∫ (−a+x)(−b+x)(−ab+x2)______ x∘ ___________ x(−a+x)(−b+x) (ab−(a+b+d)x+x2) dx

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

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

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Rubi [F] time = 7.40, 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] time = 8.50, size = 504, normalized size = 7.52 \begin {gather*} \frac {\sqrt {x (x-a) (x-b)} \left (\frac {i \left (-a \left (\sqrt {a^2-2 a (b-d)+(b+d)^2}+2 b-2 d\right )+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+2 b d+d^2\right ) \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {2 a}{a-b-d+\sqrt {(-a+b+d)^2+4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (b-x) \sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {i \left (a \left (\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 b+2 d\right )-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+2 b d+d^2\right ) \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {2 a}{-a+b+d+\sqrt {(-a+b+d)^2+4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (b-x) \sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {2 i d \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (x-b)}+2\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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IntegrateAlgebraic [A] time = 0.18, size = 67, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

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fricas [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+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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )} x}\,{d x} \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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maple [C] time = 0.38, size = 3422, normalized size = 51.07


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

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

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

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

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

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

a*x)^(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))*Ellipti

cPi((-(-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*(

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

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

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

*(1/a*x)^(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))*Ell

ipticPi((-(-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*(1-1/a*x)^(1/2)*(-b/(a-b)+1/(a-b)*x)^(1/2)*(1/a*x)^(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*(1-1/a*x)^(1/2)*(-b/(a-b

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

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