3.14.1 \(\int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} (-a^2 d+2 a d x+(b^2-d) x^2-2 b x^3+x^4)} \, dx\)

Optimal. Leaf size=94 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{x (x-b)}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{x (x-b)}\right )}{d^{3/4}} \]

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Rubi [F]  time = 10.49, 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}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\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)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x + (b^2 - d)*x^2
- 2*b*x^3 + x^4)),x]

[Out]

(4*a*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^2])/(Sqrt[-b + x^2]*(a^2*d - 2
*a*d*x^2 - b^2*(1 - d/b^2)*x^4 + 2*b*x^6 - x^8)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*a*b*Sqrt[x]*Sq
rt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][Sqrt[-a + x^2]/(Sqrt[-b + x^2]*(-(a^2*d) + 2*a*d*x^2 + b^2*(1
- d/b^2)*x^4 - 2*b*x^6 + x^8)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x
]*Defer[Subst][Defer[Int][(x^4*Sqrt[-a + x^2])/(Sqrt[-b + x^2]*(-(a^2*d) + 2*a*d*x^2 + b^2*(1 - d/b^2)*x^4 - 2
*b*x^6 + x^8)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x]

Rubi steps

\begin {align*} \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\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 x+x^2\right )}{\sqrt {x} \sqrt {-b+x} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\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 x^2+x^4\right )}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+\left (b^2-d\right ) x^4-2 b x^6+x^8\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 {2 a x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^2 d-2 a d x^2-b^2 \left (1-\frac {d}{b^2}\right ) x^4+2 b x^6-x^8\right )}+\frac {a b \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )}+\frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\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^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (4 a \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^2 d-2 a d x^2-b^2 \left (1-\frac {d}{b^2}\right ) x^4+2 b x^6-x^8\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^2 d+2 a d x^2+b^2 \left (1-\frac {d}{b^2}\right ) x^4-2 b x^6+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [C]  time = 7.41, size = 4636, normalized size = 49.32 \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 - 3*a*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x + (b^2 - d
)*x^2 - 2*b*x^3 + x^4)),x]

[Out]

((-2*I)*(a - x)*Sqrt[(-b + x)/(a - b)]*((EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b
^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b
)]*(a*(a - b) - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^
2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1]^2))/((Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*
a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2
*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2])*(Root[a^4 - 2*a^3*b + a^2
*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Ro
ot[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#
1^3 + #1^4 & , 3])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)
*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*
a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4])) + ((EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2
+ (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2], I*ArcSin
h[Sqrt[-1 + x/a]], a/(a - b)]*(a*(-a + b) + Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (
6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2]^2))/(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 +
6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b
+ a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2]
) + (a*b*EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 -
d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(Root[a^4 - 2*a^3*b + a^2*b^2
+ (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a
^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3
+ #1^4 & , 3])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^
2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2
- 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3]) + a^2*EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (
-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4], I*ArcSinh[S
qrt[-1 + x/a]], a/(a - b)]*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b +
b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*
#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3])*(-Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^
3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] + Root[a^4 - 2*a^
3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 &
, 3]) + a^2*EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2
- d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(Root[a^4 - 2*a^3*b + a^2*
b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Roo
t[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1
^3 + #1^4 & , 4])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*
#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a
^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4]) + EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (-
4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4], I*ArcSinh[Sq
rt[-1 + x/a]], a/(a - b)]*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b
^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#
1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3
+ 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] - Root[a^4 - 2*a^3*
b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & ,
3])*Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a +
2*b)*#1^3 + #1^4 & , 4]^2 + a*b*EllipticPi[a/Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 +
(6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(Root[
a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3
+ #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1
^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4])*(-Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^
2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] + Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b
- 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4]) + EllipticPi[a/Root[a^4 - 2
*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4
& , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1
+ (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3]^2*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 +
6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b
+ a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4
])*(-Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a +
2*b)*#1^3 + #1^4 & , 2] + Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b
^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4]))/((Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)
*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^
3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3])*(Root[a^4 - 2*a^
3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 &
, 1] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a
+ 2*b)*#1^3 + #1^4 & , 4])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b +
b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 3] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)
*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4])))/((Root[a^4 - 2*a^3*b + a^2*b^2 + (-4
*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 2] - Root[a^4 - 2
*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4
& , 3])*(Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*b + b^2 - d)*#1^2 + (-
4*a + 2*b)*#1^3 + #1^4 & , 2] - Root[a^4 - 2*a^3*b + a^2*b^2 + (-4*a^3 + 6*a^2*b - 2*a*b^2)*#1 + (6*a^2 - 6*a*
b + b^2 - d)*#1^2 + (-4*a + 2*b)*#1^3 + #1^4 & , 4]))))/(Sqrt[1 - a/x]*Sqrt[x*(-a + x)*(-b + x)])

