3.11.61 \(\int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} (a-(1+b d) x+d x^2)} \, dx\)

Optimal. Leaf size=80 \[ 2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )+\frac {2 \sqrt {x^2 (-a-b)+a b x+x^3}}{x (x-b)} \]

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Rubi [F]  time = 12.22, 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-3 a x^2+x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x + d*x^2)
),x]

[Out]

(-4*Sqrt[(a - x)*(b - x)*x])/(b^2*d*x) + (2*Sqrt[(a - x)*(b - x)*x])/(b*d*(b - x)*x) - (4*Sqrt[a]*Sqrt[(a - x)
*(b - x)*x]*Sqrt[1 - x/a]*EllipticE[ArcSin[Sqrt[x]/Sqrt[a]], a/b])/(b^2*d*(a - x)*Sqrt[x]*Sqrt[1 - x/b]) + (2*
Sqrt[a]*Sqrt[(a - x)*(b - x)*x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[a]], a/b])/(b*d*(a -
x)*(b - x)*Sqrt[x]) + ((1 - 2*a*d + b*d + Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[(a - x)*(b - x)*x]*Defer[Int][Sqrt
[a - x]/((b - x)^(3/2)*x^(3/2)*(-1 - b*d - Sqrt[1 - 4*a*d + 2*b*d + b^2*d^2] + 2*d*x)), x])/(d*Sqrt[a - x]*Sqr
t[b - x]*Sqrt[x]) + ((1 - 2*a*d + b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[(a - x)*(b - x)*x]*Defer[Int][Sqrt[a
- x]/((b - x)^(3/2)*x^(3/2)*(-1 - b*d + Sqrt[1 - 4*a*d + 2*b*d + b^2*d^2] + 2*d*x)), x])/(d*Sqrt[a - x]*Sqrt[b
- x]*Sqrt[x])

Rubi steps

\begin {align*} \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx &=\int \frac {\sqrt {(a-x) (b-x) x} \left (a b-2 a x+x^2\right )}{(b-x)^2 x^2 \left (a-(1+b d) x+d x^2\right )} \, dx\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \left (a b-2 a x+x^2\right )}{(b-x)^{3/2} x^{3/2} \left (a-(1+b d) x+d x^2\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\sqrt {a-x}}{d (b-x)^{3/2} x^{3/2}}-\frac {\sqrt {a-x} (a-a b d-(1-2 a d+b d) x)}{d (b-x)^{3/2} x^{3/2} \left (a+(-1-b d) x+d x^2\right )}\right ) \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2}} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} (a-a b d-(1-2 a d+b d) x)}{(b-x)^{3/2} x^{3/2} \left (a+(-1-b d) x+d x^2\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=\frac {2 \sqrt {(a-x) (b-x) x}}{b d (b-x) x}-\frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )}+\frac {\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {-a+\frac {x}{2}}{\sqrt {a-x} \sqrt {b-x} x^{3/2}} \, dx}{b d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {4 \sqrt {(a-x) (b-x) x}}{b^2 d x}+\frac {2 \sqrt {(a-x) (b-x) x}}{b d (b-x) x}+\frac {\left (4 \sqrt {(a-x) (b-x) x}\right ) \int \frac {-\frac {a b}{4}+\frac {a x}{2}}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{a b^2 d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {4 \sqrt {(a-x) (b-x) x}}{b^2 d x}+\frac {2 \sqrt {(a-x) (b-x) x}}{b d (b-x) x}-\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {b-x}}{\sqrt {a-x} \sqrt {x}} \, dx}{b^2 d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}+\frac {\sqrt {(a-x) (b-x) x} \int \frac {1}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{b d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {4 \sqrt {(a-x) (b-x) x}}{b^2 d x}+\frac {2 \sqrt {(a-x) (b-x) x}}{b d (b-x) x}-\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (2 \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}\right ) \int \frac {\sqrt {1-\frac {x}{b}}}{\sqrt {x} \sqrt {1-\frac {x}{a}}} \, dx}{b^2 d (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}+\frac {\left (\sqrt {(a-x) (b-x) 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}{b d (a-x) (b-x) \sqrt {x}}\\ &=-\frac {4 \sqrt {(a-x) (b-x) x}}{b^2 d x}+\frac {2 \sqrt {(a-x) (b-x) x}}{b d (b-x) x}-\frac {4 \sqrt {a} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{b^2 d (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}+\frac {2 \sqrt {a} \sqrt {(a-x) (b-x) 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 )}{b d (a-x) (b-x) \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x}}{(b-x)^{3/2} x^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ \end {align*}

