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

Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}{(a-x)^2}\right )}{\sqrt {d}} \]

________________________________________________________________________________________

Rubi [F]  time = 7.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 4.99, size = 2730, normalized size = 62.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*I)*(a - x)*Sqrt[(-b + x)/(a - b)]*(-5*a^2*d*EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2
+ #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + 5*a*b*d*EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d
)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - 2*b^2*d*EllipticPi[a/Root[a^2*d - a*b*d +
(-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - 3*a^2*d*EllipticPi[a/Root[a
^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + 3*a*b*d*Ellip
ticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] +
3*a*(a - b)*(EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 +
x/a]], a/(a - b)] - 2*EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sq
rt[-1 + x/a]], a/(a - b)])*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2] - EllipticPi[a/Root[a
^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[a^2*d - a*
b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]^2
+ 3*a*(a - b)*(2*EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-
1 + x/a]], a/(a - b)] - EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[
Sqrt[-1 + x/a]], a/(a - b)])*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3] - 2*(2*a - b)*(Elli
pticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]
+ EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a
- b)])*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 +
d*#1^2 + #1^3 & , 3] - 2*(2*a - b)*(2*EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & ,
2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^
3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]
*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3] + EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b
*d)*#1 + d*#1^2 + #1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 +
d*#1^2 + #1^3 & , 2]^2*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3] + EllipticPi[a/Root[a^2*d
- a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[a^2*d - a*b*d
+ (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3]^2 - E
llipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b
)]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1
^2 + #1^3 & , 3]^2 + EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(d*(-3*a^2 + 3*a*b - 2*a*d + b*d) + 2*Roo
t[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #
1^3 & , 2]^2 + 2*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]^2*Root[a^2*d - a*b*d + (-2*a*d
+ b*d)*#1 + d*#1^2 + #1^3 & , 3] + 2*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]*Root[a^2*d
- a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3]^2) + EllipticPi[a/Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 +
d*#1^2 + #1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(d*(11*a^2 - 11*a*b + 2*b^2 + 2*a*d - b*d) + d*Roo
t[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]^2 + Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2
+ #1^3 & , 2]^3 + (3*a*(-a + b) + (8*a - 4*b)*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1] -
2*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]^2)*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d
*#1^2 + #1^3 & , 3] - Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1]*Root[a^2*d - a*b*d + (-2*a
*d + b*d)*#1 + d*#1^2 + #1^3 & , 3]^2 + Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2]*(3*a*(a
- b) + (4*a - 2*b)*Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 3] - Root[a^2*d - a*b*d + (-2*a*
d + b*d)*#1 + d*#1^2 + #1^3 & , 3]^2))))/(Sqrt[1 - a/x]*Sqrt[x*(-a + x)*(-b + x)]*(Root[a^2*d - a*b*d + (-2*a*
d + b*d)*#1 + d*#1^2 + #1^3 & , 1] - Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2])*(Root[a^2*
d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 1] - Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3
& , 3])*(Root[a^2*d - a*b*d + (-2*a*d + b*d)*#1 + d*#1^2 + #1^3 & , 2] - Root[a^2*d - a*b*d + (-2*a*d + b*d)*#
1 + d*#1^2 + #1^3 & , 3]))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.62, 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)^2}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.00, size = 407, normalized size = 9.25 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} - 6 \, {\left (a - d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} - 6 \, {\left (3 \, a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + b d^{2} - 9 \, {\left (a^{2} + a b\right )} d\right )} x^{3} + {\left (15 \, a^{4} + b^{2} d^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} - {\left (4 \, a - d\right )} x^{3} + x^{4} + {\left (6 \, a^{2} - {\left (a + b\right )} d\right )} x^{2} - {\left (4 \, a^{3} - a b d\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} - a^{3} b d\right )} x}{a^{6} - 2 \, {\left (3 \, a + d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} + 2 \, {\left (3 \, a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + b d^{2} + 3 \, {\left (a^{2} + a b\right )} d\right )} x^{3} + {\left (15 \, a^{4} + b^{2} d^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} + {\left (3 \, a - d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} - b d\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((a^6 - 6*(a - d)*x^5 + x^6 + (15*a^2 - 6*(3*a + b)*d + d^2)*x^4 - 2*(10*a^3 + b*d^2 - 9*(a^2 + a*b)*d
)*x^3 + (15*a^4 + b^2*d^2 - 6*(a^3 + 3*a^2*b)*d)*x^2 - 4*(a^4 - (4*a - d)*x^3 + x^4 + (6*a^2 - (a + b)*d)*x^2
- (4*a^3 - a*b*d)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 6*(a^5 - a^3*b*d)*x)/(a^6 - 2*(3*a + d)*x^5 + x
^6 + (15*a^2 + 2*(3*a + b)*d + d^2)*x^4 - 2*(10*a^3 + b*d^2 + 3*(a^2 + a*b)*d)*x^3 + (15*a^4 + b^2*d^2 + 2*(a^
3 + 3*a^2*b)*d)*x^2 - 2*(3*a^5 + a^3*b*d)*x))/sqrt(d), sqrt(-d)*arctan(1/2*(a^3 + (3*a - d)*x^2 - x^3 - (3*a^2
- b*d)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a^2*b*d*x + (2*a + b)*d*x^3 - d*x^4 - (a^2 + 2*a*b)*d*x^2
))/d]

