Integrand size = 74, antiderivative size = 98 \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+k^2 x^2}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+k^2 x^2}\right )}{d^{3/4}} \]
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Timed out. \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\text {\$Aborted} \]
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Rubi steps Aborted
\[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx \]
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\[\int \frac {\left (-3 k^{2}+1\right ) x +2 k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{4}} \left (-1+d +\left (3 k^{2}-d \right ) x^{2}-3 k^{4} x^{4}+k^{6} x^{6}\right )}d x\]
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Timed out. \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\int { \frac {2 \, k^{2} x^{3} - {\left (3 \, k^{2} - 1\right )} x}{{\left (k^{6} x^{6} - 3 \, k^{4} x^{4} + {\left (3 \, k^{2} - d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\int { \frac {2 \, k^{2} x^{3} - {\left (3 \, k^{2} - 1\right )} x}{{\left (k^{6} x^{6} - 3 \, k^{4} x^{4} + {\left (3 \, k^{2} - d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\int -\frac {2\,k^2\,x^3-x\,\left (3\,k^2-1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/4}\,\left (3\,k^4\,x^4-d-k^6\,x^6+x^2\,\left (d-3\,k^2\right )+1\right )} \,d x \]
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