Integrand size = 85, antiderivative size = 235 \[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}} \]
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\[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(-b+x)^2 \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx \\ & = \frac {\left (x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{x^{3/4} (-a+x)^{3/2} \sqrt [4]{-b+x} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}} \\ & = \frac {\left (x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \int \frac {a b-3 b x+2 x^2}{x^{3/4} \sqrt {-a+x} \sqrt [4]{-b+x} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \text {Subst}\left (\int \frac {a b-3 b x^4+2 x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+\left (a^2-d\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \text {Subst}\left (\int \left (\frac {3 b x^4}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (-b d-a^2 \left (1-\frac {d}{a^2}\right ) x^4+2 a x^8-x^{12}\right )}+\frac {a b}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )}+\frac {2 x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}} \\ & = \frac {\left (8 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}+\frac {\left (12 b x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (-b d-a^2 \left (1-\frac {d}{a^2}\right ) x^4+2 a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}+\frac {\left (4 a b x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}} \\ \end{align*}
\[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx \]
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\[\int \frac {\left (b^{2}-2 b x +x^{2}\right ) \left (-a^{2} b +4 a b x -\left (2 a +3 b \right ) x^{2}+2 x^{3}\right )}{\left (x \left (-a +x \right )^{2} \left (-b +x \right )^{3}\right )^{\frac {3}{4}} \left (b d +\left (a^{2}-d \right ) x -2 a \,x^{2}+x^{3}\right )}d x\]
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Timed out. \[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\int { \frac {{\left (a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}\right )} {\left (b^{2} - 2 \, b x + x^{2}\right )}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (2 \, a x^{2} - x^{3} - b d - {\left (a^{2} - d\right )} x\right )}} \,d x } \]
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\[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\int { \frac {{\left (a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}\right )} {\left (b^{2} - 2 \, b x + x^{2}\right )}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (2 \, a x^{2} - x^{3} - b d - {\left (a^{2} - d\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx=\int -\frac {\left (b^2-2\,b\,x+x^2\right )\,\left (x^2\,\left (2\,a+3\,b\right )+a^2\,b-2\,x^3-4\,a\,b\,x\right )}{{\left (-x\,{\left (a-x\right )}^2\,{\left (b-x\right )}^3\right )}^{3/4}\,\left (x^3-2\,a\,x^2+\left (a^2-d\right )\,x+b\,d\right )} \,d x \]
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