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

Optimal. Leaf size=40 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}{x^2}\right )}{\sqrt {d}} \]

________________________________________________________________________________________

Rubi [F]  time = 16.43, 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 {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[b]*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/Sqrt[(a - x)*(b - x)*x
] - (2*(3*a*b - a*d - b*d)*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][x^2/(Sqrt[-a + x^2]*Sqrt[
-b + x^2]*(a*b*d - a*(1 + b/a)*d*x^2 + d*x^4 - x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*(2*a + 2*b
- d)*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][x^4/(Sqrt[-a + x^2]*Sqrt[-b + x^2]*(a*b*d - a*
(1 + b/a)*d*x^2 + d*x^4 - x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*a*b*d*Sqrt[x]*Sqrt[-a + x]*Sqrt
[-b + x]*Defer[Subst][Defer[Int][1/(Sqrt[-a + x^2]*Sqrt[-b + x^2]*(-(a*b*d) + a*(1 + b/a)*d*x^2 - d*x^4 + x^6)
), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x]

Rubi steps

\begin {align*} \int \frac {x \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x} \left (3 a b-2 (a+b) x+x^2\right )}{\sqrt {-a+x} \sqrt {-b+x} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3 a b-2 (a+b) x^2+x^4\right )}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}}+\frac {a b d+(3 a b-a d-b d) x^2-(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {a b d+(3 a b-a d-b d) x^2-(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+(a+b) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {(-3 a b+a d+b d) x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )}+\frac {(2 a+2 b-d) x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )}+\frac {a b d}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 (2 a+2 b-d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a (3 b-d)+b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {\left (2 (2 a+2 b-d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b d+a \left (1+\frac {b}{a}\right ) d x^2-d x^4+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a (3 b-d)+b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a b d-a \left (1+\frac {b}{a}\right ) d x^2+d x^4-x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 5.45, size = 2258, normalized size = 56.45 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*I)*x*Sqrt[1 - x/b]*(2*a^2*d*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[
Sqrt[-(x/a)]], a/b] + a*b*d*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt
[-(x/a)]], a/b] + 2*b^2*d*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-
(x/a)]], a/b] + 3*a*b*d*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-(x
/a)]], a/b] + 3*a*b*(EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)
]], a/b] - 2*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-(x/a)]], a/b]
)*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2] - EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2
+ #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)]], a/b]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]*Root[-(a*b
*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]^2 + 3*a*b*(2*EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2
+ #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)]], a/b] - EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & ,
3], I*ArcSinh[Sqrt[-(x/a)]], a/b])*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3] - 2*(a + b)*(Ellipti
cPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)]], a/b] + EllipticPi[a/Root
[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-(x/a)]], a/b])*Root[-(a*b*d) + (a*d + b*d)*
#1 - d*#1^2 + #1^3 & , 1]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3] - 2*(a + b)*(2*EllipticPi[a/Ro
ot[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)]], a/b] - EllipticPi[a/Root[-(a*b*d
) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-(x/a)]], a/b])*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#
1^2 + #1^3 & , 2]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3] + EllipticPi[a/Root[-(a*b*d) + (a*d +
b*d)*#1 - d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-(x/a)]], a/b]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 &
, 2]^2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3] + EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 -
d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-(x/a)]], a/b]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]*Root
[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3]^2 - EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #
1^3 & , 3], I*ArcSinh[Sqrt[-(x/a)]], a/b]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]*Root[-(a*b*d)
+ (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3]^2 + EllipticF[I*ArcSinh[Sqrt[-(x/a)]], a/b]*(-(d*(-3*a*b + a*d + b*d))
+ 2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & ,
2]^2 + 2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]^2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^
3 & , 3] + 2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #
1^3 & , 3]^2) + EllipticPi[a/Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[-(x/a)]], a
/b]*(d*(-2*a^2 + a*(-7*b + d) + b*(-2*b + d)) - d*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]^2 + Ro
ot[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]^3 + (-3*a*b + 4*(a + b)*Root[-(a*b*d) + (a*d + b*d)*#1 - d
*#1^2 + #1^3 & , 1] - 2*Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]^2)*Root[-(a*b*d) + (a*d + b*d)*#
1 - d*#1^2 + #1^3 & , 3] - Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1]*Root[-(a*b*d) + (a*d + b*d)*#
1 - d*#1^2 + #1^3 & , 3]^2 + Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2]*(3*a*b + 2*(a + b)*Root[-(a
*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3] - Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3]^2))))/(S
qrt[x/(-a + x)]*Sqrt[x*(-a + x)*(-b + x)]*(Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1] - Root[-(a*b*
d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2])*(Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 1] - Root[-(a*
b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3])*(Root[-(a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 2] - Root[-(
a*b*d) + (a*d + b*d)*#1 - d*#1^2 + #1^3 & , 3]))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.52, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{x^2}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.85, size = 312, normalized size = 7.80 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} b^{2} d^{2} + 6 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} - {\left (6 \, {\left (a + b\right )} d - d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x + 2 \, {\left (3 \, a b d - {\left (a + b\right )} d^{2}\right )} x^{3} - 4 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3} + x^{4}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d}}{a^{2} b^{2} d^{2} - 2 \, d x^{5} + x^{6} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} x^{2} + {\left (2 \, {\left (a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d^{2} x - 2 \, {\left (a b d + {\left (a + b\right )} d^{2}\right )} x^{3}}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a b d - {\left (a + b\right )} d x + d x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x^{2} - {\left (a + b\right )} d x^{3} + d x^{4}\right )}}\right )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((a^2*b^2*d^2 + 6*d*x^5 + x^6 + (a^2 + 4*a*b + b^2)*d^2*x^2 - (6*(a + b)*d - d^2)*x^4 - 2*(a^2*b + a*b
^2)*d^2*x + 2*(3*a*b*d - (a + b)*d^2)*x^3 - 4*(a*b*d*x - (a + b)*d*x^2 + d*x^3 + x^4)*sqrt(a*b*x - (a + b)*x^2
+ x^3)*sqrt(d))/(a^2*b^2*d^2 - 2*d*x^5 + x^6 + (a^2 + 4*a*b + b^2)*d^2*x^2 + (2*(a + b)*d + d^2)*x^4 - 2*(a^2
*b + a*b^2)*d^2*x - 2*(a*b*d + (a + b)*d^2)*x^3))/sqrt(d), sqrt(-d)*arctan(1/2*(a*b*d - (a + b)*d*x + d*x^2 +
x^3)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a*b*d*x^2 - (a + b)*d*x^3 + d*x^4))/d]

