Integrand size = 65, antiderivative size = 243 \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2-2 k^2 x^2+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 d^{2/3}}+\frac {\log \left (-1+k^2 x^2+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{2 d^{2/3}}-\frac {\log \left (1-2 k^2 x^2+k^4 x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{4 d^{2/3}} \]
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\[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (1-2 k^2+k^2 x^2\right )}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx \\ & = \int \frac {x \left (1-2 k^2+k^2 x^2\right )}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1-2 k^2+k^2 x}{\sqrt [3]{1+\left (-1-k^2\right ) x+k^2 x^2} \left (1-d+\left (d-2 k^2\right ) x+k^4 x^2\right )} \, dx,x,x^2\right ) \\ \end{align*}
Time = 19.71 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.84 \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\frac {\sqrt [3]{-1+x^2} \sqrt [3]{-1+k^2 x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-1+x^2}}{\sqrt [3]{d} \sqrt [3]{-1+x^2}-2 \left (-1+k^2 x^2\right )^{2/3}}\right )+2 \log \left (\sqrt [3]{d} \sqrt [3]{-1+x^2}+\left (-1+k^2 x^2\right )^{2/3}\right )-\log \left (d^{2/3} \left (-1+x^2\right )^{2/3}-\sqrt [3]{d} \sqrt [3]{-1+x^2} \left (-1+k^2 x^2\right )^{2/3}+\left (-1+k^2 x^2\right )^{4/3}\right )\right )}{4 d^{2/3} \sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
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\[\int \frac {\left (-2 k^{2}+1\right ) x +k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (1-d +\left (-2 k^{2}+d \right ) x^{2}+k^{4} x^{4}\right )}d x\]
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Timed out. \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{2} x^{3} - {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{2} x^{3} - {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=-\int -\frac {k^2\,x^3-x\,\left (2\,k^2-1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (k^4\,x^4-d+x^2\,\left (d-2\,k^2\right )+1\right )} \,d x \]
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