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

Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]

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Rubi [F]  time = 7.75, 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^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*x - (1
+ 3*a*d)*x^2 + d*x^3)),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^3*d - b*(1
+ (3*a^2*d)/b)*x^2 + (1 + 3*a*d)*x^4 - d*x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] - (4*(a - b)*Sqrt[x]*
Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^2])/(Sqrt[-b + x^2]*(a^3*d - b*(1 + (3*a^2*
d)/b)*x^2 + (1 + 3*a*d)*x^4 - d*x^6)), 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^3*d) + b*(1 + (3*a^2*d)/b)*x^2 - (1
+ 3*a*d)*x^4 + d*x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x]

Rubi steps

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

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Mathematica [C]  time = 5.18, size = 1571, normalized size = 35.70

result too large to display

Warning: Unable to verify antiderivative.

[In]

Integrate[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*
x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

((2*I)*(a - x)*Sqrt[(-b + x)/(a - b)]*(EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(2*a - b + 3*a^2*d - 3*
a*b*d - 2*d^2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1]*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*
#1^3 & , 2]^2 - 2*d^2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1]^2*Root[a^2 - a*b + (-2*a + b)*#1 +
#1^2 + d*#1^3 & , 3] - 2*d^2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 2]*Root[a^2 - a*b + (-2*a + b
)*#1 + #1^2 + d*#1^3 & , 3]^2) - EllipticPi[a/Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1], I*ArcSinh
[Sqrt[-1 + x/a]], a/(a - b)]*(2*a - b + 11*a^2*d - 11*a*b*d + 2*b^2*d + d*Root[a^2 - a*b + (-2*a + b)*#1 + #1^
2 + d*#1^3 & , 2]^2 + d^2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 2]^3 + d^2*(3*a*(-a + b) + (8*a -
4*b)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1] - 2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3
& , 1]^2)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3] - d^2*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 +
d*#1^3 & , 1]*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3]^2 + d^2*Root[a^2 - a*b + (-2*a + b)*#1 +
#1^2 + d*#1^3 & , 2]*(3*a*(a - b) + (4*a - 2*b)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3] - Root[a
^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3]^2)) + d*(EllipticPi[a/Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 +
d*#1^3 & , 2], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(5*a^2 - 5*a*b + 2*b^2 + d*Root[a^2 - a*b + (-2*a + b)*#1
+ #1^2 + d*#1^3 & , 2]^2*(Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1] - Root[a^2 - a*b + (-2*a + b)
*#1 + #1^2 + d*#1^3 & , 3]) - 2*d*(3*a*(a - b) + (-2*a + b)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & ,
1])*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3] + d*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3
& , 2]*(3*a*(-a + b) + (8*a - 4*b)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3])) + EllipticPi[a/Root
[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3], I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]*(3*a*(a - b) + d*(3*a
*(a - b) + (4*a - 2*b)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1])*Root[a^2 - a*b + (-2*a + b)*#1 +
#1^2 + d*#1^3 & , 3] - d*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1]*Root[a^2 - a*b + (-2*a + b)*#1
+ #1^2 + d*#1^3 & , 3]^2 + d*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 2]*(6*a*(a - b) + (-4*a + 2*b
)*Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3] + Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3
]^2)))))/(d^3*Sqrt[1 - a/x]*Sqrt[x*(-a + x)*(-b + x)]*(Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1] -
Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 2])*(Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 1]
- Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3])*(Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & ,
2] - Root[a^2 - a*b + (-2*a + b)*#1 + #1^2 + d*#1^3 & , 3]))

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IntegrateAlgebraic [A]  time = 0.65, size = 44, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b +
3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