________________________________________________________________________________________

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 + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d\right )} x\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((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3+(3*a^2+b*d)*x-(3*a+d)*x^2+x^3),x,
algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.11, size = 370, normalized size = 8.41

method	result	size
default	\(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+b d \right ) \textit {\_Z} -a^{3}\right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}+\underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha b d +a^{3}+a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) b}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-b d \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\)	\(370\)
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+b d \right ) \textit {\_Z} -a^{3}\right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha b d -a^{3}-a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) b}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-b d \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3+(3*a^2+b*d)*x-(3*a+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*b*sum((4*_alpha^2*a-2*_alpha^2*b+_alpha^2*d-5*_alpha*a^2+_alpha*a*b-_alpha*b*d+a^3+a^
2*b)/(-3*_alpha^2+6*_alpha*a+2*_alpha*d-3*a^2-b*d)*(_alpha^2-3*_alpha*a+_alpha*b-_alpha*d+3*a^2-3*a*b+b^2)/(a^
3-3*a^2*b+3*a*b^2-b^3)*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(x*(a*b-a*x-b*x+x^2))^(1/2)*Ellipti
cPi((-(-b+x)/b)^(1/2),-(_alpha^2-3*_alpha*a+_alpha*b-_alpha*d+3*a^2-3*a*b+b^2)*b/(a^3-3*a^2*b+3*a*b^2-b^3),(b/
(-a+b))^(1/2)),_alpha=RootOf(_Z^3+(-3*a-d)*_Z^2+(3*a^2+b*d)*_Z-a^3))

________________________________________________________________________________________

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 + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d\right )} x\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((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3+(3*a^2+b*d)*x-(3*a+d)*x^2+x^3),x,
algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.69, size = 589, normalized size = 13.39 \begin {gather*} \left (\sum _{k=1}^3\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-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a^2\,b+a^3+4\,a\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2-5\,a^2\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-2\,b\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2+d\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2+a\,b\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )\right )}{\left (\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (3\,a^2-6\,a\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )+3\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )+b\,d\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((a^2*b + x^2*(a - 2*b) + x^3 - a*x*(2*a - b))/((x*(a - x)*(b - x))^(1/2)*(x*(b*d + 3*a^2) - x^2*(3*a + d)
- a^3 + x^3)),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 - z^2*(3*a + d) + z*
(b*d + 3*a^2) - a^3, z, k) - b), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a^2*b + a^3 + 4*a*root(z^3 - z^2*(3*a +
d) + z*(b*d + 3*a^2) - a^3, z, k)^2 - 5*a^2*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - 2*b*roo
t(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k)^2 + d*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z
, k)^2 + a*b*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - b*d*root(z^3 - z^2*(3*a + d) + z*(b*d +
3*a^2) - a^3, z, k)))/((root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - b)*(x*(a - x)*(b - x))^(1/2
)*(b*d + 3*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k)^2 - 6*a*root(z^3 - z^2*(3*a + d) + z*(b*d +
3*a^2) - a^3, z, k) - 2*d*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) + 3*a^2)), 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((a**2*b-a*(2*a-b)*x-(-a+2*b)*x**2+x**3)/(x*(-a+x)*(-b+x))**(1/2)/(-a**3+(3*a**2+b*d)*x-(3*a+d)*x**2+
x**3),x)

[Out]

Timed out

________________________________________________________________________________________