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

Optimal. Leaf size=69 \[ \frac {2 \sqrt {x^2 (-a-b)+a b x+x^3}}{x^2}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} x^2}\right ) \]

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Rubi [F]  time = 23.71, 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 (3 a b-2 (a+b) x+x^2\right )}{x^2 \sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(a - x)*(b - x))/(x*Sqrt[(a - x)*(b - x)*x]) + (2*(a + b)*(a - x)*Sqrt[x]*Sqrt[1 - x/b]*EllipticE[ArcSin[Sq
rt[x]/Sqrt[b]], b/a])/(a*Sqrt[b]*Sqrt[(a - x)*(b - x)*x]*Sqrt[1 - x/a]) + (2*(a - b)*Sqrt[x]*Sqrt[1 - x/a]*Sqr
t[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(Sqrt[b]*Sqrt[(a - x)*(b - x)*x]) - (2*(a^2 - b^2)*Sqrt[x]
*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(a*Sqrt[b]*Sqrt[(a - x)*(b - x)*x]) + (2
*(a + b - 3*a*b*d)*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^2]*Sqrt[-b + x^2
])/(a*b - a*(1 + b/a)*x^2 + x^4 - d*x^6), x], x, Sqrt[x]])/(a*b*Sqrt[(a - x)*(b - x)*x]) - (2*(a + b)*d*Sqrt[x
]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^4*Sqrt[-a + x^2]*Sqrt[-b + x^2])/(a*b - a*(1 + b/a)*x^2
+ x^4 - d*x^6), x], x, Sqrt[x]])/(a*b*Sqrt[(a - x)*(b - x)*x]) + (2*(a^2 + b^2)*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b
+ x]*Defer[Subst][Defer[Int][(Sqrt[-a + x^2]*Sqrt[-b + x^2])/(-(a*b) + a*(1 + b/a)*x^2 - x^4 + d*x^6), x], x,
Sqrt[x]])/(a*b*Sqrt[(a - x)*(b - x)*x])

