n)/3)*(4*b*n))) + (3*c)/(x^(n/3)*(b^2*n)) + (sqrt(3)*c^(4/3)*atan((c^(1/3) - (2*b^(1/3))/x^(n/3))/(sqrt(3)*c^(1/3))))/(b^(7/3)*n) - (c^(4/3)*log(c^(1/3) + b^(1/3)/x^(n/3)))/(b^(7/3)*n) + (c^(4/3)*log(c^(2/3) + b^(2/3)/x^((2*n)/3) - (b^(1/3)*c^(1/3))/x^(n/3)))/(2*b^(7/3)*n), -(3/(x^((4*n)/3)*(4*b*n))) + (3*c)/(x^(n/3)*(b^2*n)) + (sqrt(3)*c^(4/3)*atan((1 - (2*b^(1/3))/(x^(n/3)*c^(1/3)))/sqrt(3)))/(b^(7/3)*n) - (c^(4/3)*log(c^(1/3) + b^(1/3)/x^(n/3)))/(b^(7/3)*n) + (c^(4/3)*log(c^(2/3) + b^(2/3)/x^((2*n)/3) - (b^(1/3)*c^(1/3))/x^(n/3)))/(2*b^(7/3)*n))]
    @test_int [x^(-n/4 - 1)/(b*x^n + c*x^(2*n)), x, 14, -(4/(x^((5*n)/4)*(5*b*n))) + (4*c)/(x^(n/4)*(b^2*n)) + (sqrt(2)*c^(5/4)*atan(1 - (sqrt(2)*b^(1/4))/(x^(n/4)*c^(1/4))))/(b^(9/4)*n) - (sqrt(2)*c^(5/4)*atan(1 + (sqrt(2)*b^(1/4))/(x^(n/4)*c^(1/4))))/(b^(9/4)*n) + (c^(5/4)*log(sqrt(c) + sqrt(b)/x^(n/2) - (sqrt(2)*b^(1/4)*c^(1/4))/x^(n/4)))/(sqrt(2)*b^(9/4)*n) - (c^(5/4)*log(sqrt(c) + sqrt(b)/x^(n/2) + (sqrt(2)*b^(1/4)*c^(1/4))/x^(n/4)))/(sqrt(2)*b^(9/4)*n)]

    @test_int [x^(-n*(p - 1) - 1)*(b*x^n + c*x^(2*n))^p, x, 1, (b*x^n + c*x^(2*n))^(1 + p)/(x^(n*(1 + p))*(c*n*(1 + p)))]
    @test_int [x^(-n*(2*p + 1) - 1)*(b*x^n + c*x^(2*n))^p, x, 1, -((b*x^n + c*x^(2*n))^(1 + p)/(x^(2*n*(1 + p))*(b*n*(1 + p))))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^(2*n-1)*(a^2+2*a*b*x^n+b^2*x^(2*n))^(p/2)=#

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

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

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

    @test_int [sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))*(d*x)^m, x, 5, (a*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/(d*(1 + m)*(a + b*x^n)) + (b^2*x^(1 + n)*(d*x)^m*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/((1 + m + n)*(a*b + b^2*x^n))]

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

    @test_int [(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)*(d*x)^m, x, 9, (a^3*(d*x)^(1 + m)*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/(d*(1 + m)*(a + b*x^n)) + (3*a^2*b^2*x^(1 + n)*(d*x)^m*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/((1 + m + n)*(a*b + b^2*x^n)) + (3*a*b^3*x^(1 + 2*n)*(d*x)^m*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/((1 + m + 2*n)*(a*b + b^2*x^n)) + (b^4*x^(1 + 3*n)*(d*x)^m*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))/((1 + m + 3*n)*(a*b + b^2*x^n))]

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

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

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

    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))*x^2, x, 2, (x^3*(a + b*x^n)*HypergeometricFunctions._₂F₁(1, 3/n, (3 + n)/n, -((b*x^n)/a)))/(3*a*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))*x^1, x, 2, (x^2*(a + b*x^n)*HypergeometricFunctions._₂F₁(1, 2/n, (2 + n)/n, -((b*x^n)/a)))/(2*a*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))*x^0, x, 2, (x*(a + b*x^n)*HypergeometricFunctions._₂F₁(1, 1/n, 1 + (1/n), -((b*x^n)/a)))/(a*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))/x^1, x, 5, ((a + b*x^n)*log(x))/(a*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))) - ((a + b*x^n)*log(a + b*x^n))/(a*n*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))/x^2, x, 2, -(((a + b*x^n)*HypergeometricFunctions._₂F₁(1, -(1/n), -((1 - n)/n), -((b*x^n)/a)))/(a*x*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))))]
    @test_int [1/sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))/x^3, x, 2, -(((a + b*x^n)*HypergeometricFunctions._₂F₁(1, -(2/n), -((2 - n)/n), -((b*x^n)/a)))/(2*a*x^2*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))))]