________________________________________________________________________________________

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 )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.10, size = 291, normalized size = 7.28

method	result	size
default	\(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +b d \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} d +3 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d +a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha d +a^{2}+b d \right ) \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha d +a^{2}+b d}{a^{2}}, \sqrt {\frac {a}{a -b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -b d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{a^{2}}\)	\(291\)
elliptic	\(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-d \,\textit {\_Z}^{2}+\left (a d +b d \right ) \textit {\_Z} -a b d \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d -a b d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha d +a^{2}+b d \right ) \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha d +a^{2}+b d}{a^{2}}, \sqrt {\frac {a}{a -b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha d -a d -b d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{a^{2}}\)	\(291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(-(-a+x)/a)^(1/2)*((-b+x)/(a-b))^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)*EllipticF((-(-a+x)/a)^
(1/2),(a/(a-b))^(1/2))+2/a^2*sum((-2*_alpha^2*a-2*_alpha^2*b+_alpha^2*d+3*_alpha*a*b-_alpha*a*d-_alpha*b*d+a*b
*d)/(-3*_alpha^2+2*_alpha*d-a*d-b*d)*(_alpha^2+_alpha*a-_alpha*d+a^2+b*d)*(-(-a+x)/a)^(1/2)*((-b+x)/(a-b))^(1/
2)*(1/a*x)^(1/2)/(x*(a*b-a*x-b*x+x^2))^(1/2)*EllipticPi((-(-a+x)/a)^(1/2),(_alpha^2+_alpha*a-_alpha*d+a^2+b*d)
/a^2,(a/(a-b))^(1/2)),_alpha=RootOf(_Z^3-d*_Z^2+(a*d+b*d)*_Z-a*b*d))

________________________________________________________________________________________

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 )} x}{{\left (a b d - {\left (a + b\right )} d x + d x^{2} - x^{3}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.75, size = 457, normalized size = 11.42 \begin {gather*} \left (\sum _{k=1}^3\left (-\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2+2\,b\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-d\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-3\,a\,b\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+b\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-a\,b\,d\right )}{\left (\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (3\,{\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-d\,z^2+d\,z\,\left (a+b\right )-a\,b\,d,z,k\right )+a\,d+b\,d\right )}\right )\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(-(2*b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-b/(root(z^3 - d*z^2 + d*z*(a +
b) - a*b*d, z, k) - b), asin(((b - x)/b)^(1/2)), -b/(a - b))*(2*a*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k
)^2 + 2*b*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^2 - d*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^2
- 3*a*b*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) + a*d*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) + b*
d*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) - a*b*d))/((root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k) - b)
*(x*(a - x)*(b - x))^(1/2)*(a*d + b*d + 3*root(z^3 - d*z^2 + d*z*(a + b) - a*b*d, z, k)^2 - 2*d*root(z^3 - d*z
^2 + d*z*(a + b) - a*b*d, z, k))), k, 1, 3) - (2*b*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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________