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IntegrateAlgebraic [A]  time = 1.20, size = 94, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x +
(b^2 - d)*x^2 - 2*b*x^3 + x^4)),x]

[Out]

ArcTan[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x))]/d^(3/4) - ArcTanh[(d^(1/4)*Sqrt[a*b*x + (-a -
b)*x^2 + x^3])/(x*(-b + x))]/d^(3/4)

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fricas [B]  time = 1.23, size = 379, normalized size = 4.03 \begin {gather*} \frac {1}{d^{3}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} d \frac {1}{d^{3}}^{\frac {1}{4}}}{b x - x^{2}}\right ) - \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} 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*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x+(b^2-d)*x^2-2*b*x^3+x^4),x,
algorithm="fricas")

[Out]

(d^(-3))^(1/4)*arctan(-sqrt(a*b*x - (a + b)*x^2 + x^3)*d*(d^(-3))^(1/4)/(b*x - x^2)) - 1/4*(d^(-3))^(1/4)*log(
(2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^2 + d)*x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((a*d^3 - d^3*x)*(d^(-3))
^(3/4) + (b*d*x - d*x^2)*(d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*x^3 -
x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)) + 1/4*(d^(-3))^(1/4)*log((2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^2 + d)*
x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((a*d^3 - d^3*x)*(d^(-3))^(3/4) + (b*d*x - d*x^2)*(d^(-3))^(1/4)) - 2*
(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*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}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\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-3*a*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x+(b^2-d)*x^2-2*b*x^3+x^4),x,
algorithm="giac")

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)*sqrt((a -
x)*(b - x)*x)), x)

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maple [C]  time = 0.10, size = 251, normalized size = 2.67

method	result	size
default	\(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha d +2 a d -b d \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha d +2 a d -b d \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(251\)
elliptic	\(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha d +2 a d -b d \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha d +2 a d -b d \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(251\)

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*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x+(b^2-d)*x^2-2*b*x^3+x^4),x,method
=_RETURNVERBOSE)

[Out]

b/d*sum((-_alpha^3+3*_alpha^2*a-2*_alpha*a^2-_alpha*a*b+a^2*b)/(-2*_alpha^3+3*_alpha^2*b-_alpha*b^2+_alpha*d-a
*d)*(_alpha^3-_alpha^2*b-_alpha*d+2*a*d-b*d)/(a^2-2*a*b+b^2)*(-(-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)*EllipticPi((-(-b+x)/b)^(1/2),-(_alpha^3-_alpha^2*b-_alpha*d+2*a*d-b*d)*b/d/(a^2
-2*a*b+b^2),(b/(-a+b))^(1/2)),_alpha=RootOf(_Z^4-2*b*_Z^3+(b^2-d)*_Z^2+2*a*d*_Z-a^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}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\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-3*a*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x+(b^2-d)*x^2-2*b*x^3+x^4),x,
algorithm="maxima")

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)*sqrt((a -
x)*(b - x)*x)), x)

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mupad [B]  time = 1.44, size = 453, normalized size = 4.82 \begin {gather*} \sum _{k=1}^4\frac {2\,b\,\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-a\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left ({\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2-2\,a\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+a\,b\right )}{\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (2\,b^2\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-6\,b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2+4\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3-2\,d\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+2\,a\,d\right )} \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*(a - x)*(b - x))^(1/2)*(x^2*(d - b^2) + a^2*d + 2*b*x^3 - x^4
- 2*a*d*x)),x)

[Out]

symsum((2*b*(root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) - a)*(x/b)^(1/2)*((b - x)/b)^(1/2)*((
a - x)/(a - b))^(1/2)*ellipticPi(-b/(root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) - b), asin(((
b - x)/b)^(1/2)), -b/(a - b))*(a*b - 2*a*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) + root(z^
4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^2))/((root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*
d, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(2*a*d - 2*d*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z,
k) - 6*b*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^2 + 2*b^2*root(z^4 - 2*b*z^3 - z^2*(d - b
^2) + 2*a*d*z - a^2*d, z, k) + 4*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^3)), k, 1, 4)

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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*(-a+x)*(-b+x))**(1/2)/(-a**2*d+2*a*d*x+(b**2-d)*x**2-2*b*x**3
+x**4),x)

[Out]

Timed out

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