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Mathematica [C]  time = 4.76, size = 277, normalized size = 3.46 \begin {gather*} \frac {2 (x-a) \left (i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {2 a d}{2 a d-b d+\sqrt {(b d+1)^2-4 a d}-1};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {2 a d}{-2 a d+b d+\sqrt {(b d+1)^2-4 a d}+1};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )-i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+\sqrt {\frac {x}{a}-1}\right )}{\sqrt {\frac {x}{a}-1} \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x +
d*x^2)),x]

[Out]

(2*(-a + x)*(Sqrt[-1 + x/a] - I*Sqrt[x/a]*Sqrt[(-b + x)/(a - b)]*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b
)] + I*Sqrt[x/a]*Sqrt[(-b + x)/(a - b)]*EllipticPi[(2*a*d)/(-1 + 2*a*d - b*d + Sqrt[-4*a*d + (1 + b*d)^2]), I*
ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + I*Sqrt[x/a]*Sqrt[(-b + x)/(a - b)]*EllipticPi[(-2*a*d)/(1 - 2*a*d + b*d
+ Sqrt[-4*a*d + (1 + b*d)^2]), I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/(Sqrt[x*(-a + x)*(-b + x)]*Sqrt[-1 + x/
a])

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IntegrateAlgebraic [A]  time = 1.03, size = 83, normalized size = 1.04 \begin {gather*} \frac {2 \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} x (-b+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a - (1 +
b*d)*x + d*x^2)),x]

[Out]

(2*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x)) - 2*Sqrt[d]*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d
]*x*(-b + x))]

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fricas [A]  time = 0.88, size = 312, normalized size = 3.90 \begin {gather*} \left [\frac {{\left (b x - x^{2}\right )} \sqrt {d} \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 ) - 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (b x - x^{2}\right )}}, \frac {{\left (b x - x^{2}\right )} \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 ) - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{b x - x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, algorithm="
fricas")

[Out]

[1/2*((b*x - x^2)*sqrt(d)*log((d^2*x^4 - 2*(b*d^2 - 3*d)*x^3 + (b^2*d^2 - 6*(a + b)*d + 1)*x^2 + a^2 - 4*sqrt(
a*b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b*d - 1)*x - a)*sqrt(d) + 2*(3*a*b*d - a)*x)/(d^2*x^4 - 2*(b*d^2 + d)*x^3
+ (b^2*d^2 + 2*(a + b)*d + 1)*x^2 + a^2 - 2*(a*b*d + a)*x)) - 4*sqrt(a*b*x - (a + b)*x^2 + x^3))/(b*x - x^2),
((b*x - x^2)*sqrt(-d)*arctan(1/2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b*d - 1)*x - a)*sqrt(-d)/(a*b*d*x
- (a + b)*d*x^2 + d*x^3)) - 2*sqrt(a*b*x - (a + b)*x^2 + x^3))/(b*x - x^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )} {\left (b - x\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, algorithm="
giac")

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/(sqrt((a - x)*(b - x)*x)*(d*x^2 - (b*d + 1)*x + a)*(b - x)*x
), x)

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maple [C]  time = 0.39, size = 2557, normalized size = 31.96

method	result	size
elliptic	\(-\frac {2 \left (a b -a x -b x +x^{2}\right )}{b \sqrt {x \left (a b -a x -b x +x^{2}\right )}}+\frac {-2 a x +2 x^{2}}{b \sqrt {\left (-b +x \right ) \left (-a x +x^{2}\right )}}-\frac {2 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {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 {b^{3} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) 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 (\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 b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\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 (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\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 {b^{3} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) 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 (\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 b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\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 (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-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 (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}\)	\(2557\)
risch	\(-\frac {2 \left (a -x \right ) \left (b -x \right )}{b \sqrt {x \left (-a +x \right ) \left (-b +x \right )}}+\frac {-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \EllipticE \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-b \left (\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}\right )+\left (a -b \right ) b \left (\frac {-2 a x +2 x^{2}}{b \left (a -b \right ) \sqrt {\left (-b +x \right ) \left (-a x +x^{2}\right )}}-\frac {2 \left (-\frac {1}{b}+\frac {a}{b \left (a -b \right )}\right ) 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 )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \EllipticE \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{b \left (a -b \right ) \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )}{b}\)	\(2838\)
default	\(\left (a -b \right ) \left (\frac {-2 a x +2 x^{2}}{b \left (a -b \right ) \sqrt {\left (-b +x \right ) \left (-a x +x^{2}\right )}}+\frac {2 \left (\frac {1}{b}-\frac {a}{b \left (a -b \right )}\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 )}{\left (a -b \right ) \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )+a \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 )-\frac {b^{3} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) 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 (\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 b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\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 (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}-\frac {1}{2 d}-\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b -\frac {b d +1+\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\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 {b^{3} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) 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 (\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 b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-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 (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {b^{2} \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}-\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (\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 b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-a +b}}\right ) a}{\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 (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right )}+\frac {b \sqrt {1-\frac {x}{b}}\, \sqrt {-\frac {a}{-a +b}+\frac {x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {b}{b +\frac {-b d +\sqrt {b^{2} d^{2}-4 a d +2 b d +1}-1}{2 d}}, \sqrt {\frac {b}{-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 (\frac {b}{2}+\frac {\sqrt {b^{2} d^{2}-4 a d +2 b d +1}}{2 d}-\frac {1}{2 d}\right ) d}\)	\(2937\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x,method=_RETURNVERB
OSE)

