Integrand size = 46, antiderivative size = 173 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \]
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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 {-2 a b+(a+b) x}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a b+(a+b) 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 {a+b-\sqrt {a^2-2 a b+b^2+4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}+\frac {a+b+\sqrt {a^2-2 a b+b^2+4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (\left (a+b-\sqrt {a^2-2 a b+b^2+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 (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (\left (a+b+\sqrt {a^2-2 a b+b^2+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 (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}
Time = 15.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{2 \sqrt [3]{d}} \]
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Time = 1.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+\ln \left (\frac {-d^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}}{d^{\frac {1}{3}}}\) | \(119\) |
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Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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\[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\int \frac {2\,a\,b-x\,\left (a+b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \]
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