Integrand size = 53, antiderivative size = 81 \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}} \]
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\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (-2 a b+(a+b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \\ & = \frac {\left (x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {\sqrt {x} (-2 a b+(a+b) x)}{(-a+x)^{3/4} (-b+x)^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ & = \frac {\left (x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \left (\frac {\left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}\right ) \sqrt {x}}{(-a+x)^{3/4} (-b+x)^{3/4} \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 {x}}{(-a+x)^{3/4} (-b+x)^{3/4} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ & = \frac {\left (\left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}\right ) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/4} (-b+x)^{3/4} \left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (\left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}\right ) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {\sqrt {x}}{(-a+x)^{3/4} (-b+x)^{3/4} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ \end{align*}
\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \]
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Time = 3.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )}{d^{\frac {3}{4}}}\) | \(81\) |
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Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {x^3\,\left (a+b\right )-2\,a\,b\,x^2}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \]
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