Integrand size = 49, antiderivative size = 174 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x} (-2 a b+(a+b) x)}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-a b+(a+b) 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 (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}\right ) \sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}+\frac {\left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}\right ) \sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (\left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}}+\frac {\left (\left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ \end{align*}
Time = 15.66 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.78 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]
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Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 d^{\frac {2}{3}}}\) | \(120\) |
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Timed out. \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x - {\left (a + b\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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\[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x - {\left (a + b\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {x^2\,\left (a+b\right )-2\,a\,b\,x}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \]
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