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

    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)*x^2, x, 2, (x^3*(a + b*x^n)*HypergeometricFunctions._₂F₁(3, 3/n, (3 + n)/n, -((b*x^n)/a)))/(3*a^3*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)*x^1, x, 2, (x^2*(a + b*x^n)*HypergeometricFunctions._₂F₁(3, 2/n, (2 + n)/n, -((b*x^n)/a)))/(2*a^3*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)*x^0, x, 2, (x*(a + b*x^n)^3*HypergeometricFunctions._₂F₁(3, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^3*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2))]
    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x^1, x, 4, 1/(a^2*n*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))) + 1/(2*a*n*(a + b*x^n)*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))) + ((a + b*x^n)*log(x))/(a^3*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))) - ((a + b*x^n)*log(a + b*x^n))/(a^3*n*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n)))]
    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x^2, x, 2, -(((a + b*x^n)*HypergeometricFunctions._₂F₁(3, -(1/n), -((1 - n)/n), -((b*x^n)/a)))/(a^3*x*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))))]
    @test_int [1/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x^3, x, 2, -(((a + b*x^n)*HypergeometricFunctions._₂F₁(3, -(2/n), -((2 - n)/n), -((b*x^n)/a)))/(2*a^3*x^2*sqrt(a^2 + 2*a*b*x^n + b^2*x^(2*n))))]

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

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

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

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*x^n+c*x^(2*n))^p*with*m=k*n-1=#

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

    @test_int [x^(n/4 - 1)/(a + b*x^n + c*x^(2*n)), x, 8, (2*2^(3/4)*c^(3/4)*atan((2^(1/4)*c^(1/4)*x^(n/4))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*n) - (2*2^(3/4)*c^(3/4)*atan((2^(1/4)*c^(1/4)*x^(n/4))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*n) + (2*2^(3/4)*c^(3/4)*atanh((2^(1/4)*c^(1/4)*x^(n/4))/(-b - sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*n) - (2*2^(3/4)*c^(3/4)*atanh((2^(1/4)*c^(1/4)*x^(n/4))/(-b + sqrt(b^2 - 4*a*c))^(1/4)))/(sqrt(b^2 - 4*a*c)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*n)]
    @test_int [x^(n/3 - 1)/(a + b*x^n + c*x^(2*n)), x, 14, -((2^(2/3)*sqrt(3)*c^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b - sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n)) + (2^(2/3)*sqrt(3)*c^(2/3)*atan((1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b + sqrt(b^2 - 4*a*c))^(1/3))/sqrt(3)))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n) + (2^(2/3)*c^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n) - (2^(2/3)*c^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n) - (c^(2/3)*log((b - sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)))/(2^(1/3)*sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n) + (c^(2/3)*log((b + sqrt(b^2 - 4*a*c))^(2/3) - 2^(1/3)*c^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)))/(2^(1/3)*sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n)]
    @test_int [x^(n/2 - 1)/(a + b*x^n + c*x^(2*n)), x, 4, (2*sqrt(2)*sqrt(c)*atan((sqrt(2)*sqrt(c)*x^(n/2))/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))*n) - (2*sqrt(2)*sqrt(c)*atan((sqrt(2)*sqrt(c)*x^(n/2))/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c))*n)]
    @test_int [x^(-n/2 - 1)/(a + b*x^n + c*x^(2*n)), x, 6, -(2/(x^(n/2)*(a*n))) + (sqrt(2)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(a))/(x^(n/2)*sqrt(b - sqrt(b^2 - 4*a*c)))))/(a^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))*n) + (sqrt(2)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(a))/(x^(n/2)*sqrt(b + sqrt(b^2 - 4*a*c)))))/(a^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*n)]
    @test_int [x^(-n/3 - 1)/(a + b*x^n + c*x^(2*n)), x, 16, -(3/(x^(n/3)*(a*n))) - (sqrt(3)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*a^(1/3))/(x^(n/3)*(b - sqrt(b^2 - 4*a*c))^(1/3)))/sqrt(3)))/(2^(1/3)*a^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n) - (sqrt(3)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((1 - (2*2^(1/3)*a^(1/3))/(x^(n/3)*(b + sqrt(b^2 - 4*a*c))^(1/3)))/sqrt(3)))/(2^(1/3)*a^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)))/(2^(1/3)*a^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)))/(2^(1/3)*a^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n) - ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b - sqrt(b^2 - 4*a*c))^(2/3) + (2^(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b - sqrt(b^2 - 4*a*c))^(1/3))/x^(n/3)))/(2*2^(1/3)*a^(4/3)*(b - sqrt(b^2 - 4*a*c))^(2/3)*n) - ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*log((b + sqrt(b^2 - 4*a*c))^(2/3) + (2^(2/3)*a^(2/3))/x^((2*n)/3) - (2^(1/3)*a^(1/3)*(b + sqrt(b^2 - 4*a*c))^(1/3))/x^(n/3)))/(2*2^(1/3)*a^(4/3)*(b + sqrt(b^2 - 4*a*c))^(2/3)*n)]
    @test_int [x^(-n/4 - 1)/(a + b*x^n + c*x^(2*n)), x, 10, -(4/(x^(n/4)*(a*n))) - (2^(3/4)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*a^(1/4))/(x^(n/4)*(-b - sqrt(b^2 - 4*a*c))^(1/4))))/(a^(5/4)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*n) - (2^(3/4)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atan((2^(1/4)*a^(1/4))/(x^(n/4)*(-b + sqrt(b^2 - 4*a*c))^(1/4))))/(a^(5/4)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*n) - (2^(3/4)*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*a^(1/4))/(x^(n/4)*(-b - sqrt(b^2 - 4*a*c))^(1/4))))/(a^(5/4)*(-b - sqrt(b^2 - 4*a*c))^(3/4)*n) - (2^(3/4)*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*atanh((2^(1/4)*a^(1/4))/(x^(n/4)*(-b + sqrt(b^2 - 4*a*c))^(1/4))))/(a^(5/4)*(-b + sqrt(b^2 - 4*a*c))^(3/4)*n)]

