3.26.87 \(\int \frac {-3+2 (1+k^2) x+(1+k^2) x^2-4 k^2 x^3+k^2 x^4}{((1-x^2) (1-k^2 x^2))^{2/3} (-1+d+(-1-2 d) x+(d+k^2) x^2+k^2 x^3)} \, dx\)

Optimal. Leaf size=224 \[ \frac {\log \left (\sqrt [3]{d} x-\sqrt [3]{d}+\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2-2 d^{2/3} x+d^{2/3}+\left (\sqrt [3]{d}-\sqrt [3]{d} x\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x-\sqrt {3} \sqrt [3]{d}}{\sqrt [3]{d} x-\sqrt [3]{d}-2 \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{d^{2/3}} \]

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Rubi [F]  time = 8.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (-1
- 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x]

[Out]

-(((5 + d/k^2)*x*(1 - x^2)^(2/3)*(1 - k^2*x^2)^(2/3)*AppellF1[1/2, 2/3, 2/3, 3/2, x^2, k^2*x^2])/((1 - x^2)*(1
- k^2*x^2))^(2/3)) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*Sqrt[(-1 - k^2 + 2*k^2*x^2)^2]*((-1 + k^2)^(2/3) + 2^(2/3)*k^
(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))*Sqrt[((-1 + k^2)^(4/3) - 2^(2/3)*k^(2/3)*(-1 + k^2)^(2/3)*((1 - x^2)*(1
- k^2*x^2))^(1/3) + 2*2^(1/3)*k^(4/3)*((1 - x^2)*(1 - k^2*x^2))^(2/3))/((1 + Sqrt[3])*(-1 + k^2)^(2/3) + 2^(2
/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(-1 + k^2)^(2/3) + 2^(2/3)*k^(
2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))/((1 + Sqrt[3])*(-1 + k^2)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^
2))^(1/3))], -7 - 4*Sqrt[3]])/(2^(2/3)*k^(2/3)*(1 + k^2 - 2*k^2*x^2)*Sqrt[(-1 - k^2*(1 - 2*x^2))^2]*Sqrt[((-1
+ k^2)^(2/3)*((-1 + k^2)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3)))/((1 + Sqrt[3])*(-1 + k^2)^(
2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))^2]) - ((d^2 - 8*k^2 - d*(1 - 5*k^2))*(1 - x^2)^(2/3)*(
1 - k^2*x^2)^(2/3)*Defer[Int][1/((1 - x^2)^(2/3)*(1 - k^2*x^2)^(2/3)*(1 - d + (1 + 2*d)*x - (d + k^2)*x^2 - k^
2*x^3)), x])/(k^2*((1 - x^2)*(1 - k^2*x^2))^(2/3)) + ((d + 2*d^2 + 2*k^2 + 11*d*k^2 - 2*k^4)*(1 - x^2)^(2/3)*(
1 - k^2*x^2)^(2/3)*Defer[Int][x/((1 - x^2)^(2/3)*(1 - k^2*x^2)^(2/3)*(1 - d + (1 + 2*d)*x - (d + k^2)*x^2 - k^
2*x^3)), x])/(k^2*((1 - x^2)*(1 - k^2*x^2))^(2/3)) - ((d^2 + 2*k^2 + 8*d*k^2 + 6*k^4)*(1 - x^2)^(2/3)*(1 - k^2
*x^2)^(2/3)*Defer[Int][x^2/((1 - x^2)^(2/3)*(1 - k^2*x^2)^(2/3)*(1 - d + (1 + 2*d)*x - (d + k^2)*x^2 - k^2*x^3
)), x])/(k^2*((1 - x^2)*(1 - k^2*x^2))^(2/3))

