Integrand size = 63, antiderivative size = 232 \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{-2 b x+2 x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (b x-x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (b^2 x^2-2 b x^3+x^4+\left (-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x} (-a b+(2 a-b) x)}{x^{2/3} (-b+x)^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {\left (2 a-b+\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{-a+x}}{x^{2/3} (-b+x)^{2/3} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}+\frac {\left (2 a-b-\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{-a+x}}{x^{2/3} (-b+x)^{2/3} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (\left (2 a-b-\sqrt {b^2+4 a^2 d-4 a b d}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{x^{2/3} (-b+x)^{2/3} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}}+\frac {\left (\left (2 a-b+\sqrt {b^2+4 a^2 d-4 a b d}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{x^{2/3} (-b+x)^{2/3} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ \end{align*}
\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \]
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\[\int \frac {a^{2} b -2 a^{2} x +\left (2 a -b \right ) x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{2} d +\left (-2 a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
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Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {x^2\,\left (2\,a-b\right )+a^2\,b-2\,a^2\,x}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (a^2\,d+x\,\left (b-2\,a\,d\right )+x^2\,\left (d-1\right )\right )} \,d x \]
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