Rubi steps

\begin {align*} \int \frac {(-a+x) (-b+x) \left (3 a b-2 (a+b) x+x^2\right )}{x^2 \sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \sqrt {-b+x} \left (3 a b-2 (a+b) x+x^2\right )}{x^{5/2} \left (-a b+(a+b) x-x^2+d x^3\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 (3 a b-2 (a+b) x^2+x^4\right )}{x^4 \left (-a b+(a+b) x^2-x^4+d x^6\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 {3 \sqrt {-a+x^2} \sqrt {-b+x^2}}{x^4}+\frac {(-a-b) \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b x^2}+\frac {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a^2-b^2+(a+b-3 a b d) x^2-(a+b) d x^4\right )}{a b \left (a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=-\frac {\left (6 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{x^4} \, 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 {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a^2-b^2+(a+b-3 a b d) x^2-(a+b) d x^4\right )}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \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}}{x^2} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a+b) (a-x) (b-x)}{a b \sqrt {(a-x) (b-x) x}}+\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}-\frac {\left (4 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} (-a-b)+x^2}{x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}} \, 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 {(b+a (1-3 b d)) x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6}+\frac {(-a-b) d x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6}+\frac {a^2 \left (1+\frac {b^2}{a^2}\right ) \sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6}\right ) \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} (-a-b)+x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (4 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {-a b+\frac {1}{2} (a+b) x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \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 b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{b \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}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \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 b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \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 b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b) \sqrt {x} \sqrt {-a+x} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a+b) \sqrt {x} (-a+x) \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a}}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)} \sqrt {1-\frac {x}{a}}}-\frac {\left (2 (a-b) (a+b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {4 (a+b) (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}}-\frac {2 \left (a^2-b^2\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \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 b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b) \sqrt {x} (-a+x) \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a}}}{\sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)} \sqrt {1-\frac {x}{a}}}+\frac {\left (2 (a-b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x) (b-x)}{x \sqrt {(a-x) (b-x) x}}+\frac {2 (a+b) (a-x) \sqrt {x} \sqrt {1-\frac {x}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}}+\frac {2 (a-b) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b} \sqrt {(a-x) (b-x) x}}-\frac {2 \left (a^2-b^2\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \sqrt {b} \sqrt {(a-x) (b-x) x}}+\frac {\left (2 \left (a^2+b^2\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a b+a \left (1+\frac {b}{a}\right ) x^2-x^4+d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 (a+b) d \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 b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (a+b-3 a b d) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a b-a \left (1+\frac {b}{a}\right ) x^2+x^4-d x^6} \, dx,x,\sqrt {x}\right )}{a b \sqrt {x (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [C]  time = 9.72, size = 4112, normalized size = 59.59 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[x*(-a + x)*(-b + x)]*(1 - (I*x*Sqrt[(-b + x)/(a - b)]*(3*a^6*d^2*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]],
a/(a - b)] - a^3*(-1 + 3*a*d)*(-a + b + 3*a^2*d)*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + a^3*(-a + b
+ 3*a^2*d)*(-1 + 2*a*d + 2*b*d)*(EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1
^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3
*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]) + a^3*(-1 + 3*a*d)*(b + 4*a^2*d + a*(-1 + b
*d))*(EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[-1
+ x/a]], a/(a - b)] - EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3],
I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]) - 3*a^5*(2*a - b)*d^2*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#
1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + a^2*(2*a - b)*(-1 + 3*a*d)*(-a +
b + 3*a^2*d)*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh
[Sqrt[-1 + x/a]], a/(a - b)] - 2*a^2*(2*a - b)*d^2*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3
*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 -
3*a*d)*#1^2 + d*#1^3 & , 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]^2 - a^2*
(2*a - b)*d^2*(EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSin
h[Sqrt[-1 + x/a]], a/(a - b)] + EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3
& , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#
1^3 & , 1]^2*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3] + a^2*(2*a - b)*d^2*(Ell
ipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]],
a/(a - b)] - EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh
[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]^2*Root[
-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3] - d*(b + 4*a^2*d + a*(-1 + b*d))*(Elliptic
Pi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a
- b)] + EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt
[-1 + x/a]], a/(a - b)] - 2*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & ,
3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3
& , 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]^2*Root[-(a^3*d) + (-a + b + 3*
a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3] + 2*d^2*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[-(a^
3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1]^2*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 -
3*a*d)*#1^2 + d*#1^3 & , 2]^3*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3] + a^2*
(2*a - b)*d^2*(EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSin
h[Sqrt[-1 + x/a]], a/(a - b)] - EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3
& , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#
1^3 & , 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^2 - a^2*(2*a - b)*d^2*(Ell
ipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]],
a/(a - b)] + EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh
[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]*Root[-(
a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^2 + d*(b + 4*a^2*d + a*(-1 + b*d))*(Elliptic
Pi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a
- b)] - 2*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sq
rt[-1 + x/a]], a/(a - b)] + EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & ,
3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3
& , 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]*Root[-(a^3*d) + (-a + b + 3*a^
2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^2 - d*(-1 + 2*a*d + 2*b*d)*(2*EllipticPi[a/Root[-(a^3*d) + (-a + b
+ 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - EllipticPi[a/Root[-(
a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] - Ell
ipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]],
a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1]^2*Root[-(a^3*d) + (-a +
b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 +
d*#1^3 & , 3]^2 + 2*d^2*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1
+ (1 - 3*a*d)*#1^2 + d*#1^3 & , 1]^3*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]*
Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^2 - d*(-1 + 2*a*d + 2*b*d)*(EllipticP
i[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1], I*ArcSinh[Sqrt[-1 + x/a]], a/(a
- b)] - 2*EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2], I*ArcSinh[Sqr
t[-1 + x/a]], a/(a - b)] + EllipticPi[a/Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & ,
3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 &
, 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]^2*Root[-(a^3*d) + (-a + b + 3*a
^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^2 + 2*d^2*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*Root[-(a
^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1]*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 -
3*a*d)*#1^2 + d*#1^3 & , 2]^2*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3]^3))/(d^
2*Sqrt[1 - a/x]*(b - x)*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1]*(Root[-(a^3*d
) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1] - Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*
a*d)*#1^2 + d*#1^3 & , 2])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2]*(Root[-(a^
3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 1] - Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 -
3*a*d)*#1^2 + d*#1^3 & , 3])*(Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 2] - Root
[-(a^3*d) + (-a + b + 3*a^2*d)*#1 + (1 - 3*a*d)*#1^2 + d*#1^3 & , 3])*Root[-(a^3*d) + (-a + b + 3*a^2*d)*#1 +
(1 - 3*a*d)*#1^2 + d*#1^3 & , 3])))/x^2

