Integrand size = 75, antiderivative size = 94 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}} \]
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\[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \left (a b-2 a x+x^2\right )}{\sqrt {x} \sqrt {-b+x} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {\sqrt {-a+x^2} \left (a b-2 a x^2+x^4\right )}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+\left (b^2-d\right ) x^4-2 b x^6+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \left (\frac {2 a x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^2 d-2 a d x^2-b^2 \left (1-\frac {d}{b^2}\right ) x^4+2 b x^6-x^8\right )}+\frac {a b \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )}+\frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (4 a \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^2 d-2 a d x^2-b^2 \left (1-\frac {d}{b^2}\right ) x^4+2 b x^6-x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ \end{align*}
Time = 11.97 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.70 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {-\arctan \left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )}{d^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.67
method | result | size |
default | \(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(251\) |
elliptic | \(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(251\) |
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Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 673, normalized size of antiderivative = 7.16 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) + \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (i \, a d^{3} - i \, d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (-i \, b d x + i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) - \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (-i \, a d^{3} + i \, d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (i \, b d x - i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) \]
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Timed out. \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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\[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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Time = 6.52 (sec) , antiderivative size = 453, normalized size of antiderivative = 4.82 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\sum _{k=1}^4\frac {2\,b\,\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-a\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left ({\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2-2\,a\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+a\,b\right )}{\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (2\,b^2\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-6\,b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2+4\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3-2\,d\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+2\,a\,d\right )} \]
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