Optimal. Leaf size=69 \[ \frac {2 \sqrt {x^2 (-a-b)+a b x+x^3}}{x^2}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} x^2}\right ) \]
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Rubi [F] time = 23.71, 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+x) (-b+x) \left (3 a b-2 (a+b) x+x^2\right )}{x^2 \sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-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+x) (-b+x) \left (3 a b-2 (a+b) x+x^2\right )}{x^2 \sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \sqrt {-b+x} \left (3 a b-2 (a+b) x+x^2\right )}{x^{5/2} \left (-a b+(a+b) x-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 ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (3 a b-2 (a+b) x^2+x^4\right )}{x^4 \left (-a b+(a+b) x^2-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 ) \operatorname {Subst}\left (\int \left (-\frac {3 \sqrt {-a+x^2} \sqrt {-b+x^2}}{x^4}+\frac {(-a-b) \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b x^2}+\frac {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a^2-b^2+(a+b-3 a b d) x^2-(a+b) d x^4\right )}{a b \left (a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=-\frac {\left (6 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a^2-b^2+(a+b-3 a b d) x^2-(a+b) d x^4\right )}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{x^2} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a+b) (a-x) (b-x)}{a b \sqrt {(a-x) (b-x) x}}+\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}-\frac {\left (4 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} (-a-b)+x^2}{x^2 \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 ) \operatorname {Subst}\left (\int \left (\frac {(b+a (1-3 b d)) x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6}+\frac {(-a-b) d x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6}+\frac {a^2 \left (1+\frac {b^2}{a^2}\right ) \sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6}\right ) \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} (-a-b)+x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (4 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {-a b+\frac {1}{2} (a+b) x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} (-a+x) \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a}}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)} \sqrt {1-\frac {x}{a}}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {4 (a+b) (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}}-\frac {2 \left (a^2-b^2\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b) \sqrt {x} (-a+x) \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a}}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)} \sqrt {1-\frac {x}{a}}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {2 (a+b) (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}}+\frac {2 (a-b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b} \sqrt {(a-x) (b-x) x}}-\frac {2 \left (a^2-b^2\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [C] time = 9.72, size = 4112, normalized size = 59.59 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.95, size = 338, normalized size = 4.90 \begin {gather*} \left [\frac {\sqrt {d} x^{2} \log \left (\frac {d^{2} x^{6} + 6 \, d x^{5} - {\left (6 \, {\left (a + b\right )} d - 1\right )} x^{4} + a^{2} b^{2} + 2 \, {\left (3 \, a b d - a - b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 4 \, {\left (d x^{4} + a b x - {\left (a + b\right )} x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}{d^{2} x^{6} - 2 \, d x^{5} + {\left (2 \, {\left (a + b\right )} d + 1\right )} x^{4} + a^{2} b^{2} - 2 \, {\left (a b d + a + b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, x^{2}}, \frac {\sqrt {-d} x^{2} \arctan \left (\frac {{\left (d x^{3} + a b - {\left (a + b\right )} x + x^{2}\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 ) + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{x^{2}}\right ] \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*} \int \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 321, normalized size = 4.65
method | result | size |
elliptic | \(\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{x^{2}}-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -3 \underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2} d}\) | \(321\) |
risch | \(\frac {2 \left (a -x \right ) \left (b -x \right )}{x \sqrt {x \left (-a +x \right ) \left (-b +x \right )}}+d \left (-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b +a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha +b}{a^{2} d}, \sqrt {\frac {a}{a -b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2} a^{2}}\right )\) | \(328\) |
default | \(\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b +a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2} d}-3 a b \left (-\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 a b \,x^{2}}-\frac {4 \left (a b -a x -b x +x^{2}\right ) \left (a +b \right )}{3 a^{2} b^{2} \sqrt {x \left (a b -a x -b x +x^{2}\right )}}-\frac {2 \left (-\frac {1}{3 a b}+\frac {2 \left (a +b \right )^{2}}{3 a^{2} b^{2}}+\frac {2 \left (-a -b \right ) \left (a +b \right )}{3 a^{2} b^{2}}\right ) 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 )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {4 \left (a +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \left (\left (-a +b \right ) \EllipticE \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )+a \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )\right )}{3 a^{2} b \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )-\left (-2 a -2 b \right ) \left (-\frac {2 \left (a b -a x -b x +x^{2}\right )}{a b \sqrt {x \left (a b -a x -b x +x^{2}\right )}}-\frac {2 \left (\frac {a +b}{a b}+\frac {-a -b}{a b}\right ) 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 )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \left (\left (-a +b \right ) \EllipticE \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )+a \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )\right )}{a \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )\) | \(818\) |
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*} \int \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.61, size = 97, normalized size = 1.41 \begin {gather*} \sqrt {d}\,\ln \left (\frac {a\,b-a\,x-b\,x+d\,x^3+x^2-2\,\sqrt {d}\,x\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}}{a\,x-a\,b+b\,x+d\,x^3-x^2}\right )+\frac {2\,\sqrt {x^3-b\,x^2-a\,x^2+a\,b\,x}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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