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

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

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

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

    @test_int [x^2/(a + b*x^n + c*x^(2*n)), x, 3, -((2*c*x^3*HypergeometricFunctions._₂F₁(1, 3/n, (3 + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/(3*(b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c)))) - (2*c*x^3*HypergeometricFunctions._₂F₁(1, 3/n, (3 + n)/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(3*(b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c)))]
    @test_int [x^1/(a + b*x^n + c*x^(2*n)), x, 3, -((c*x^2*HypergeometricFunctions._₂F₁(1, 2/n, (2 + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/(b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c))) - (c*x^2*HypergeometricFunctions._₂F₁(1, 2/n, (2 + n)/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c))]
    @test_int [x^0/(a + b*x^n + c*x^(2*n)), x, 3, -((2*c*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/(b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c))) - (2*c*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c))]
    @test_int [1/(x^1*(a + b*x^n + c*x^(2*n))), x, 7, (b*atanh((b + 2*c*x^n)/sqrt(b^2 - 4*a*c)))/(a*sqrt(b^2 - 4*a*c)*n) + log(x)/a - log(a + b*x^n + c*x^(2*n))/(2*a*n)]
    @test_int [1/(x^2*(a + b*x^n + c*x^(2*n))), x, 3, (2*c*HypergeometricFunctions._₂F₁(1, -(1/n), -((1 - n)/n), -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/((b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c))*x) + (2*c*HypergeometricFunctions._₂F₁(1, -(1/n), -((1 - n)/n), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/((b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c))*x)]
    @test_int [1/(x^3*(a + b*x^n + c*x^(2*n))), x, 3, (c*HypergeometricFunctions._₂F₁(1, -(2/n), -((2 - n)/n), -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/((b^2 - 4*a*c - b*sqrt(b^2 - 4*a*c))*x^2) + (c*HypergeometricFunctions._₂F₁(1, -(2/n), -((2 - n)/n), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/((b^2 - 4*a*c + b*sqrt(b^2 - 4*a*c))*x^2)]

