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

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

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Rubi [F]  time = 9.68, 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 {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 7.60, size = 226, normalized size = 2.94 \begin {gather*} \frac {2 \left (x (a-x)^2 (x-b)-\frac {i d (x-a)^3 \sqrt {\frac {x-b}{a-b}} \left (-\Pi \left (-\frac {2 a}{-2 a+b+d-\sqrt {b^2+2 d b+d^2-4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )-\Pi \left (-\frac {2 a}{-2 a+b+d+\sqrt {b^2+2 d b+d^2-4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{\sqrt {1-\frac {a}{x}}}\right )}{(a-x)^2 \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.76, size = 77, 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} (a-x)}\right )-\frac {2 \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.05, size = 2792, normalized size = 36.26

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*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)*EllipticE((-(-a
+x)/a)^(1/2),(a/(a-b))^(1/2))+b*EllipticF((-(-a+x)/a)^(1/2),(a/(a-b))^(1/2)))+2*a^2*(-(-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*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))-d*(-4/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2
),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*Elli
pticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))+1/(-4*a*d+b^2+2*b
*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*
d+d^2)^(1/2)),(a/(a-b))^(1/2))*b^2+2/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(
-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b*d+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*(-4*a*d+b^2+2*b*d+d^2)^
(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b+1/(-
4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4
*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d^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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(
a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d+4/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^
2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d
+d^2)^(1/2)),(a/(a-b))^(1/2))-1/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)
/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b^2-2/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2
)),(a/(a-b))^(1/2))*b*d+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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*
d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b-1/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*Ellipt
icPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d^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*(-4*a*d+b^2+2*b
*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2)
)*d)-a*(a-b)*(-2*(-b*x+x^2)/a/(a-b)/((-a+x)*(-b*x+x^2))^(1/2)-2*(-1/a-1/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)
*(-(-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 {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a d - {\left (b + d\right )} x + x^{2}\right )} {\left (a - x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.87, size = 754, normalized size = 9.79 \begin {gather*} \frac {2\,a\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\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}}-\frac {2\,b\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\left (a-b\right )\,\mathrm {E}\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}}-\frac {2\,b\,d\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\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}}+\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d-\frac {b\,d}{2}+\frac {d\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}-\frac {d^2}{2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )}+\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {d}{2}-\frac {b}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (\frac {b\,d}{2}-a\,d+\frac {d\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}+\frac {d^2}{2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {d}{2}-\frac {b}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )}+\frac {2\,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}}\,\left (a\,b-a^2\right )\,\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*(b - x)*(a*b - 2*a*x + x^2))/((a - x)*(x*(a - x)*(b - x))^(1/2)*(a*d + x^2 - x*(b + d))),x)

[Out]

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

[Out]

Timed out

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