Integrand size = 51, antiderivative size = 233 \[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {-2 a+b-\sqrt {b^2+4 a^2 d-4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}+\frac {-2 a+b+\sqrt {b^2+4 a^2 d-4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (\left (-2 a+b-\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (\left (-2 a+b+\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}
\[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \]
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\[\int \frac {a b +\left (-2 a +b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2} d +\left (-2 a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
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Timed out. \[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a b - {\left (2 \, a - b\right )} x}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a b - {\left (2 \, a - b\right )} x}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a\,b-x\,\left (2\,a-b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (a^2\,d+x\,\left (b-2\,a\,d\right )+x^2\,\left (d-1\right )\right )} \,d x \]
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