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

Optimal. Leaf size=79 \[ \frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{(x-a) (x-b)}-2 \sqrt {d} \tanh ^{-1}\left (\frac {x}{\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right ) \]

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Rubi [F]  time = 13.14, 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 b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.96, size = 280, normalized size = 3.54 \begin {gather*} \frac {2 i x^{3/2} \sqrt {1-\frac {a}{x}} \sqrt {1-\frac {b}{x}} \Pi \left (\frac {2 b d}{a d+b d-\sqrt {(a d+b d+1)^2-4 a b d^2}+1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )+2 i x^{3/2} \sqrt {1-\frac {a}{x}} \sqrt {1-\frac {b}{x}} \Pi \left (\frac {2 b d}{a d+b d+\sqrt {(a d+b d+1)^2-4 a b d^2}+1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )-2 i x^{3/2} \sqrt {1-\frac {a}{x}} \sqrt {1-\frac {b}{x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )+2 \sqrt {-a} x}{\sqrt {-a} \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-a]*x - (2*I)*Sqrt[1 - a/x]*Sqrt[1 - b/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-a]/Sqrt[x]], b/a] + (2*I)*
Sqrt[1 - a/x]*Sqrt[1 - b/x]*x^(3/2)*EllipticPi[(2*b*d)/(1 + a*d + b*d - Sqrt[-4*a*b*d^2 + (1 + a*d + b*d)^2]),
I*ArcSinh[Sqrt[-a]/Sqrt[x]], b/a] + (2*I)*Sqrt[1 - a/x]*Sqrt[1 - b/x]*x^(3/2)*EllipticPi[(2*b*d)/(1 + a*d + b
*d + Sqrt[-4*a*b*d^2 + (1 + a*d + b*d)^2]), I*ArcSinh[Sqrt[-a]/Sqrt[x]], b/a])/(Sqrt[-a]*Sqrt[x*(-a + x)*(-b +
x)])

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IntegrateAlgebraic [A]  time = 0.29, size = 79, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{(-a+x) (-b+x)}-2 \sqrt {d} \tanh ^{-1}\left (\frac {x}{\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.16, size = 395, normalized size = 5.00 \begin {gather*} \left [\frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {d} \log \left (\frac {a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} - 3 \, d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} - 4 \, {\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} + 2 \, {\left (3 \, a b d - {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}{a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} + d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} - 2 \, {\left (a b d + {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}, \frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right ) + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{a b - {\left (a + b\right )} x + x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.38, size = 4026, normalized size = 50.96

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4026\)
default	\(\text {Expression too large to display}\)	\(4442\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/((a*b+(-a-b)*x+x^2)*x)^(1/2)-2*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(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/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)*
b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a-1/2/d-1/
2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(a*d+b*d+1+(a^2*d
^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a^2*d+2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1
)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*
a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(a*d+b*
d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a*d-2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d
+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*
b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(
a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a-1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2
*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)
/(1/2*b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1
/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*d-2/(a^2*d^2-2*a*b*d^2+b^2
*d^2+2*a*d+2*b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)
^(1/2)/(1/2*b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),
b/(b-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))-1/(a^2*d^2-2*a*b*d^2
+b^2*d^2+2*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^
3)^(1/2)/(1/2*b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2
),b/(b-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))/d-b*(1-x/b)^(1/2)*
(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a
*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2
*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a-b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(
a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*Ellipti
cPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2)
)-b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a-1/2/d-
1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(a*d+b*d+1+(a^2
*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))/d+1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^
(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a+1/
2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-a*d-b*d+(a^
2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a^2*d-2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+
2*b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2
*b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-
a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a*d+2/(a^2*d^2-2*a*b*d^2+b^2*d
^2+2*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/
2)/(1/2*b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b
+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a+1/(a^2*d^2-2*a*b*d^2+
b^2*d^2+2*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x
^3)^(1/2)/(1/2*b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/
2),b/(b+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*d+2/(a^2*d^2-2*a
*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-
b*x^2+x^3)^(1/2)/(1/2*b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)
/b)^(1/2),b/(b+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))-b*(1-x/b)
^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a+1/2/d*(a^2*d^2-2*
a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2
+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))*a-b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)
^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)
*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+
b))^(1/2))-b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2
*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-a*d-b*
d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))/d+1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d
+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*
b-1/2*a+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-b+x)/b)^(1/2),b/(b+1/2*(-a
*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(b/(-a+b))^(1/2))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.87, size = 695, normalized size = 8.80 \begin {gather*} -\frac {2\,a\,\sqrt {\frac {x}{a}}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\middle |\frac {a}{b}\right )-\frac {a\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\right )}{2\,b\,\sqrt {1-\frac {x}{b}}}\right )\,\sqrt {\frac {a-x}{a}}\,\sqrt {\frac {b-x}{b}}}{\left (\frac {a}{b}-1\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,b\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )+\frac {b\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\right )}{2\,\sqrt {\frac {b-x}{a-b}+1}\,\left (a-b\right )}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\left (\frac {b}{a-b}+1\right )\,\left (a-b\right )\,\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((x^3 - a*b*x)/((a - x)*(b - x)*(d*x^2 - x*(a*d + b*d + 1) + a*b*d)*(x*(a - x)*(b - x))^(1/2)),x)

[Out]

- (2*a*(x/a)^(1/2)*(ellipticE(asin((x/a)^(1/2)), a/b) - (a*sin(2*asin((x/a)^(1/2))))/(2*b*(1 - x/b)^(1/2)))*((
a - x)/a)^(1/2)*((b - x)/b)^(1/2))/((a/b - 1)*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (b*(x/b)^(1/2)*((b - x)/b)^
(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b - (a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 +
1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2
*a*b*d^2 + 1)^(1/2) + 1))/(d*(b - (a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/
(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticP
i(b/(b - (a*d + b*d - (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^
(1/2)), -b/(a - b))*(a*d + b*d - (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1))/(d*(b - (a*d
+ b*d - (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2
)) - (2*a*b*(ellipticE(asin(((b - x)/b)^(1/2)), -b/(a - b)) + (b*sin(2*asin(((b - x)/b)^(1/2))))/(2*((b - x)/(
a - b) + 1)^(1/2)*(a - b)))*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/((b/(a - b) + 1)*(a - b)*(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*b*x+x**3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))**(1/2)/(a*b*d-(a*d+b*d+1)*x+d*x**2),x)

[Out]

Timed out

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