Integrand size = 55, antiderivative size = 179 \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 \sqrt [3]{b} x^3+\sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} x^3+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b^{2/3} x^6+\sqrt [3]{b} x^3 \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]
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\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {x^{5/3} \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{1-x} \sqrt [3]{1-k x} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7 \left (8-7 (1+k) x^3+6 k x^6\right )}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+(1+k) x^3-k x^6+b x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \left (\frac {7 (1+k) x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-(1+k) x^3+k x^6-b x^{24}\right )}+\frac {8 x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+(1+k) x^3-k x^6+b x^{24}\right )}+\frac {6 k x^{13}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+(1+k) x^3-k x^6+b x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (24 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+(1+k) x^3-k x^6+b x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (18 k \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{13}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+(1+k) x^3-k x^6+b x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (21 (1+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-(1+k) x^3+k x^6-b x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}
\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx \]
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\[\int \frac {x^{2} \left (8-7 \left (1+k \right ) x +6 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-1+\left (1+k \right ) x -k \,x^{2}+b \,x^{8}\right )}d x\]
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Timed out. \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int \frac {x^{2} \cdot \left (6 k x^{2} - 7 k x - 7 x + 8\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (b x^{8} - k x^{2} + k x + x - 1\right )}\, dx \]
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\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{2}}{{\left (b x^{8} - k x^{2} + {\left (k + 1\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{2}}{{\left (b x^{8} - k x^{2} + {\left (k + 1\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^8\right )} \, dx=\int \frac {x^2\,\left (6\,k\,x^2-7\,x\,\left (k+1\right )+8\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (b\,x^8-k\,x^2+\left (k+1\right )\,x-1\right )} \,d x \]
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