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

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fricas [B]  time = 1.22, size = 441, normalized size = 10.02 \begin {gather*} \left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} - 9 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} + {\left (6 \, a^{2} d - a - b\right )} x^{2} - {\left (4 \, a^{3} d - a b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} d^{2} - a^{3} b d\right )} x}{a^{6} d^{2} + d^{2} x^{6} - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} + 3 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} d^{2} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - {\left (3 \, a^{2} d - b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((a^6*d^2 + d^2*x^6 - 6*(a*d^2 - d)*x^5 + (15*a^2*d^2 - 6*(3*a + b)*d + 1)*x^4 - 2*(10*a^3*d^2 - 9*(a^
2 + a*b)*d + b)*x^3 + (15*a^4*d^2 + b^2 - 6*(a^3 + 3*a^2*b)*d)*x^2 - 4*(a^4*d + d*x^4 - (4*a*d - 1)*x^3 + (6*a
^2*d - a - b)*x^2 - (4*a^3*d - a*b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 6*(a^5*d^2 - a^3*b*d)*x)/(a^6
*d^2 + d^2*x^6 - 2*(3*a*d^2 + d)*x^5 + (15*a^2*d^2 + 2*(3*a + b)*d + 1)*x^4 - 2*(10*a^3*d^2 + 3*(a^2 + a*b)*d
+ b)*x^3 + (15*a^4*d^2 + b^2 + 2*(a^3 + 3*a^2*b)*d)*x^2 - 2*(3*a^5*d^2 + a^3*b*d)*x))/sqrt(d), sqrt(-d)*arctan
(1/2*(a^3*d - d*x^3 + (3*a*d - 1)*x^2 - (3*a^2*d - b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a^2*b*d*x +
(2*a + b)*d*x^3 - d*x^4 - (a^2 + 2*a*b)*d*x^2))/d]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\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^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.11, size = 399, normalized size = 9.07

method	result	size
default	\(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -a^{3} d \right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +a^{3} d +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\)	\(399\)
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -a^{3} d \right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -a^{3} d -a^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*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))-2/d^2*b*sum((4*_alpha^2*a*d-2*_alpha^2*b*d-5*_alpha*a^2*d+_alpha*a*b*d+a^3*d+a^2*b*d+
_alpha^2-_alpha*b)/(-3*_alpha^2*d+6*_alpha*a*d-3*a^2*d+2*_alpha-b)*(_alpha^2*d-3*_alpha*a*d+_alpha*b*d+3*a^2*d
-3*a*b*d+b^2*d-_alpha)/(a^3-3*a^2*b+3*a*b^2-b^3)*(-(-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^2*d-3*_alpha*a*d+_alpha*b*d+3*a^2*d-3*a*b*d+b^2*d-_al
pha)*b/d/(a^3-3*a^2*b+3*a*b^2-b^3),(b/(-a+b))^(1/2)),_alpha=RootOf(d*_Z^3+(-3*a*d-1)*_Z^2+(3*a^2*d+b)*_Z-a^3*d
))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\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^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.92, size = 368, normalized size = 8.36 \begin {gather*} \frac {\ln \left (\frac {\left (a-b+x+a^2\,d-2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+d\,x^2-2\,a\,d\,x\right )\,\left (a\,x^2-a^4\,d-2\,b\,x^2+b^2\,x-2\,d\,x^4+x^3-a^5\,d^2+d^2\,x^5+2\,a^2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a^2\,d\,x^2-5\,a\,d^2\,x^4+5\,a^4\,d^2\,x-a\,b\,x+10\,a^2\,d^2\,x^3-10\,a^3\,d^2\,x^2+a^3\,b\,d+4\,a\,d\,x^3+2\,a^3\,d\,x+2\,b\,d\,x^3-2\,a\,b\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a\,b\,d\,x^2\right )}{\left (-d\,a^3+3\,d\,a^2\,x-3\,d\,a\,x^2+d\,x^3-x^2+b\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2-2\,a\,b-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2+2\,b\,d\,x^2-2\,b\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(((a - b + x + a^2*d - 2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) + d*x^2 - 2*a*d*x)*(a*x^2 - a^4*d - 2*b*x^2 + b^
2*x - 2*d*x^4 + x^3 - a^5*d^2 + d^2*x^5 + 2*a^2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a^2*d*x^2 - 5*a*d^2*x^4
+ 5*a^4*d^2*x - a*b*x + 10*a^2*d^2*x^3 - 10*a^3*d^2*x^2 + a^3*b*d + 4*a*d*x^3 + 2*a^3*d*x + 2*b*d*x^3 - 2*a*b*
d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a*b*d*x^2))/((b*x - a^3*d + d*x^3 - x^2 - 3*a*d*x^2 + 3*a^2*d*x)*(2*a*x
- 2*a*b - 2*b*x + 2*a^3*d - 2*d*x^3 + a^2 + b^2 + x^2 + a^4*d^2 + d^2*x^4 - 4*a*d^2*x^3 - 4*a^3*d^2*x + 6*a^2*
d^2*x^2 - 2*a^2*b*d + 2*a*d*x^2 - 2*a^2*d*x + 2*b*d*x^2)))/d^(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**2*b-a*(2*a-b)*x-(-a+2*b)*x**2+x**3)/(x*(-a+x)*(-b+x))**(1/2)/(-a**3*d+(3*a**2*d+b)*x-(3*a*d+1)*x
**2+d*x**3),x)

[Out]

Timed out

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