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

Optimal. Leaf size=77 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}} \]

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Rubi [F]  time = 19.43, 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 {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

Rubi steps

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

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Mathematica [C]  time = 13.75, size = 22729, normalized size = 295.18 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.36, size = 77, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.85, size = 406, normalized size = 5.27 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} d^{\frac {1}{4}}}{a b - {\left (a + b\right )} x + x^{2}}\right )}{d^{\frac {1}{4}}} - \frac {\log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {3}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {3}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b
)*x^3+x^4),x, algorithm="fricas")

[Out]

-arctan(sqrt(a*b*x - (a + b)*x^2 + x^3)*d^(1/4)/(a*b - (a + b)*x + x^2))/d^(1/4) - 1/4*log((a^2*b^2 - 2*(a + b
)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x
+ (a*b*d - (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*
x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4) + 1/4*log((a^2*b^2 - 2*(a + b)*x^3 + x
^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x + (a*b*d
- (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4
+ (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4)

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

[Out]

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

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maple [C]  time = 0.10, size = 425, normalized size = 5.52

method	result	size
default	\(-\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}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\left (-2 a -2 b \right ) \textit {\_Z}^{3}+\left (a^{2}+4 a b +b^{2}-d \right ) \textit {\_Z}^{2}+\left (-2 a^{2} b -2 a \,b^{2}\right ) \textit {\_Z} +a^{2} b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3} a +\underline {\hspace {1.25 ex}}\alpha ^{3} b -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 a^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} d \right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha 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 {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +b d}{b d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-4 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+a^{2} b +a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b d}\)	\(425\)
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}}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\left (-2 a -2 b \right ) \textit {\_Z}^{3}+\left (a^{2}+4 a b +b^{2}-d \right ) \textit {\_Z}^{2}+\left (-2 a^{2} b -2 a \,b^{2}\right ) \textit {\_Z} +a^{2} b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a -\underline {\hspace {1.25 ex}}\alpha ^{3} b +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 a^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} d \right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha 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 {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +b d}{b d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-4 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+a^{2} b +a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b d}\)	\(427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*EllipticF((-(-b+x)/b)^(
1/2),(b/(-a+b))^(1/2))+1/b/d*sum((_alpha^3*a+_alpha^3*b-_alpha^2*a^2-4*_alpha^2*a*b-_alpha^2*b^2+3*_alpha*a^2*
b+3*_alpha*a*b^2-2*a^2*b^2+_alpha^2*d)/(-2*_alpha^3+3*_alpha^2*a+3*_alpha^2*b-_alpha*a^2-4*_alpha*a*b-_alpha*b
^2+a^2*b+a*b^2+_alpha*d)*(-_alpha^3+2*_alpha^2*a+_alpha^2*b-_alpha*a^2-2*_alpha*a*b+a^2*b+_alpha*d+b*d)*(-(-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),(-_alph
a^3+2*_alpha^2*a+_alpha^2*b-_alpha*a^2-2*_alpha*a*b+a^2*b+_alpha*d+b*d)/b/d,(b/(-a+b))^(1/2)),_alpha=RootOf(_Z
^4+(-2*a-2*b)*_Z^3+(a^2+4*a*b+b^2-d)*_Z^2+(-2*a^2*b-2*a*b^2)*_Z+a^2*b^2))

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

[Out]

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

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mupad [B]  time = 1.78, size = 1150, normalized size = 14.94 \begin {gather*} \left (\sum _{k=1}^4\left (-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (-2\,a^2\,b^2+a^2\,b\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )\,3-a^2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+3\,a\,b^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-4\,a\,b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+a\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3-b^2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3+d\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2\right )}{\left (\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (a^2\,b-a^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+a\,b^2-4\,a\,b\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+3\,a\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2-b^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+3\,b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2-2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3+d\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )\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(-((a - x)*(b - x)*(a*b - x^2))/((x*(a - x)*(b - x))^(1/2)*(x^4 - 2*x^3*(a + b) + a^2*b^2 + x^2*(4*a*b - d
+ a^2 + b^2) - 2*a*b*x*(a + b))),x)

[Out]

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

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

[Out]

Timed out

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