Integrand size = 52, antiderivative size = 216 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (a b+(-a-b) x+x^2\right )^{2/3}}{-2 b \sqrt [3]{d}+2 \sqrt [3]{d} x+\left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b \sqrt {d}-\sqrt {d} x+\sqrt [6]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^2 d-2 b d x+d x^2+\left (-b d^{2/3}+d^{2/3} x\right ) \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{4/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-a (a-2 b)-2 b x+x^2}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx \\ & = \int \left (\frac {1}{\left (a b+(-a-b) x+x^2\right )^{2/3}}-\frac {2 a^2-2 a b+b d-(2 a-2 b+d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2+b d+(-2 a-d) x+x^2\right )}\right ) \, dx \\ & = \int \frac {1}{\left (a b+(-a-b) x+x^2\right )^{2/3}} \, dx-\int \frac {2 a^2-2 a b+b d+(-2 a+2 b-d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2+b d+(-2 a-d) x+x^2\right )} \, dx \\ & = \frac {\left (3 \sqrt {(-a-b+2 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {(-a-b)^2-4 a b+4 x^3}} \, dx,x,\sqrt [3]{a b+(-a-b) x+x^2}\right )}{-a-b+2 x}-\int \frac {2 a^2-2 a b+b d+(-2 a+2 b-d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2+b d+(-2 a-d) x+x^2\right )} \, dx \\ & = -\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \left ((a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right ) \sqrt {(-a-b+2 x)^2} \sqrt {\frac {(a-b)^{4/3}-2^{2/3} (a-b)^{2/3} \sqrt [3]{a b-(a+b) x+x^2}+2 \sqrt [3]{2} \left (a b-(a+b) x+x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}}{\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {(a-b)^{2/3} \left ((a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )}{\left (\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )^2}} (a+b-2 x) \sqrt {(a+b-2 x)^2}}-\int \frac {2 a^2-2 a b+b d+(-2 a+2 b-d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2+b d+(-2 a-d) x+x^2\right )} \, dx \\ \end{align*}
Time = 3.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.80 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (a-x)^{2/3}}{(a-x)^{2/3}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )-2 \log \left ((a-x)^{2/3}+\sqrt [3]{d} \sqrt [3]{b-x}\right )+\log \left ((a-x)^{4/3}-\sqrt [3]{d} (a-x)^{2/3} \sqrt [3]{b-x}+d^{2/3} (b-x)^{2/3}\right )\right )}{2 \sqrt [3]{d} ((-a+x) (-b+x))^{2/3}} \]
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\[\int \frac {-a \left (a -2 b \right )-2 b x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{2}+b d -\left (2 a +d \right ) x +x^{2}\right )}d x\]
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Timed out. \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=-\int \frac {-x^2+2\,b\,x+a\,\left (a-2\,b\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (b\,d-x\,\left (2\,a+d\right )+a^2+x^2\right )} \,d x \]
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