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IntegrateAlgebraic [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 1.95, size = 338, normalized size = 4.90 \begin {gather*} \left [\frac {\sqrt {d} x^{2} \log \left (\frac {d^{2} x^{6} + 6 \, d x^{5} - {\left (6 \, {\left (a + b\right )} d - 1\right )} x^{4} + a^{2} b^{2} + 2 \, {\left (3 \, a b d - a - b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 4 \, {\left (d x^{4} + a b x - {\left (a + b\right )} x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}{d^{2} x^{6} - 2 \, d x^{5} + {\left (2 \, {\left (a + b\right )} d + 1\right )} x^{4} + a^{2} b^{2} - 2 \, {\left (a b d + a + b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, x^{2}}, \frac {\sqrt {-d} x^{2} \arctan \left (\frac {{\left (d x^{3} + a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x^{2} - {\left (a + b\right )} d x^{3} + d x^{4}\right )}}\right ) + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(d)*x^2*log((d^2*x^6 + 6*d*x^5 - (6*(a + b)*d - 1)*x^4 + a^2*b^2 + 2*(3*a*b*d - a - b)*x^3 + (a^2 +
4*a*b + b^2)*x^2 - 4*(d*x^4 + a*b*x - (a + b)*x^2 + x^3)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 2*(a^2*b +
a*b^2)*x)/(d^2*x^6 - 2*d*x^5 + (2*(a + b)*d + 1)*x^4 + a^2*b^2 - 2*(a*b*d + a + b)*x^3 + (a^2 + 4*a*b + b^2)*x
^2 - 2*(a^2*b + a*b^2)*x)) + 4*sqrt(a*b*x - (a + b)*x^2 + x^3))/x^2, (sqrt(-d)*x^2*arctan(1/2*(d*x^3 + a*b - (
a + b)*x + x^2)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a*b*d*x^2 - (a + b)*d*x^3 + d*x^4)) + 2*sqrt(a*b*x -
(a + b)*x^2 + x^3))/x^2]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+x)*(-b+x)*(3*a*b-2*(a+b)*x+x^2)/x^2/(x*(-a+x)*(-b+x))^(1/2)/(-a*b+(a+b)*x-x^2+d*x^3),x, algorith
m="giac")

[Out]

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

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maple [C]  time = 0.18, size = 321, normalized size = 4.65

method	result	size
elliptic	\(\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{x^{2}}-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -3 \underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2} d}\)	\(321\)
risch	\(\frac {2 \left (a -x \right ) \left (b -x \right )}{x \sqrt {x \left (-a +x \right ) \left (-b +x \right )}}+d \left (-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b +a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha +b}{a^{2} d}, \sqrt {\frac {a}{a -b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2} a^{2}}\right )\)	\(328\)
default	\(\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b +a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \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 ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{b^{2} d}-3 a b \left (-\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 a b \,x^{2}}-\frac {4 \left (a b -a x -b x +x^{2}\right ) \left (a +b \right )}{3 a^{2} b^{2} \sqrt {x \left (a b -a x -b x +x^{2}\right )}}-\frac {2 \left (-\frac {1}{3 a b}+\frac {2 \left (a +b \right )^{2}}{3 a^{2} b^{2}}+\frac {2 \left (-a -b \right ) \left (a +b \right )}{3 a^{2} b^{2}}\right ) 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 {4 \left (a +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \left (\left (-a +b \right ) \EllipticE \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )+a \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )\right )}{3 a^{2} b \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )-\left (-2 a -2 b \right ) \left (-\frac {2 \left (a b -a x -b x +x^{2}\right )}{a b \sqrt {x \left (a b -a x -b x +x^{2}\right )}}-\frac {2 \left (\frac {a +b}{a b}+\frac {-a -b}{a b}\right ) 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 {2 \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \left (\left (-a +b \right ) \EllipticE \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )+a \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )\right )}{a \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right )\)	\(818\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, a b - 2 \, {\left (a + b\right )} x + x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (d x^{3} - a b + {\left (a + b\right )} x - x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+x)*(-b+x)*(3*a*b-2*(a+b)*x+x^2)/x^2/(x*(-a+x)*(-b+x))^(1/2)/(-a*b+(a+b)*x-x^2+d*x^3),x, algorith
m="maxima")

[Out]

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

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mupad [B]  time = 4.61, size = 97, normalized size = 1.41 \begin {gather*} \sqrt {d}\,\ln \left (\frac {a\,b-a\,x-b\,x+d\,x^3+x^2-2\,\sqrt {d}\,x\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}}{a\,x-a\,b+b\,x+d\,x^3-x^2}\right )+\frac {2\,\sqrt {x^3-b\,x^2-a\,x^2+a\,b\,x}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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