Integrand size = 80, antiderivative size = 297 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
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\[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx \]
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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 {x^{5/3} (-2 a b+(a+b) x)}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) 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^7 \left (-2 a b+(a+b) x^3\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 b^2+2 a b (a+b) x^3-\left (a^2+4 a b+b^2\right ) x^6+2 (a+b) 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^7}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+(1-d) x^{12}\right )}+\frac {(-a-b) x^{10}}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\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 (-a-b) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (6 a b \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}
Time = 10.85 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 (x (-a+x) (-b+x))^{2/3}}{\sqrt [3]{d} x^2}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )-2 \log \left (\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )+\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{4 d^{2/3}} \]
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Time = 0.54 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.45
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}} x +\left (b -x \right ) \left (a -x \right ) \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{3}}{x^{3}}\right )}{4 d^{\frac {2}{3}}}\) | \(135\) |
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Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{2}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{2}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\int \frac {x^2\,\left (2\,a\,b-x\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (2\,x^3\,\left (a+b\right )-x^2\,\left (a^2+4\,a\,b+b^2\right )-a^2\,b^2+x^4\,\left (d-1\right )+2\,a\,b\,x\,\left (a+b\right )\right )} \,d x \]
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