Integrand size = 68, antiderivative size = 92 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt [4]{d} (a-x)}\right )}{\sqrt [4]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{\sqrt [4]{d}} \]
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\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x} \sqrt {-b+x} \left (a b-2 a x+x^2\right )}{\sqrt {-a+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 {x^2 \sqrt {-b+x^2} \left (a b-2 a x^2+x^4\right )}{\sqrt {-a+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^4 \sqrt {-b+x^2}}{\sqrt {-a+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 x^2 \sqrt {-b+x^2}}{\sqrt {-a+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^6 \sqrt {-b+x^2}}{\sqrt {-a+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^6 \sqrt {-b+x^2}}{\sqrt {-a+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^4 \sqrt {-b+x^2}}{\sqrt {-a+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 {x^2 \sqrt {-b+x^2}}{\sqrt {-a+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 = 15.84 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {x (-b+x)}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.37 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.88
method | result | size |
default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3} a +\underline {\hspace {1.25 ex}}\alpha ^{3} b +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d +a^{2} d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(357\) |
elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a -\underline {\hspace {1.25 ex}}\alpha ^{3} b -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha a d -a^{2} d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(357\) |
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Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 601, normalized size of antiderivative = 6.53 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\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 {\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 (\frac {a d - d x}{d^{\frac {1}{4}}} + \frac {b d x - d x^{2}}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\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 (\frac {a d - d x}{d^{\frac {1}{4}}} + \frac {b d x - d x^{2}}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} - \frac {i \, \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 (\frac {i \, a d - i \, d x}{d^{\frac {1}{4}}} + \frac {-i \, b d x + i \, d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {i \, \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 (\frac {-i \, a d + i \, d x}{d^{\frac {1}{4}}} + \frac {i \, b d x - i \, d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} \]
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Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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Time = 6.31 (sec) , antiderivative size = 705, normalized size of antiderivative = 7.66 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\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=\left (\sum _{k=1}^4\left (-\frac {2\,b\,\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 (-d\,a^2+a\,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 )-3\,a\,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+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 )}^3+2\,d\,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 )+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 )}^2-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 )}^3-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\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 )}\right )\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
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