Integrand size = 91, antiderivative size = 223 \[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d}+\sqrt [3]{d} x+2 \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}-\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}+\frac {\log \left (d^{2/3}+2 d^{2/3} x+d^{2/3} x^2+\left (\sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \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-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ & = \int \left (\frac {-5-\frac {d}{k^2}}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\frac {x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\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-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx \\ & = \left (-5-\frac {d}{k^2}\right ) \int \frac {1}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx+\frac {\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-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}-\int \frac {x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ & = -\left (\frac {1}{2} \text {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 )\right )+\frac {\int \left (\frac {d^2 \left (1-\frac {d+8 k^2-5 d k^2}{d^2}\right )}{\left (-1+d+(1+2 d) x+\left (d+k^2\right ) x^2-k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {\left (-d-2 d^2-2 k^2-11 d k^2+2 k^4\right ) x}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) x^2}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx}{k^2}+\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 (1+\left (-1-k^2\right ) x^2+k^2 x^4\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} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},\frac {2}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {\left (-d^2-2 k^2-8 d k^2-6 k^4\right ) \int \frac {x^2}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}+\frac {\left (-d-2 d^2-2 k^2-11 d k^2+2 k^4\right ) \int \frac {x}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}+\frac {\left (d^2-8 k^2-d \left (1-5 k^2\right )\right ) \int \frac {1}{\left (-1+d+(1+2 d) x+\left (d+k^2\right ) x^2-k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}-\frac {\left (3 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \text {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} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},\frac {2}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\left (1-\left (1+k^2\right ) x^2+k^2 x^4\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}} \operatorname {EllipticF}\left (\arcsin \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^2-2 k^2-8 d k^2-6 k^4\right ) \int \frac {x^2}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}+\frac {\left (-d-2 d^2-2 k^2-11 d k^2+2 k^4\right ) \int \frac {x}{\left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2}+\frac {\left (d^2-8 k^2-d \left (1-5 k^2\right )\right ) \int \frac {1}{\left (-1+d+(1+2 d) x+\left (d+k^2\right ) x^2-k^2 x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{k^2} \\ \end{align*}
\[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx \]
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\[\int \frac {3+2 \left (k^{2}+1\right ) x -\left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}-k^{2} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d -\left (1+2 d \right ) x -\left (k^{2}+d \right ) x^{2}+k^{2} x^{3}\right )}d x\]
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Timed out. \[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int { -\frac {k^{2} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} - {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int { -\frac {k^{2} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} - {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {3+2 \left (1+k^2\right ) x-\left (1+k^2\right ) x^2-4 k^2 x^3-k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) x-\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int \frac {4\,k^2\,x^3-2\,x\,\left (k^2+1\right )+k^2\,x^4+x^2\,\left (k^2+1\right )-3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (d-k^2\,x^3+x^2\,\left (k^2+d\right )+x\,\left (2\,d+1\right )-1\right )} \,d x \]
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