Optimal. Leaf size=44 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{\sqrt {d}} \]
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Rubi [C] time = 7.29, antiderivative size = 376, normalized size of antiderivative = 8.55, number of steps used = 15, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6718, 6728, 117, 116, 169, 538, 537} \begin {gather*} \frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {x (a-x) (b-x)}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \left (-\sqrt {(b d+1)^2-4 a d}-2 a d+b d+1\right ) \Pi \left (\frac {2 a d}{b d-\sqrt {(b d+1)^2-4 a d}+1};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \left (-\sqrt {(b d+1)^2-4 a d}+b d+1\right ) \sqrt {x (a-x) (b-x)}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \left (\sqrt {(b d+1)^2-4 a d}-2 a d+b d+1\right ) \Pi \left (\frac {2 a d}{b d+\sqrt {(b d+1)^2-4 a d}+1};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \left (\sqrt {(b d+1)^2-4 a d}+b d+1\right ) \sqrt {x (a-x) (b-x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 116
Rule 117
Rule 169
Rule 537
Rule 538
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a b-2 a x+x^2}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a-(1+b d) x+d x^2\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {1}{d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}}-\frac {a-a b d-(1-2 a d+b d) x}{d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a+(-1-b d) x+d x^2\right )}\right ) \, dx}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a-a b d-(1-2 a d+b d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a+(-1-b d) x+d x^2\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )}+\frac {-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}-\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (1+b d+\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (1+b d-\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}} \left (1+b d+\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}} \left (1+b d-\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \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}} \left (1+b d+\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \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}} \left (1+b d-\sqrt {-4 a d+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}-\frac {2 \sqrt {a} \left (1-2 a d+b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a d}{1+b d-\sqrt {-4 a d+(1+b d)^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \left (1+b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}}-\frac {2 \sqrt {a} \left (1-2 a d+b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a d}{1+b d+\sqrt {-4 a d+(1+b d)^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \left (1+b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}}\\ \end {align*}
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Mathematica [C] time = 7.22, size = 238, normalized size = 5.41 \begin {gather*} \frac {2 i (a-x)^{3/2} \sqrt {\frac {x}{x-a}} \sqrt {\frac {b-x}{a-x}} \left (-\Pi \left (\frac {2 (a-b) d}{2 a d-b d+\sqrt {(b d+1)^2-4 a d}-1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {a-x}}\right )|1-\frac {b}{a}\right )-\Pi \left (-\frac {2 (a-b) d}{-2 a d+b d+\sqrt {(b d+1)^2-4 a d}+1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {a-x}}\right )|1-\frac {b}{a}\right )+F\left (i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {a-x}}\right )|1-\frac {b}{a}\right )\right )}{\sqrt {-a} d \sqrt {x (x-a) (x-b)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 44, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{\sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.65, size = 231, normalized size = 5.25 \begin {gather*} \left [\frac {\log \left (\frac {d^{2} x^{4} - 2 \, {\left (b d^{2} - 3 \, d\right )} x^{3} + {\left (b^{2} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {d} + 2 \, {\left (3 \, a b d - a\right )} x}{d^{2} x^{4} - 2 \, {\left (b d^{2} + d\right )} x^{3} + {\left (b^{2} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left (a b d + a\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right )}{d}\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 {a b - 2 \, a x + x^{2}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 2508, normalized size = 57.00
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 {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b^{2} d}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}+\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b^{2} d}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}-\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}}{d}\) | \(2508\) |
elliptic | \(-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {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 {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}+\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d^{2}}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right ) d^{2}}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d^{2}}-\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {b^{2} d^{2}-4 a d +2 b d +1}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d^{2}}\) | \(2530\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 437, normalized size = 9.93 \begin {gather*} \frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,d-b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}-1\right )}{d^2\,\left (b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\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}}}{d\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (b\,d-2\,a\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1\right )}{d^2\,\left (b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \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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