Integrand size = 99, antiderivative size = 135 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{3/4}}{(b-x) (-c+x)}\right )}{d^{3/4}} \]
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\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a+x)^2 \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \\ & = \frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} \left (-3 b c+a (b+c)-2 (a-b-c) x-x^2\right )}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right ) (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {2 (-a+b+c) x (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )}+\frac {x^2 (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )}\right ) \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \frac {\left ((-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {x^2 (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (2 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {x (-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left ((-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4}}{(-b+x)^{3/4} (-c+x)^{3/4} \left (b c+a^3 d-\left (b+c+3 a^2 d\right ) x+(1+3 a d) x^2-d x^3\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \left (a+x^4\right )^2}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \left (a+x^4\right )}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {1+2 a d}{d^2 \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {x^4}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {(a-b) (a-c) (1+2 a d)-\left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) x^4+\left (1+4 a d-b d-c d+a^2 d^2\right ) x^8}{d^2 \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {1}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}}+\frac {(a-b) (a-c)+(2 a-b-c) x^4+(1+a d) x^8}{d \left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}-\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = -\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {(a-b) (a-c) (1+2 a d)-\left (b+c-5 a^2 d-b c d-a (2-3 b d-3 c d)\right ) x^4+\left (1+4 a d-b d-c d+a^2 d^2\right ) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {(a-b) (a-c)+(2 a-b-c) x^4+(1+a d) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (1+2 a d) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = -\frac {\left (4 (-3 b c+a (b+c)) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (-a^2+a \left (b+c-2 x^4\right )+b \left (-c+x^4\right )+x^4 \left (c-x^4+d x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {(a-b) (a-c) (-1-2 a d)}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {\left (c-5 a^2 d-a (2-3 b d-3 c d)+b (1-c d)\right ) x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {\left (-1-4 a d+b d+c d-a^2 d^2\right ) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} (-b+x)^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {(a-b) (-a+c)}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {(-2 a+b+c) x^4}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}+\frac {(-1-a d) x^8}{\left (a-b+x^4\right )^{3/4} \left (a-c+x^4\right )^{3/4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right )+2 a \left (1-\frac {b+c}{2 a}\right ) x^4+x^8-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (8 (a-b-c) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d ((-a+x) (-b+x) (-c+x))^{3/4}}+\frac {\left (4 (1+2 a d) (-a+x)^{3/4} \left (\frac {-b+x}{a-b}\right )^{3/4} (-c+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a-c+x^4\right )^{3/4} \left (1+\frac {x^4}{a-b}\right )^{3/4}} \, dx,x,\sqrt [4]{-a+x}\right )}{d^2 ((-a+x) (-b+x) (-c+x))^{3/4}} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \]
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\[\int \frac {\left (a^{2}-2 a x +x^{2}\right ) \left (-a b -a c +3 b c +2 \left (a -b -c \right ) x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {3}{4}} \left (-b c -a^{3} d +\left (3 a^{2} d +b +c \right ) x -\left (3 a d +1\right ) x^{2}+d \,x^{3}\right )}d x\]
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Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} + b c - {\left (3 \, a^{2} d + b + c\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b-a c+3 b c+2 (a-b-c) x+x^2\right )}{((-a+x) (-b+x) (-c+x))^{3/4} \left (-b c-a^3 d+\left (b+c+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int \frac {\left (a^2-2\,a\,x+x^2\right )\,\left (a\,b+a\,c-3\,b\,c+2\,x\,\left (b-a+c\right )-x^2\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{3/4}\,\left (b\,c-x\,\left (3\,d\,a^2+b+c\right )+a^3\,d-d\,x^3+x^2\,\left (3\,a\,d+1\right )\right )} \,d x \]
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