Integrand size = 54, antiderivative size = 81 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}} \]
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\[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 a b-2 (a+b) x^4+x^8\right )}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (-a b d+(a+b) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {2 (a+b) x^6}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )}+\frac {3 a b x^2}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^4-d x^8+x^{12}\right )}+\frac {x^{10}}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^4-d x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)}}+\frac {\left (12 a b \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)}}+\frac {\left (8 (a+b) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{-a+x^4} \sqrt [4]{-b+x^4} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)}} \\ \end{align*}
Time = 11.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )\right )}{d^{3/4}} \]
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\[\int \frac {3 a b -2 \left (a +b \right ) x +x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{4}} \left (-a b d +\left (a +b \right ) d x -d \,x^{2}+x^{3}\right )}d x\]
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Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\int \frac {3\,a\,b+x^2-2\,x\,\left (a+b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/4}\,\left (-x^3+d\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\right )} \,d x \]
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