Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]
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Rubi [F]
time = 5.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\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-2 b) x+x^2\right )}{\sqrt {x} \sqrt {-b+x} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d 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 {\sqrt {-a+x^2} \left (-a b+(2 a-2 b) x^2+x^4\right )}{\sqrt {-b+x^2} \left (-a^3 d+\left (b+3 a^2 d\right ) x^2-(1+3 a d) x^4+d 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 {a b \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )}+\frac {2 (-a+b) x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )}+\frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^2-(1+3 a d) x^4+d 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 {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^2-(1+3 a d) x^4+d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a-b) \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^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\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^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 21.16, size = 1571, normalized size = 35.70 \begin {gather*} \frac {2 i (a-x) \sqrt {\frac {-b+x}{a-b}} \left (F\left (i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right ) \left (2 a-b+3 a^2 d-3 a b d-2 d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ] \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]^2-2 d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]-2 d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ] \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]^2\right )-\Pi \left (\frac {a}{\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]};i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right ) \left (2 a-b+11 a^2 d-11 a b d+2 b^2 d+d \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]^2+d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]^3+d^2 \left (3 a (-a+b)+(8 a-4 b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]-2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]^2\right ) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]-d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ] \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]^2+d^2 \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ] \left (3 a (a-b)+(4 a-2 b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]-\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]^2\right )\right )+d \left (\Pi \left (\frac {a}{\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]};i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right ) \left (5 a^2-5 a b+2 b^2+d \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]^2 \left (\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]-\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]\right )-2 d \left (3 a (a-b)+(-2 a+b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]\right ) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]+d \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ] \left (3 a (-a+b)+(8 a-4 b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]\right )\right )+\Pi \left (\frac {a}{\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]};i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right ) \left (3 a (a-b)+d \left (3 a (a-b)+(4 a-2 b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]\right ) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]-d \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ] \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]^2+d \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ] \left (6 a (a-b)+(-4 a+2 b) \text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]+\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]^2\right )\right )\right )\right )}{d^3 \sqrt {1-\frac {a}{x}} \sqrt {x (-a+x) (-b+x)} \left (\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]-\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]\right ) \left (\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,1\right ]-\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]\right ) \left (\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,2\right ]-\text {Root}\left [a^2-a b+(-2 a+b) \text {$\#$1}+\text {$\#$1}^2+d \text {$\#$1}^3\&,3\right ]\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.20, size = 399, normalized size = 9.07
method | result | size |
default | \(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{d \sqrt {b x a -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -a^{3} d \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +a^{3} d +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\) | \(399\) |
elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {b x a -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -a^{3} d \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -a^{3} d -a^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\) | \(401\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (35) = 70\).
time = 0.82, size = 441, normalized size = 10.02 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} - 9 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} + {\left (6 \, a^{2} d - a - b\right )} x^{2} - {\left (4 \, a^{3} d - a b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} d^{2} - a^{3} b d\right )} x}{a^{6} d^{2} + d^{2} x^{6} - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} + 3 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} d^{2} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - {\left (3 \, a^{2} d - b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.92, size = 368, normalized size = 8.36 \begin {gather*} \frac {\ln \left (\frac {\left (a-b+x+a^2\,d-2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+d\,x^2-2\,a\,d\,x\right )\,\left (a\,x^2-a^4\,d-2\,b\,x^2+b^2\,x-2\,d\,x^4+x^3-a^5\,d^2+d^2\,x^5+2\,a^2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a^2\,d\,x^2-5\,a\,d^2\,x^4+5\,a^4\,d^2\,x-a\,b\,x+10\,a^2\,d^2\,x^3-10\,a^3\,d^2\,x^2+a^3\,b\,d+4\,a\,d\,x^3+2\,a^3\,d\,x+2\,b\,d\,x^3-2\,a\,b\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a\,b\,d\,x^2\right )}{\left (-d\,a^3+3\,d\,a^2\,x-3\,d\,a\,x^2+d\,x^3-x^2+b\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2-2\,a\,b-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2+2\,b\,d\,x^2-2\,b\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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