Integrand size = 91, antiderivative size = 387 \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [6]{d}-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (-a \sqrt [6]{d}+\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (-a \sqrt [6]{d}+\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
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\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a+x)^2 (-a b+(2 a-b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{5/3} (-a b+(2 a-b) x)}{\sqrt [3]{x} \sqrt [3]{-b+x} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x \left (-a+x^3\right )^{5/3} \left (-a b+(2 a-b) x^3\right )}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3+\left (-b^2+6 a^2 d\right ) x^6+2 (b-2 a d) x^9+(-1+d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {(2 a-b) x^4 \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6+2 b \left (1-\frac {2 a d}{b}\right ) x^9-(1-d) x^{12}\right )}+\frac {a b x \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (-a^4 d+4 a^3 d x^3+b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6-2 b \left (1-\frac {2 a d}{b}\right ) x^9+(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 (2 a-b) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6+2 b \left (1-\frac {2 a d}{b}\right ) x^9-(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (3 a b \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (-a^4 d+4 a^3 d x^3+b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6-2 b \left (1-\frac {2 a d}{b}\right ) x^9+(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}
\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \]
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\[\int \frac {\left (-a b +\left (2 a -b \right ) x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{4} d -4 a^{3} d x +\left (6 a^{2} d -b^{2}\right ) x^{2}+2 \left (-2 a d +b \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}d x\]
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Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=-\int \frac {\left (a\,b-x\,\left (2\,a-b\right )\right )\,\left (a^2-2\,a\,x+x^2\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (2\,x^3\,\left (b-2\,a\,d\right )+x^2\,\left (6\,a^2\,d-b^2\right )+a^4\,d+x^4\,\left (d-1\right )-4\,a^3\,d\,x\right )} \,d x \]
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