Integrand size = 55, antiderivative size = 40 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{x^2}\right )}{\sqrt {d}} \]
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\[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx \]
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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} \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {-a+x} \sqrt {-b+x} \left (-a b d+(a+b) d x-d x^2+x^3\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 \left (3 a b-2 (a+b) x^2+x^4\right )}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\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 {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}}+\frac {a b d+(3 a b-a d-b d) x^2-(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\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 {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, 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 {a b d+(3 a b-a d-b d) x^2-(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\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 {(-3 a b+a d+b d) x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )}+\frac {(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )}+\frac {a b d}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 (2 a+2 b-d) \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 b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a (3 b-d)+b d) \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 b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right ),\frac {b}{a}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {\left (2 (2 a+2 b-d) \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 b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a (3 b-d)+b d) \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 b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ \end{align*}
Time = 12.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x)}}{x^2}\right )}{\sqrt {d}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.82 (sec) , antiderivative size = 291, normalized size of antiderivative = 7.28
method | result | size |
default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +d b \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} d +3 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d +a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}}{b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -d b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2}}\) | \(291\) |
elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +d b \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d -a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +a d +b^{2}}{b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -d b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2}}\) | \(291\) |
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).
Time = 0.47 (sec) , antiderivative size = 312, normalized size of antiderivative = 7.80 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} b^{2} d^{2} + 6 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} - {\left (6 \, {\left (a + b\right )} d - d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x + 2 \, {\left (3 \, a b d - {\left (a + b\right )} d^{2}\right )} x^{3} - 4 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3} + x^{4}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d}}{a^{2} b^{2} d^{2} - 2 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} + {\left (2 \, {\left (a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x - 2 \, {\left (a b d + {\left (a + b\right )} d^{2}\right )} x^{3}}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a b d - {\left (a + b\right )} d x + d x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x^{2} - {\left (a + b\right )} d x^{3} + d x^{4}\right )}}\right )}{d}\right ] \]
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Timed out. \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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\[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\int { -\frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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Time = 5.28 (sec) , antiderivative size = 457, normalized size of antiderivative = 11.42 \[ \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\left (\sum _{k=1}^3\left (-\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2+2\,b\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-d\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-3\,a\,b\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+b\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-a\,b\,d\right )}{\left (\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (3\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d+b\,d\right )}\right )\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
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