3.11.18 \(\int \frac {(-a+x) (-b+x) (-a b+x^2)}{\sqrt {x (-a+x) (-b+x)} (a^2 b^2 d-2 a b (a+b) d x+(-1+a^2 d+4 a b d+b^2 d) x^2-2 (a+b) d x^3+d x^4)} \, dx\)

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

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Rubi [F]  time = 31.54, 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 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d 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*d - 2*a*b*(a + b)*d*x + (-1 + a^2*d
+ 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*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^2*d)
+ 2*a^2*b*(1 + b/a)*d*x^2 + (1 - (a^2 + 4*a*b + b^2)*d)*x^4 + 2*a*(1 + b/a)*d*x^6 - d*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]*Sqr
t[-b + x^2])/(a^2*b^2*d - 2*a^2*b*(1 + b/a)*d*x^2 - (1 - (a^2 + 4*a*b + b^2)*d)*x^4 - 2*a*(1 + b/a)*d*x^6 + d*
x^8), 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 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d 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 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d 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 d-2 a b (a+b) d x^2+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^4-2 (a+b) d x^6+d 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 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) d x^6-d x^8}+\frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) d x^6+d 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 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) d x^6+d 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 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) d x^6-d x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [C]  time = 13.85, size = 24546, normalized size = 318.78 \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*d - 2*a*b*(a + b)*d*x + (-1 +
a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*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 {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}} \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*d - 2*a*b*(a + b)*d*
x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]

[Out]

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

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fricas [B]  time = 1.47, size = 483, normalized size = 6.27 \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^{2} \frac {1}{d^{3}}^{\frac {3}{4}}}{a b - {\left (a + b\right )} x + x^{2}}\right ) - \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d \frac {1}{d^{3}}^{\frac {1}{4}} x + {\left (a b d^{3} - {\left (a + b\right )} d^{3} x + d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{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}}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d \frac {1}{d^{3}}^{\frac {1}{4}} x + {\left (a b d^{3} - {\left (a + b\right )} d^{3} x + d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{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}}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right ) \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*d-2*a*b*(a+b)*d*x+(a^2*d+4*a*b*d+b^2*d-1)*
x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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maple [C]  time = 0.10, size = 451, normalized size = 5.86

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 )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3} a d +\underline {\hspace {1.25 ex}}\alpha ^{3} b d -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2} d -2 a^{2} b^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{d b}\)	\(451\)
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 )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b d +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} d +4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b d -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2} d +2 a^{2} b^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a -3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha \,a^{2} d +4 \underline {\hspace {1.25 ex}}\alpha a d b +\underline {\hspace {1.25 ex}}\alpha \,b^{2} d -a^{2} b d -a \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{d b}\)	\(455\)

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

[Out]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (a-x\right )\,\left (b-x\right )\,\left (a\,b-x^2\right )}{\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (x^2\,\left (d\,a^2+4\,d\,a\,b+d\,b^2-1\right )+d\,x^4+a^2\,b^2\,d-2\,d\,x^3\,\left (a+b\right )-2\,a\,b\,d\,x\,\left (a+b\right )\right )} \,d 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^2*(a^2*d + b^2*d + 4*a*b*d - 1) + d*x^4 + a^2
*b^2*d - 2*d*x^3*(a + b) - 2*a*b*d*x*(a + b))),x)

[Out]

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

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

[Out]

Timed out

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