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

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

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

    @test_int [x^3*(a + b*x^n + c*x^(2*n))^(3/2), x, 2, (a*x^4*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(4/n, -(3/2), -(3/2), (4 + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(4*sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [x^2*(a + b*x^n + c*x^(2*n))^(3/2), x, 2, (a*x^3*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(3/n, -(3/2), -(3/2), (3 + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(3*sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [x^1*(a + b*x^n + c*x^(2*n))^(3/2), x, 2, (a*x^2*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(2/n, -(3/2), -(3/2), (2 + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(2*sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(a + b*x^n + c*x^(2*n))^(3/2), x, 2, (a*x*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(1/n, -(3/2), -(3/2), 1 + 1/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c))))]
    @test_int [(a + b*x^n + c*x^(2*n))^(3/2)/x^1, x, 8, ((b^2 + 8*a*c + 2*b*c*x^n)*sqrt(a + b*x^n + c*x^(2*n)))/(8*c*n) + (a + b*x^n + c*x^(2*n))^(3/2)/(3*n) - (a^(3/2)*atanh((2*a + b*x^n)/(2*sqrt(a)*sqrt(a + b*x^n + c*x^(2*n)))))/n - (b*(b^2 - 12*a*c)*atanh((b + 2*c*x^n)/(2*sqrt(c)*sqrt(a + b*x^n + c*x^(2*n)))))/(16*c^(3/2)*n)]
    @test_int [(a + b*x^n + c*x^(2*n))^(3/2)/x^2, x, 2, -((a*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(-(1/n), -(3/2), -(3/2), -((1 - n)/n), -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(x*sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))]
    @test_int [(a + b*x^n + c*x^(2*n))^(3/2)/x^3, x, 2, -((a*sqrt(a + b*x^n + c*x^(2*n))*AppellF1(-(2/n), -(3/2), -(3/2), -((2 - n)/n), -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))), -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(2*x^2*sqrt(1 + (2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))]

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

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

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

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

    @test_int [(d*x)^m*(a + b*x^n + c*x^(2*n))^3, x, 14, (3*a^2*b*x^(1 + n)*(d*x)^m)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(1 + 2*n)*(d*x)^m)/(1 + m + 2*n) + (b*(b^2 + 6*a*c)*x^(1 + 3*n)*(d*x)^m)/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(1 + 4*n)*(d*x)^m)/(1 + m + 4*n) + (3*b*c^2*x^(1 + 5*n)*(d*x)^m)/(1 + m + 5*n) + (c^3*x^(1 + 6*n)*(d*x)^m)/(1 + m + 6*n) + (a^3*(d*x)^(1 + m))/(d*(1 + m))]
    @test_int [(d*x)^m*(a + b*x^n + c*x^(2*n))^2, x, 10, (2*a*b*x^(1 + n)*(d*x)^m)/(1 + m + n) + ((b^2 + 2*a*c)*x^(1 + 2*n)*(d*x)^m)/(1 + m + 2*n) + (2*b*c*x^(1 + 3*n)*(d*x)^m)/(1 + m + 3*n) + (c^2*x^(1 + 4*n)*(d*x)^m)/(1 + m + 4*n) + (a^2*(d*x)^(1 + m))/(d*(1 + m))]
    @test_int [(d*x)^m*(a + b*x^n + c*x^(2*n))^1, x, 6, (b*x^(1 + n)*(d*x)^m)/(1 + m + n) + (c*x^(1 + 2*n)*(d*x)^m)/(1 + m + 2*n) + (a*(d*x)^(1 + m))/(d*(1 + m))]
    @test_int [(d*x)^m/(a + b*x^n + c*x^(2*n))^1, x, 3, (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)) - (2*c*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m))]
    @test_int [(d*x)^m/(a + b*x^n + c*x^(2*n))^2, x, 5, ((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^(2*n))) + (c*((4*a*c*(1 + m - 2*n) - b^2*(1 + m - n))/sqrt(b^2 - 4*a*c) - b*(1 + m - n))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c)))))/(a*(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)*n) - (c*(4*a*c*(1 + m - 2*n) - b^2*(1 + m - n) + b*sqrt(b^2 - 4*a*c)*(1 + m - n))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))/(a*(b^2 - 4*a*c)^(3/2)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m)*n)]
    @test_int [(d*x)^m/(a + b*x^n + c*x^(2*n))^3, x, 6, ((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(2*a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^(2*n))^2) - ((d*x)^(1 + m)*(4*a^2*c^2*(1 + m - 4*n) - 5*a*b^2*c*(1 + m - 3*n) + b^4*(1 + m - 2*n) - b*c*(2*a*c*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*d*n^2*(a + b*x^n + c*x^(2*n))) - (1/(2*a^2*(b^2 - 4*a*c)^(5/2)*(b - sqrt(b^2 - 4*a*c))*d*(1 + m)*n^2))*(c*(b*sqrt(b^2 - 4*a*c)*(2*a*c*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) - b^4*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*n^2) + 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) - 8*a^2*c^2*(1 + m^2 + m*(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b - sqrt(b^2 - 4*a*c))))) - (1/(2*a^2*(b^2 - 4*a*c)^(5/2)*(b + sqrt(b^2 - 4*a*c))*d*(1 + m)*n^2))*(c*(b*sqrt(b^2 - 4*a*c)*(2*a*c*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) + b^4*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*n^2) - 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) + 8*a^2*c^2*(1 + m^2 + m*(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((2*c*x^n)/(b + sqrt(b^2 - 4*a*c)))))]

