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

Optimal. Leaf size=84 \[ 2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} x (x-b)}\right )+\frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{x (b-x)} \]

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Rubi [F]  time = 12.66, 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+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x^{3/2} \sqrt {-a+x} (-b+x)^{3/2} \left (a+(-1-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 \frac {\sqrt {-a+x} \left (-a b+2 a x-x^2\right )}{x^{3/2} (-b+x)^{3/2} \left (a+(-1-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 {-a+x}}{d x^{3/2} (-b+x)^{3/2}}+\frac {\sqrt {-a+x} (a-a b d-(1-2 a d+b d) x)}{d x^{3/2} (-b+x)^{3/2} \left (a+(-1-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 {-a+x}}{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 {-a+x} (a-a b d-(1-2 a d+b d) x)}{x^{3/2} (-b+x)^{3/2} \left (a+(-1-b d) x+d x^2\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x)}{b d \sqrt {(a-x) (b-x) x}}+\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )}+\frac {\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+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 {1}{2} (2 a-b)-\frac {x}{2}}{\sqrt {x} \sqrt {-a+x} (-b+x)^{3/2}} \, dx}{b d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x)}{b d \sqrt {(a-x) (b-x) x}}-\frac {4 (a-x) x}{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} (a-b) b+\frac {1}{2} (-a+b) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{(a-b) b^2 d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x)}{b d \sqrt {(a-x) (b-x) x}}-\frac {4 (a-x) x}{b^2 d \sqrt {(a-x) (b-x) x}}-\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{b d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a+b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-b+x}}{\sqrt {x} \sqrt {-a+x}} \, dx}{(a-b) b^2 d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x)}{b d \sqrt {(a-x) (b-x) x}}-\frac {4 (a-x) x}{b^2 d \sqrt {(a-x) (b-x) x}}+\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 (-a+b) \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) 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}{b d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {2 (a-x)}{b d \sqrt {(a-x) (b-x) x}}-\frac {4 (a-x) x}{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 )}{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 )}{b d \sqrt {(a-x) (b-x) x}}+\frac {\left (\left (-1+2 a d-b d-\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d-\sqrt {1-4 a d+2 b d+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1+2 a d-b d+\sqrt {-4 a d+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x}}{x^{3/2} (-b+x)^{3/2} \left (-1-b d+\sqrt {1-4 a d+2 b d+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 = 2.57, size = 277, normalized size = 3.30 \begin {gather*} -\frac {2 (x-a) \left (i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {2 a d}{2 a d-b d+\sqrt {(b d+1)^2-4 a d}-1};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {2 a d}{-2 a d+b d+\sqrt {(b d+1)^2-4 a d}+1};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )-i \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+\sqrt {\frac {x}{a}-1}\right )}{\sqrt {\frac {x}{a}-1} \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.00, size = 84, normalized size = 1.00 \begin {gather*} 2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} x (x-b)}\right )+\frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{x (b-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.82, size = 313, normalized size = 3.73 \begin {gather*} \left [\frac {{\left (b x - x^{2}\right )} \sqrt {d} \log \left (\frac {d^{2} x^{4} - 2 \, {\left (b d^{2} - 3 \, d\right )} x^{3} + {\left (b^{2} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {d} + 2 \, {\left (3 \, a b d - a\right )} x}{d^{2} x^{4} - 2 \, {\left (b d^{2} + d\right )} x^{3} + {\left (b^{2} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left (a b d + a\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (b x - x^{2}\right )}}, -\frac {{\left (b x - x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \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}}}{b x - x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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maple [C]  time = 0.03, size = 2956, normalized size = 35.19 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^
2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*d+
1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b^2*d+2/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(1-1/a*x)^(1/2)*(-1
/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*
d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b-
4/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*
x^2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*
d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))+1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-
b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)
^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))/d-2*a^
2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/
d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))
),(a/(a-b))^(1/2))+a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/
(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(b*d+1+(b^2*d^2-4*
a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b+a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*
x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2
/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))/d-1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(1-1/a*x)^(1/2)
*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+
1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2
))*b^2*d-2/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a
*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/
2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b-2*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1
/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi
((-(-a+x)/a)^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))+a*(1-1/a*x)^(1/2)*(-1/(
a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/
2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b+a
*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*
d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d
),(a/(a-b))^(1/2))/d+4/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^
(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)
^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))-1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*a*(
1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b+1/2/d*(b^2*d^
2-4*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-b*d+(b^2*d^2-4*a*d+2*b*d+1)^(1/2)-1)/d),
(a/(a-b))^(1/2))/d-(a-b)*(2*(-a*x+x^2)/(a-b)/b/((-b+x)*(-a*x+x^2))^(1/2)-2*(-1/b+a/(a-b)/b)*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/(a-b)/b*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)*((a-b)*Ellipt
icE((-(-a+x)/a)^(1/2),(a/(a-b))^(1/2))+b*EllipticF((-(-a+x)/a)^(1/2),(a/(a-b))^(1/2))))-a*(-2*(a*b-a*x-b*x+x^2
)/a/b/(x*(a*b-a*x-b*x+x^2))^(1/2)-2*((a+b)/a/b+(-a-b)/a/b)*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/b*(-(-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)*((a-b)*EllipticE((-(-a+x)/a)^(1/2),(a/(a-b))^(1/2))+b*
EllipticF((-(-a+x)/a)^(1/2),(a/(a-b))^(1/2))))

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

[Out]

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

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mupad [B]  time = 0.18, size = 628, normalized size = 7.48 \begin {gather*} \frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (b\,d-2\,a\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1\right )}{d\,\left (b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,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 {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,d-b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}-1\right )}{d\,\left (b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,\left (a-b\right )\,\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}}}{b\,\left (\frac {a}{b}-1\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,\left (\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\frac {\sqrt {\frac {b-x}{a-b}+1}\,\sqrt {\frac {b-x}{b}}}{\sqrt {1-\frac {b-x}{b}}}}{\frac {b}{a-b}+1}-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\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^2*b + 3*a*x^2 - x^3 - a*x*(2*a + b))/(x*(b - x)*(x*(a - x)*(b - x))^(1/2)*(a - x*(b*d + 1) + d*x^2)),x
)

[Out]

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

[Out]

Timed out

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