Rubi steps

\begin {align*} \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {3-2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2+4 k^2 x^3-k^2 x^4}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \left (\frac {-5-\frac {d}{k^2}}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}}+\frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}}+\frac {d-d^2+8 k^2-5 d k^2+\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) x-\left (d^2+2 k^2+8 d k^2+6 k^4\right ) x^2}{k^2 \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )}\right ) \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}} \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-5-\frac {d}{k^2}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}} \, dx}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {d-d^2+8 k^2-5 d k^2+\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) x-\left (d^2+2 k^2+8 d k^2+6 k^4\right ) x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {\left (5+\frac {d}{k^2}\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} \left (1-k^2 x\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \left (\frac {d \left (1-d-5 k^2+\frac {8 k^2}{d}\right )}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )}+\frac {\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )}+\frac {\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )}\right ) \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {\left (5+\frac {d}{k^2}\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (1+\left (-1-k^2\right ) x+k^2 x^2\right )^{2/3}} \, dx,x,x^2\right )+\frac {\left (d \left (1-d-5 k^2+\frac {8 k^2}{d}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ &=-\frac {\left (5+\frac {d}{k^2}\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (d \left (1-d-5 k^2+\frac {8 k^2}{d}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (3 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 k^2+\left (-1-k^2\right )^2+4 k^2 x^3}} \, dx,x,\sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}\right )}{2 \left (-1-k^2+2 k^2 x^2\right )}\\ &=-\frac {\left (5+\frac {d}{k^2}\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {2}{3};\frac {3}{2};x^2,k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right ) \sqrt {\frac {\left (-1+k^2\right )^{4/3}-2^{2/3} k^{2/3} \left (-1+k^2\right )^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}+2 \sqrt [3]{2} k^{4/3} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} k^{2/3} \left (1+k^2-2 k^2 x^2\right ) \sqrt {\left (-1-k^2 \left (1-2 x^2\right )\right )^2} \sqrt {\frac {\left (-1+k^2\right )^{2/3} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}}}+\frac {\left (d \left (1-d-5 k^2+\frac {8 k^2}{d}\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}+\frac {\left (\left (d+2 d^2+2 k^2+11 d k^2-2 k^4\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \left (1-d+(1+2 d) x-\left (d+k^2\right ) x^2-k^2 x^3\right )} \, dx}{k^2 \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 3.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d
+ (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x]

[Out]

Integrate[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d
+ (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x^3)), x]

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IntegrateAlgebraic [A]  time = 8.06, size = 224, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {-\sqrt {3} \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d} x}{-\sqrt [3]{d}+\sqrt [3]{d} x-2 \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)
*(-1 + d + (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x]

[Out]

(Sqrt[3]*ArcTan[(-(Sqrt[3]*d^(1/3)) + Sqrt[3]*d^(1/3)*x)/(-d^(1/3) + d^(1/3)*x - 2*(1 + (-1 - k^2)*x^2 + k^2*x
^4)^(1/3))])/d^(2/3) + Log[-d^(1/3) + d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) - Log[d^(2/3)
- 2*d^(2/3)*x + d^(2/3)*x^2 + (d^(1/3) - d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2
+ k^2*x^4)^(2/3)]/(2*d^(2/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d
)*x^2+k^2*x^3),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} + {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d
)*x^2+k^2*x^3),x, algorithm="giac")

[Out]

integrate((k^2*x^4 - 4*k^2*x^3 + (k^2 + 1)*x^2 + 2*(k^2 + 1)*x - 3)/((k^2*x^3 + (k^2 + d)*x^2 - (2*d + 1)*x +
d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-3+2 \left (k^{2}+1\right ) x +\left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}+k^{2} x^{4}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {2}{3}} \left (-1+d +\left (-1-2 d \right ) x +\left (k^{2}+d \right ) x^{2}+k^{2} x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+
k^2*x^3),x)

[Out]

int((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+
k^2*x^3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} + {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d
)*x^2+k^2*x^3),x, algorithm="maxima")

[Out]

integrate((k^2*x^4 - 4*k^2*x^3 + (k^2 + 1)*x^2 + 2*(k^2 + 1)*x - 3)/((k^2*x^3 + (k^2 + d)*x^2 - (2*d + 1)*x +
d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {2\,x\,\left (k^2+1\right )-4\,k^2\,x^3+k^2\,x^4+x^2\,\left (k^2+1\right )-3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (d+k^2\,x^3+x^2\,\left (k^2+d\right )-x\,\left (2\,d+1\right )-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*(k^2 + 1) - 4*k^2*x^3 + k^2*x^4 + x^2*(k^2 + 1) - 3)/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(d + k^2*x^3 +
x^2*(d + k^2) - x*(2*d + 1) - 1)),x)

[Out]

int((2*x*(k^2 + 1) - 4*k^2*x^3 + k^2*x^4 + x^2*(k^2 + 1) - 3)/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(d + k^2*x^3 +
x^2*(d + k^2) - x*(2*d + 1) - 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+2*(k**2+1)*x+(k**2+1)*x**2-4*k**2*x**3+k**2*x**4)/((-x**2+1)*(-k**2*x**2+1))**(2/3)/(-1+d+(-1-2*
d)*x+(k**2+d)*x**2+k**2*x**3),x)

[Out]

Timed out

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