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

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

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

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

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

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

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

    @test_int [(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4), x, 3, (a*(d + e*x)^4)/(4*e) + (b*(d + e*x)^6)/(6*e) + (c*(d + e*x)^8)/(8*e)]
    @test_int [(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 4, (a^2*(d + e*x)^4)/(4*e) + (a*b*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*(d + e*x)^8)/(8*e) + (b*c*(d + e*x)^10)/(5*e) + (c^2*(d + e*x)^12)/(12*e)]
    @test_int [(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 4, (a^3*(d + e*x)^4)/(4*e) + (a^2*b*(d + e*x)^6)/(2*e) + (3*a*(b^2 + a*c)*(d + e*x)^8)/(8*e) + (b*(b^2 + 6*a*c)*(d + e*x)^10)/(10*e) + (c*(b^2 + a*c)*(d + e*x)^12)/(4*e) + (3*b*c^2*(d + e*x)^14)/(14*e) + (c^3*(d + e*x)^16)/(16*e)]

    @test_int [(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4), x, 3, (a*f^3*(d + e*x)^4)/(4*e) + (b*f^3*(d + e*x)^6)/(6*e) + (c*f^3*(d + e*x)^8)/(8*e)]
    @test_int [(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 4, (a^2*f^3*(d + e*x)^4)/(4*e) + (a*b*f^3*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*f^3*(d + e*x)^8)/(8*e) + (b*c*f^3*(d + e*x)^10)/(5*e) + (c^2*f^3*(d + e*x)^12)/(12*e)]
    @test_int [(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 4, (a^3*f^3*(d + e*x)^4)/(4*e) + (a^2*b*f^3*(d + e*x)^6)/(2*e) + (3*a*(b^2 + a*c)*f^3*(d + e*x)^8)/(8*e) + (b*(b^2 + 6*a*c)*f^3*(d + e*x)^10)/(10*e) + (c*(b^2 + a*c)*f^3*(d + e*x)^12)/(4*e) + (3*b*c^2*f^3*(d + e*x)^14)/(14*e) + (c^3*f^3*(d + e*x)^16)/(16*e)]

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

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

    @test_int [(d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, ((d + e*x)*(2*a + b*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + ((b - (b^2 + 4*a*c)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e) + ((b^2 + 4*a*c + b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, (2*a + b*(d + e*x)^2)/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (b*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*e)]
    @test_int [(d + e*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, -((d + e*x)*(b + 2*c*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(2*b - sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (sqrt(c)*(2*b + sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, -(b + 2*c*(d + e*x)^2)/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (2*c*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*e)]
    @test_int [1/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, ((d/e + x)*(b^2 - 2*a*c + b*c*e^2*(d/e + x)^2))/(2*a*(b^2 - 4*a*c)*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)) + (sqrt(c)*(b^2 - 12*a*c + b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (sqrt(c)*(b^2 - 12*a*c - b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 9, (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*(b^2 - 4*a*c)^(3/2)*e) + log(d + e*x)/(a^2*e) - log(a + b*(d + e*x)^2 + c*(d + e*x)^4)/(4*a^2*e)]
    @test_int [1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 6, -((3*b^2 - 10*a*c)/(2*a^2*(b^2 - 4*a*c)*e*(d + e*x))) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (sqrt(c)*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e) + (sqrt(c)*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e)]
    @test_int [1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 9, -((b^2 - 3*a*c)/(a^2*(b^2 - 4*a*c)*e*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)*e) - (2*b*log(d + e*x))/(a^3*e) + (b*log(a + b*(d + e*x)^2 + c*(d + e*x)^4))/(2*a^3*e)]
    @test_int [1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 7, -((5*b^2 - 14*a*c)/(6*a^2*(b^2 - 4*a*c)*e*(d + e*x)^3)) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*e*(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e) - (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e)]

