Integrand size = 86, antiderivative size = 244 \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )\right )}{2 d^{2/3}} \]
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\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{a b+(-a-b) x+x^2} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \\ & = \int \frac {a (a-5 b)+(3 a+5 b) x-4 x^2}{\sqrt [3]{a b-(a+b) x+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )+\left (-5 a^4+d\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )} \, dx \\ & = \int \left (\frac {a (a-5 b)}{\sqrt [3]{a b-(a+b) x+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}+\frac {(3 a+5 b) x}{\sqrt [3]{a b-(a+b) x+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}+\frac {4 x^2}{\sqrt [3]{a b-(a+b) x+x^2} \left (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )}\right ) \, dx \\ & = 4 \int \frac {x^2}{\sqrt [3]{a b-(a+b) x+x^2} \left (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx+(a (a-5 b)) \int \frac {1}{\sqrt [3]{a b-(a+b) x+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )} \, dx+(3 a+5 b) \int \frac {x}{\sqrt [3]{a b-(a+b) x+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )} \, dx \\ \end{align*}
Time = 11.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.81 \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}}\right )+2 \log \left (a^2-2 a x+x^2-\sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}\right )-\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\sqrt [3]{d} (a-x)^2 \sqrt [3]{(-a+x) (-b+x)}+d^{2/3} ((-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]
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\[\int \frac {-a \left (a -5 b \right )-\left (3 a +5 b \right ) x +4 x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{5}+b d -\left (-5 a^{4}+d \right ) x -10 a^{3} x^{2}+10 a^{2} x^{3}-5 a \,x^{4}+x^{5}\right )}d x\]
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Timed out. \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int \frac {\left (- a + x\right ) \left (a - 5 b + 4 x\right )}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )} \left (- a^{5} + 5 a^{4} x - 10 a^{3} x^{2} + 10 a^{2} x^{3} - 5 a x^{4} + b d - d x + x^{5}\right )}\, dx \]
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\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int { \frac {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int { \frac {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int \frac {-4\,x^2+\left (3\,a+5\,b\right )\,x+a\,\left (a-5\,b\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (5\,a\,x^4-b\,d+x\,\left (d-5\,a^4\right )+a^5-x^5-10\,a^2\,x^3+10\,a^3\,x^2\right )} \,d x \]
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