3.9.74 \(\int \frac {(-a+x) (-b+x) (-a b+x^2)}{x \sqrt {x (-a+x) (-b+x)} (a b-(a+b+d) x+x^2)} \, dx\) [874]

Optimal. Leaf size=67 \[ \frac {2 \sqrt {a b x+(-a-b) x^2+x^3}}{x}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right ) \]

[Out]

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

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Rubi [F]
time = 5.17, 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+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 19.04, size = 504, normalized size = 7.52 \begin {gather*} \frac {\sqrt {x (-a+x) (-b+x)} \left (2-\frac {2 i d \sqrt {\frac {-b+x}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (-b+x)}+\frac {i \left (a^2+b^2+2 b d+d^2+b \sqrt {a^2-2 a (b-d)+(b+d)^2}-d \sqrt {a^2-2 a (b-d)+(b+d)^2}-a \left (2 b-2 d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right )\right ) \sqrt {\frac {-b+x}{a-b}} \Pi \left (\frac {2 a}{a-b-d+\sqrt {4 a d+(-a+b+d)^2}};i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right )}{\sqrt {a^2-2 a (b-d)+(b+d)^2} \sqrt {1-\frac {a}{x}} (b-x)}-\frac {i \left (a^2+b^2+2 b d+d^2-b \sqrt {a^2-2 a (b-d)+(b+d)^2}+d \sqrt {a^2-2 a (b-d)+(b+d)^2}+a \left (-2 b+2 d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right )\right ) \sqrt {\frac {-b+x}{a-b}} \Pi \left (-\frac {2 a}{-a+b+d+\sqrt {4 a d+(-a+b+d)^2}};i \sinh ^{-1}\left (\sqrt {-1+\frac {x}{a}}\right )|\frac {a}{a-b}\right )}{\sqrt {a^2-2 a (b-d)+(b+d)^2} \sqrt {1-\frac {a}{x}} (b-x)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.96, size = 3786, normalized size = 56.51

method	result	size
risch	\(\text {Expression too large to display}\)	\(3422\)
elliptic	\(\text {Expression too large to display}\)	\(3462\)
default	\(\text {Expression too large to display}\)	\(3786\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d*a*(-(-a+x)/a)^(1/2)*((-b+x)/(a-b))^(1/2)*(x/a)^(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*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)*(
(-a+b)*EllipticE((-(-b+x)/b)^(1/2),(b/(-a+b))^(1/2))+a*EllipticF((-(-b+x)/b)^(1/2),(b/(-a+b))^(1/2)))-d*(1/(a^
2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a^3*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*
x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1
/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))-2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)
^(1/2)*a^2*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-
1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*
a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b+2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a^2*(-x/a+1)^(1/2)*
(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d
+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(
1/2)),(a/(a-b))^(1/2))*d+a^2*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(
1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b
-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))+1/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a*(
-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a
^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^
2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b^2+2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a*(-x/a+1)^(1/2)*(-1/(a-b)*b+
1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d
^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-
b))^(1/2))*b*d+a*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-
1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(
a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b+1/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a*(-x/a+1)^(1
/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2
*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^
2)^(1/2)),(a/(a-b))^(1/2))*d^2+a*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^
3)^(1/2)/(1/2*a-1/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1
/2*b-1/2*d-1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d-1/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/
2)*a^3*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*
d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+
2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))+2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a^2*(-x/a+1)^(1/2)*(-1/(a
-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2
*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),
(a/(a-b))^(1/2))*b-2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a^2*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/
a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticP
i((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d+a^2*(-x
/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2
-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+
2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))-1/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b
)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1
/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1
/2))*b^2-2/(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)*a*(-x/a+1)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*
b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a
)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b*d+a*(-x/a+1)^(1/2)*
(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(x/a)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d
+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(1/2*a-1/2*b-1/2*d+1/2*(a^2-2*a*b+2*a*d+b^2+2*b*d+d^2)^(
1/2)),(a/(a-b))^(1/2))*b-1/(a^2-2*a*b+2*a*d+b^2...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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+x)*(-b+x)*(-a*b+x^2)/x/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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+x)*(-b+x)*(-a*b+x**2)/x/(x*(-a+x)*(-b+x))**(1/2)/(a*b-(a+b+d)*x+x**2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.11, size = 722, normalized size = 10.78 \begin {gather*} \frac {b\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d-\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )}-\frac {2\,a\,b\,\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 {2\,b\,d\,\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}}-\frac {2\,b\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\left (a-b\right )\,\mathrm {E}\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\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d+\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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