    @test_int [(d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, ((d + e*x)*(2*a + b*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) - ((d + e*x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/(8*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*sqrt(c)*(3*b^2 + 4*a*c - 2*b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (3*sqrt(c)*(3*b^2 + 4*a*c + 2*b*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, (2*a + b*(d + e*x)^2)/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) - (3*b*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*b*c*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*e)]
    @test_int [(d + e*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, -(((d + e*x)*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2)) + ((d + e*x)*(b*(b^2 + 8*a*c) + c*(b^2 + 20*a*c)*(d + e*x)^2))/(8*a*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(b^2 + 20*a*c + (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))*e) + (sqrt(c)*(b^2 + 20*a*c - (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, -(b + 2*c*(d + e*x)^2)/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*c*(b + 2*c*(d + e*x)^2))/(2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (6*c^2*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*e)]
    @test_int [1/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, ((d/e + x)*(b^2 - 2*a*c + b*c*e^2*(d/e + x)^2))/(4*a*(b^2 - 4*a*c)*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)^2) + ((d/e + x)*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b*c*(b^2 - 8*a*c)*e^2*(d/e + x)^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*e^2*(d/e + x)^2 + c*e^4*(d/e + x)^4)) + (3*sqrt(c)*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b*(b^2 - 8*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (3*sqrt(c)*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b*(b^2 - 8*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 10, (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*(b^2 - 4*a*c)^(5/2)*e) + log(d + e*x)/(a^3*e) - log(a + b*(d + e*x)^2 + c*(d + e*x)^4)/(4*a^3*e)]
    @test_int [1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 7, -((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(8*a^3*(b^2 - 4*a*c)^2*e*(d + e*x))) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*(d + e*x)^2)/(8*a^2*(b^2 - 4*a*c)^2*e*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) - (5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 10, -((3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*e*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^4*(b^2 - 4*a*c)^(5/2)*e) - (3*b*log(d + e*x))/(a^4*e) + (3*b*log(a + b*(d + e*x)^2 + c*(d + e*x)^4))/(4*a^4*e)]

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

    @test_int [(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, (f^4*(d + e*x)*(2*a + b*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + ((b - (b^2 + 4*a*c)/sqrt(b^2 - 4*a*c))*f^4*atan((sqrt(2)*sqrt(c)*(d + e*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))*e) + ((b^2 + 4*a*c + b*sqrt(b^2 - 4*a*c))*f^4*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, (f^3*(2*a + b*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (b*f^3*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*e)]
    @test_int [(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, -(f^2*(d + e*x)*(b + 2*c*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(2*b - sqrt(b^2 - 4*a*c))*f^2*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (sqrt(c)*(2*b + sqrt(b^2 - 4*a*c))*f^2*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2, x, 5, -(f*(b + 2*c*(d + e*x)^2))/(2*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (2*c*f*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*e)]
    @test_int [1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 9, (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^2 - 6*a*c)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*(b^2 - 4*a*c)^(3/2)*e*f) + log(d + e*x)/(a^2*e*f) - log(a + b*(d + e*x)^2 + c*(d + e*x)^4)/(4*a^2*e*f)]
    @test_int [1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 6, -((3*b^2 - 10*a*c)/(2*a^2*(b^2 - 4*a*c)*e*f^2*(d + e*x))) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (sqrt(c)*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e*f^2) + (sqrt(c)*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e*f^2)]
    @test_int [1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 9, -((b^2 - 3*a*c)/(a^2*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)*e*f^3) - (2*b*log(d + e*x))/(a^3*e*f^3) + (b*log(a + b*(d + e*x)^2 + c*(d + e*x)^4))/(2*a^3*e*f^3)]
    @test_int [1/((d*f + e*f*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2), x, 7, -((5*b^2 - 14*a*c)/(6*a^2*(b^2 - 4*a*c)*e*f^4*(d + e*x)^3)) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*e*f^4*(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^4*(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e*f^4) - (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*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))*e*f^4)]

