Integrand size = 84, antiderivative size = 252 \[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2-4 k x+2 k^2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{2/3}}+\frac {\log \left (1-2 k x+k^2 x^2-\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}}-\frac {\log \left (1-4 k x+6 k^2 x^2-4 k^3 x^3+k^4 x^4+\left (\sqrt [3]{b}-2 \sqrt [3]{b} k x+\sqrt [3]{b} k^2 x^2\right ) \left (x+(-1-k) x^2+k x^3\right )^{2/3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{4/3}\right )}{4 b^{2/3}} \]
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\[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {(-1+x) \sqrt [3]{x} \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{(1-x)^{2/3} (1-k x)^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx}{((1-x) x (1-k x))^{2/3}} \\ & = \frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-x} \sqrt [3]{x} \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{(1-k x)^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx}{((1-x) x (1-k x))^{2/3}} \\ & = \frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-x} \sqrt [3]{x} (-1+(2-k) x) \sqrt [3]{1-k x}}{1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4} \, dx}{((1-x) x (1-k x))^{2/3}} \\ & = \frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^3} \left (-1+(2-k) x^3\right ) \sqrt [3]{1-k x^3}}{1-4 k x^3+\left (-b+6 k^2\right ) x^6+\left (2 b-4 k^3\right ) x^9+\left (-b+k^4\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}} \\ & = \frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \text {Subst}\left (\int \left (\frac {(2-k) x^6 \sqrt [3]{1-x^3} \sqrt [3]{1-k x^3}}{1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}}+\frac {x^3 \sqrt [3]{1-x^3} \sqrt [3]{1-k x^3}}{-1+4 k x^3+b \left (1-\frac {6 k^2}{b}\right ) x^6-2 b \left (1-\frac {2 k^3}{b}\right ) x^9+b \left (1-\frac {k^4}{b}\right ) x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}} \\ & = \frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^3} \sqrt [3]{1-k x^3}}{-1+4 k x^3+b \left (1-\frac {6 k^2}{b}\right ) x^6-2 b \left (1-\frac {2 k^3}{b}\right ) x^9+b \left (1-\frac {k^4}{b}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}+\frac {\left (3 (2-k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1-x^3} \sqrt [3]{1-k x^3}}{1-4 k x^3-b \left (1-\frac {6 k^2}{b}\right ) x^6+2 b \left (1-\frac {2 k^3}{b}\right ) x^9-b \left (1-\frac {k^4}{b}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}} \\ \end{align*}
\[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \]
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\[\int -\frac {\left (-1+x \right ) x \left (-1+2 x +\left (k^{2}-2 k \right ) x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (1-4 k x +\left (6 k^{2}-b \right ) x^{2}+\left (-4 k^{3}+2 b \right ) x^{3}+\left (k^{4}-b \right ) x^{4}\right )}d x\]
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Timed out. \[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\text {Timed out} \]
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\[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} - 2 \, k\right )} x^{2} + 2 \, x - 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} - 2 \, k\right )} x^{2} + 2 \, x - 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int -\frac {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=-\int \frac {x\,\left (x-1\right )\,\left (\left (2\,k-k^2\right )\,x^2-2\,x+1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b-k^4\right )\,x^4+\left (4\,k^3-2\,b\right )\,x^3+\left (b-6\,k^2\right )\,x^2+4\,k\,x-1\right )} \,d x \]
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