Integrand size = 74, antiderivative size = 222 \[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-4 x+2 x^2+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (1-2 x+x^2-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (1-4 x+6 x^2-4 x^3+x^4+\left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]
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\[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx \]
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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 {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(1-x)^{2/3} \left (1+(4-2 k) x-3 k x^2\right )}{\sqrt [3]{x} \sqrt [3]{1-k x} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\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 \left (1-x^3\right )^{2/3} \left (1+(4-2 k) x^3-3 k x^6\right )}{\sqrt [3]{1-k x^3} \left (-1+(5+b) x^3-(10+b k) x^6+10 x^9-5 x^{12}+x^{15}\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 {2 (-2+k) x^4 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-5 \left (1+\frac {b}{5}\right ) x^3+10 \left (1+\frac {b k}{10}\right ) x^6-10 x^9+5 x^{12}-x^{15}\right )}+\frac {3 k x^7 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-5 \left (1+\frac {b}{5}\right ) x^3+10 \left (1+\frac {b k}{10}\right ) x^6-10 x^9+5 x^{12}-x^{15}\right )}+\frac {x \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (-1+5 \left (1+\frac {b}{5}\right ) x^3-10 \left (1+\frac {b k}{10}\right ) x^6+10 x^9-5 x^{12}+x^{15}\right )}\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 \frac {x \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (-1+5 \left (1+\frac {b}{5}\right ) x^3-10 \left (1+\frac {b k}{10}\right ) x^6+10 x^9-5 x^{12}+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (6 (2-k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-5 \left (1+\frac {b}{5}\right ) x^3+10 \left (1+\frac {b k}{10}\right ) x^6-10 x^9+5 x^{12}-x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 k \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7 \left (1-x^3\right )^{2/3}}{\sqrt [3]{1-k x^3} \left (1-5 \left (1+\frac {b}{5}\right ) x^3+10 \left (1+\frac {b k}{10}\right ) x^6-10 x^9+5 x^{12}-x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}
\[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx \]
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\[\int \frac {1-\left (-3+2 k \right ) x -\left (4+k \right ) x^{2}+3 k \,x^{3}}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-1+\left (5+b \right ) x -\left (b k +10\right ) x^{2}+10 x^{3}-5 x^{4}+x^{5}\right )}d x\]
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Timed out. \[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int \frac {\left (x - 1\right ) \left (3 k x^{2} + 2 k x - 4 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (- b k x^{2} + b x + x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1\right )}\, dx \]
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\[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int { \frac {3 \, k x^{3} - {\left (k + 4\right )} x^{2} - {\left (2 \, k - 3\right )} x + 1}{{\left (x^{5} - 5 \, x^{4} - {\left (b k + 10\right )} x^{2} + 10 \, x^{3} + {\left (b + 5\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int { \frac {3 \, k x^{3} - {\left (k + 4\right )} x^{2} - {\left (2 \, k - 3\right )} x + 1}{{\left (x^{5} - 5 \, x^{4} - {\left (b k + 10\right )} x^{2} + 10 \, x^{3} + {\left (b + 5\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right )} \, dx=\int -\frac {-3\,k\,x^3+\left (k+4\right )\,x^2+\left (2\,k-3\right )\,x-1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (x^5-5\,x^4+10\,x^3+\left (-b\,k-10\right )\,x^2+\left (b+5\right )\,x-1\right )} \,d x \]
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