    @test_int [(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, (f^4*(d + e*x)*(2*a + b*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) - (f^4*(d + e*x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/(8*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*sqrt(c)*(3*b^2 + 4*a*c - 2*b*sqrt(b^2 - 4*a*c))*f^4*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))*e) - (3*sqrt(c)*(3*b^2 + 4*a*c + 2*b*sqrt(b^2 - 4*a*c))*f^4*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, (f^3*(2*a + b*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) - (3*b*f^3*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*b*c*f^3*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*e)]
    @test_int [(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, -((f^2*(d + e*x)*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2)) + (f^2*(d + e*x)*(b*(b^2 + 8*a*c) + c*(b^2 + 20*a*c)*(d + e*x)^2))/(8*a*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (sqrt(c)*(b^2 + 20*a*c + (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*f^2*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))*e) + (sqrt(c)*(b^2 + 20*a*c - (b*(b^2 - 52*a*c))/sqrt(b^2 - 4*a*c))*f^2*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c))*e)]
    @test_int [(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3, x, 6, -(f*(b + 2*c*(d + e*x)^2))/(4*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*c*f*(b + 2*c*(d + e*x)^2))/(2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (6*c^2*f*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*e)]
    @test_int [1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 10, (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*(b^2 - 4*a*c)^(5/2)*e*f) + log(d + e*x)/(a^3*e*f) - log(a + b*(d + e*x)^2 + c*(d + e*x)^4)/(4*a^3*e*f)]
    @test_int [1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 7, -((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(8*a^3*(b^2 - 4*a*c)^2*e*f^2*(d + e*x))) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*(d + e*x)^2)/(8*a^2*(b^2 - 4*a*c)^2*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))*e*f^2) - (3*sqrt(c)*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) - (5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2)/sqrt(b^2 - 4*a*c))*atan((sqrt(2)*sqrt(c)*(d + e*x))/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c))*e*f^2)]
    @test_int [1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3), x, 10, -((3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*atanh((b + 2*c*(d + e*x)^2)/sqrt(b^2 - 4*a*c)))/(2*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) - (3*b*log(d + e*x))/(a^4*e*f^3) + (3*b*log(a + b*(d + e*x)^2 + c*(d + e*x)^4))/(4*a^4*e*f^3)]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(a+b*(d+e*x)^3+c*(d+e*x)^6)^p=#

    @test_int [x^1/sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6), x, 7, -((d*(d + e*x)*sqrt(1 + (2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/3, 1/2, 1/2, 4/3, -((2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))))/(e^2*sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6))) + ((d + e*x)^2*sqrt(1 + (2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(2/3, 1/2, 1/2, 5/3, -((2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))))/(2*e^2*sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6))]
    @test_int [x^2/sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6), x, 10, (d^2*(d + e*x)*sqrt(1 + (2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(1/3, 1/2, 1/2, 4/3, -((2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))))/(e^3*sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6)) - (d*(d + e*x)^2*sqrt(1 + (2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c)))*sqrt(1 + (2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))*AppellF1(2/3, 1/2, 1/2, 5/3, -((2*c*(d + e*x)^3)/(b - sqrt(b^2 - 4*a*c))), -((2*c*(d + e*x)^3)/(b + sqrt(b^2 - 4*a*c)))))/(e^3*sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6)) + atanh((b + 2*c*(d + e*x)^3)/(2*sqrt(c)*sqrt(a + b*(d + e*x)^3 + c*(d + e*x)^6)))/(3*sqrt(c)*e^3)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(a+b*(d+e*x)^7+c*(d+e*x)^14)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(a+b*(d+e*x)^7+c*(d+e*x)^14)^p=#

    @test_int [(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14), x, 3, (1/21)*(2 + 3*x)^7 + (1/42)*(2 + 3*x)^14 + (1/63)*(2 + 3*x)^21]
    @test_int [(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14)^2, x, 4, (1/21)*(2 + 3*x)^7 + (1/21)*(2 + 3*x)^14 + (1/21)*(2 + 3*x)^21 + (1/42)*(2 + 3*x)^28 + (1/105)*(2 + 3*x)^35]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(a+b*(d+e*x)^7+c*(d+e*x)^14)^(p/2)=#
    end

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.jl