Integrand size = 88, antiderivative size = 121 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]
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\[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {-3 b c+a (b+c)-2 (a-b-c) x-x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \left (\frac {a (b+c) \left (1-\frac {3 b c}{a b+a c}\right )}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {2 (-a+b+c) x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )}+\frac {x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (-a^3-b c d+\left (3 a^2+(b+c) d\right ) x-(3 a+d) x^2+x^3\right )}\right ) \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {x^2}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (-a^3-b c d+\left (3 a^2+(b+c) d\right ) x-(3 a+d) x^2+x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (2 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {x}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left ((-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \int \frac {1}{\sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) x+(3 a+d) x^2-x^3\right )} \, dx}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 d+a d \left (b+c-2 x^4\right )+b d \left (-c+x^4\right )+x^4 \left (c d-d x^4+x^8\right )\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3+b c d-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \left (\frac {a^2 x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )}+\frac {2 a x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )}+\frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+x^4\right )}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a^2 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \left (\frac {a x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^4+d x^8-x^{12}\right )}+\frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^4+d x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ & = \frac {\left (4 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (8 a \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 a^2 \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (-a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d-2 a \left (1-\frac {b+c}{2 a}\right ) d x^4-d x^8+x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}-\frac {\left (8 a (a-b-c) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^2 \left (1+\frac {b c-a (b+c)}{a^2}\right ) d+2 a \left (1-\frac {b+c}{2 a}\right ) d x^4+d x^8-x^{12}\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (4 (-3 b c+a (b+c)) \sqrt [4]{-a+x} \sqrt [4]{-b+x} \sqrt [4]{-c+x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b+x^4} \sqrt [4]{a-c+x^4} \left (a^3 \left (1+\frac {b c d}{a^3}\right )-\left (3 a^2+(b+c) d\right ) \left (a+x^4\right )+(3 a+d) \left (a+x^4\right )^2-\left (a+x^4\right )^3\right )} \, dx,x,\sqrt [4]{-a+x}\right )}{\sqrt [4]{(-a+x) (-b+x) (-c+x)}} \\ \end{align*}
Time = 10.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63 \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{-a+x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(-a+x) (-b+x) (-c+x)}}{a-x}\right )\right )}{d^{3/4}} \]
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\[\int \frac {-a b -a c +3 b c +2 \left (a -b -c \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{4}} \left (-a^{3}-b c d +\left (3 a^{2}+b d +c d \right ) x -\left (3 a +d \right ) x^{2}+x^{3}\right )}d x\]
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Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { \frac {a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { \frac {a b + a c - 3 \, b c - 2 \, {\left (a - b - c\right )} x - x^{2}}{{\left (a^{3} + b c d + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d + c d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt [4]{(-a+x) (-b+x) (-c+x)} \left (-a^3-b c d+\left (3 a^2+b d+c d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int \frac {a\,b+a\,c-3\,b\,c+2\,x\,\left (b-a+c\right )-x^2}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{1/4}\,\left (x^2\,\left (3\,a+d\right )-x\,\left (3\,a^2+b\,d+c\,d\right )+a^3-x^3+b\,c\,d\right )} \,d x \]
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