Integrand size = 89, antiderivative size = 203 \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {-\sqrt {3}+\sqrt {3} k x}{-1+k x-2 \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-1+k x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (1-2 k x+k^2 x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ & = \int \left (-\frac {k \left (5 d+k^2\right )}{d^2 \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {k^2 x}{d \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {-k \left (8 d^2-k^2-d \left (5-k^2\right )\right )-\left (2 k^4-2 d^2 \left (1-k^2\right )+d k^2 \left (11+k^2\right )\right ) x+k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{d^2 \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx \\ & = \frac {\int \frac {-k \left (8 d^2-k^2-d \left (5-k^2\right )\right )-\left (2 k^4-2 d^2 \left (1-k^2\right )+d k^2 \left (11+k^2\right )\right ) x+k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {k^2 \int \frac {x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d}-\frac {\left (k \left (5 d+k^2\right )\right ) \int \frac {1}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2} \\ & = \frac {\int \left (\frac {k \left (-8 d^2+k^2+d \left (5-k^2\right )\right )}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) x}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right ) x^2}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx}{d^2}+\frac {k^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 )}{2 d}-\frac {\left (k \left (5 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}{d^2 \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \\ & = -\frac {k \left (5 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 )}{d^2 \left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\frac {\left (k \left (8 d^2-k^2-d \left (5-k^2\right )\right )\right ) \int \frac {1}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {\left (k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right )\right ) \int \frac {x^2}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) \int \frac {x}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {\left (3 k^2 \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 d \left (-1-k^2+2 k^2 x^2\right )} \\ & = -\frac {k \left (5 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 )}{d^2 \left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} k^{4/3} \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} d \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 (k \left (8 d^2-k^2-d \left (5-k^2\right )\right )\right ) \int \frac {1}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {\left (k \left (8 d k^2+k^4+2 d^2 \left (3+k^2\right )\right )\right ) \int \frac {x^2}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2}+\frac {\left (-2 k^4+2 d^2 \left (1-k^2\right )-d k^2 \left (11+k^2\right )\right ) \int \frac {x}{\left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx}{d^2} \\ \end{align*}
\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx \]
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\[\int \frac {-3 k +2 \left (k^{2}+1\right ) x +k \left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}+k^{3} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d -\left (2+d \right ) k x +\left (k^{2}+d \right ) x^{2}+d k \,x^{3}\right )}d x\]
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Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int { \frac {k^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{2} + d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int { \frac {k^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{2} + d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int \frac {2\,x\,\left (k^2+1\right )-3\,k-4\,k^2\,x^3+k^3\,x^4+k\,x^2\,\left (k^2+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (d\,k\,x^3+\left (k^2+d\right )\,x^2-k\,\left (d+2\right )\,x-d+1\right )} \,d x \]
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