Integrand size = 83, antiderivative size = 236 \[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}+2 \sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left (\sqrt [3]{d}+\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (-\frac {k}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {(4-d) k+2 \left (1-(1+d) k^2\right ) x-k \left (4+d k^2\right ) x^2}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = -\left (k \int \frac {1}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx\right )+\int \frac {(4-d) k+2 \left (1-(1+d) k^2\right ) x-k \left (4+d k^2\right ) x^2}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = -\frac {\left (k \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}\right ) \int \frac {1}{\sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}} \, dx}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\int \left (\frac {(4-d) k}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 \left (1-(1+d) k^2\right ) x}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {k \left (-4-d k^2\right ) x^2}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = -\frac {k x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\sqrt [3]{1-\left (1+k^2\right ) x^2+k^2 x^4}}+((4-d) k) \int \frac {1}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\left (k \left (4+d k^2\right )\right ) \int \frac {x^2}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\left (2 \left (1-(1+d) k^2\right )\right ) \int \frac {x}{\left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ \end{align*}
\[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx \]
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\[\int \frac {3 k +\left (-k^{2}+2\right ) x -3 k \,x^{2}-k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (1-d -\left (1+2 d \right ) k x +\left (-d \,k^{2}-1\right ) x^{2}+k \,x^{3}\right )}d x\]
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Timed out. \[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\int { -\frac {k^{2} x^{3} + 3 \, k x^{2} + {\left (k^{2} - 2\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x - {\left (d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\int { -\frac {k^{2} x^{3} + 3 \, k x^{2} + {\left (k^{2} - 2\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x - {\left (d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {3 k+\left (2-k^2\right ) x-3 k x^2-k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(1+2 d) k x+\left (-1-d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {x\,\left (k^2-2\right )-3\,k+k^2\,x^3+3\,k\,x^2}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (-k\,x^3+\left (d\,k^2+1\right )\,x^2+k\,\left (2\,d+1\right )\,x+d-1\right )} \,d x \]
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