[Out]

-2*(a*b-a*x-b*x+x^2)/b/(x*(a*b-a*x-b*x+x^2))^(1/2)+2*(-a*x+x^2)/b/((-b+x)*(-a*x+x^2))^(1/2)-2*b*(1-x/b)^(1/2)*
(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)*EllipticF((-(-b+x)/b)^(1/2),(b/(-a+b)
)^(1/2))-1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a
*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/
d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*d-2/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)
*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*
b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))
+4/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2
+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(
b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a-1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*
a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)
)*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))/d+2*b*(1-x/
b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-
4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b)
)^(1/2))*a-b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1
/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+
1)^(1/2))),(b/(-a+b))^(1/2))-b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3
)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*
d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))/d+1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(-1/(-a+b)*a+
1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*
EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*d+2/(b^2*d^2
-4*a*d+2*b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/
2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*
d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))+2*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a
*x^2-b*x^2+x^3)^(1/2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*
(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a-b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)
*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+
x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))-b*(1-x/b)^(1/2)*(-1/(-a+b)*a+
1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*
EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))/d-4/(b^2*d^2
-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)
/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*d+
2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a+1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)
*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*Elliptic
Pi((-(-b+x)/b)^(1/2),b/(b+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )} {\left (b - x\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, algorithm="
maxima")

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/(sqrt((a - x)*(b - x)*x)*(d*x^2 - (b*d + 1)*x + a)*(b - x)*x
), x)

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mupad [B]  time = 0.87, size = 628, normalized size = 7.85 \begin {gather*} \frac {2\,a\,\left (\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\frac {\sqrt {\frac {b-x}{a-b}+1}\,\sqrt {\frac {b-x}{b}}}{\sqrt {1-\frac {b-x}{b}}}}{\frac {b}{a-b}+1}-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\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}}+\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\,\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\,a\,\left (a-b\right )\,\sqrt {\frac {x}{a}}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\middle |\frac {a}{b}\right )-\frac {a\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\right )}{2\,b\,\sqrt {1-\frac {x}{b}}}\right )\,\sqrt {\frac {a-x}{a}}\,\sqrt {\frac {b-x}{b}}}{b\,\left (\frac {a}{b}-1\right )\,\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\,\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.

[In]

int((a^2*b + 3*a*x^2 - x^3 - a*x*(2*a + b))/(x*(b - x)*(x*(a - x)*(b - x))^(1/2)*(a - x*(b*d + 1) + d*x^2)),x)

[Out]

(2*a*((ellipticE(asin(((b - x)/b)^(1/2)), -b/(a - b)) - (((b - x)/(a - b) + 1)^(1/2)*((b - x)/b)^(1/2))/(1 - (
b - x)/b)^(1/2))/(b/(a - b) + 1) - ellipticF(asin(((b - x)/b)^(1/2)), -b/(a - b)))*(x/b)^(1/2)*((b - x)/b)^(1/
2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*b*x)^(1/2) + (b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a -
b))^(1/2)*ellipticPi(b/(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)),
-b/(a - b))*(2*a*d - b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) - 1))/(d*(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 +
1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) + (2*a*(a - b)*(x/a)^(1/2)*(ellipticE(asin((x/a)^(1/2
)), a/b) - (a*sin(2*asin((x/a)^(1/2))))/(2*b*(1 - x/b)^(1/2)))*((a - x)/a)^(1/2)*((b - x)/b)^(1/2))/(b*(a/b -
1)*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/
(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)), -b/(a - b))*(b*d - 2*a*d
+ (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1))/(d*(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d))*(x
^3 - x^2*(a + b) + a*b*x)^(1/2))

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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.

[In]

integrate((-a**2*b+a*(2*a+b)*x-3*a*x**2+x**3)/x/(-b+x)/(x*(-a+x)*(-b+x))**(1/2)/(a-(b*d+1)*x+d*x**2),x)

[Out]

Timed out

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