3.81 \(\int \frac{-a+(-1+2 a) x}{(-a+x) \sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx\)

Optimal. Leaf size=46 \[ \log \left (\frac{-2 \left (\sqrt{(1-x) x \left (a^2-2 a x+x\right )}+x\right )-a^2+2 a x+x^2}{(a-x)^2}\right ) \]

[Out]

Log[(-a^2 + 2*a*x + x^2 - 2*(x + Sqrt[(1 - x)*x*(a^2 + x - 2*a*x)]))/(a - x)^2]

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Rubi [C]  time = 2.93293, antiderivative size = 180, normalized size of antiderivative = 3.91, number of steps used = 7, number of rules used = 7, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.137 \[ \frac{4 (1-a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} \Pi \left (\frac{1}{a};\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}}-\frac{2 (1-2 a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} F\left (\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*a)*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*EllipticF[ArcSin[S
qrt[x]], -((1 - 2*a)/a^2)])/Sqrt[a^2*x + (1 - 2*a - a^2)*x^2 - (1 - 2*a)*x^3] +
(4*(1 - a)*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*EllipticPi[a^(-1), Ar
cSin[Sqrt[x]], -((1 - 2*a)/a^2)])/Sqrt[a^2*x + (1 - 2*a - a^2)*x^2 - (1 - 2*a)*x
^3]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-a+(-1+2*a)*x)/(-a+x)/(a**2*x-(a**2+2*a-1)*x**2+(-1+2*a)*x**3)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.590397, size = 133, normalized size = 2.89 \[ \frac{2 i (x-1)^{3/2} \sqrt{\frac{x}{x-1}} \sqrt{-\frac{a^2-2 a x+x}{(2 a-1) (x-1)}} \left (2 a \Pi \left (1-a;i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|-\frac{(a-1)^2}{2 a-1}\right )-F\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|-\frac{(a-1)^2}{2 a-1}\right )\right )}{\sqrt{-(x-1) x \left (a^2-2 a x+x\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*(-1 + x)^(3/2)*Sqrt[x/(-1 + x)]*Sqrt[-((a^2 + x - 2*a*x)/((-1 + 2*a)*(-1
+ x)))]*(-EllipticF[I*ArcSinh[1/Sqrt[-1 + x]], -((-1 + a)^2/(-1 + 2*a))] + 2*a*E
llipticPi[1 - a, I*ArcSinh[1/Sqrt[-1 + x]], -((-1 + a)^2/(-1 + 2*a))]))/Sqrt[-((
-1 + x)*x*(a^2 + x - 2*a*x))]

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Maple [C]  time = 0.079, size = 536, normalized size = 11.7 \[ 2\,{\frac{{a}^{2}}{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3}}{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3} \left ( a-1 \right ) }{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a+(-1+2*a)*x)/(-a+x)/(a^2*x-(a^2+2*a-1)*x^2+(-1+2*a)*x^3)^(1/2),x)

[Out]

2*a^2/(-1+2*a)*(-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2)*((-1+x)/(a^2/(-1+2*a)-1))^
(1/2)*(x/a^2*(-1+2*a))^(1/2)/(-a^2*x^2+2*a*x^3+a^2*x-2*a*x^2-x^3+x^2)^(1/2)*Elli
pticF((-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2),(a^2/(-1+2*a)/(a^2/(-1+2*a)-1))^(1/
2))-4*a^3/(-1+2*a)*(-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2)*((-1+x)/(a^2/(-1+2*a)-
1))^(1/2)*(x/a^2*(-1+2*a))^(1/2)/(-a^2*x^2+2*a*x^3+a^2*x-2*a*x^2-x^3+x^2)^(1/2)*
EllipticF((-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2),(a^2/(-1+2*a)/(a^2/(-1+2*a)-1))
^(1/2))-4*a^3*(a-1)/(-1+2*a)*(-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2)*((-1+x)/(a^2
/(-1+2*a)-1))^(1/2)*(x/a^2*(-1+2*a))^(1/2)/(-a^2*x^2+2*a*x^3+a^2*x-2*a*x^2-x^3+x
^2)^(1/2)/(a^2/(-1+2*a)-a)*EllipticPi((-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2),a^2
/(-1+2*a)/(a^2/(-1+2*a)-a),(a^2/(-1+2*a)/(a^2/(-1+2*a)-1))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (2 \, a - 1\right )} x - a}{\sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}{\left (a - x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.25922, size = 85, normalized size = 1.85 \[ \log \left (-\frac{a^{2} - 2 \,{\left (a - 1\right )} x - x^{2} + 2 \, \sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}}{a^{2} - 2 \, a x + x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="fricas")

[Out]

log(-(a^2 - 2*(a - 1)*x - x^2 + 2*sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x
^2))/(a^2 - 2*a*x + x^2))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-a+(-1+2*a)*x)/(-a+x)/(a**2*x-(a**2+2*a-1)*x**2+(-1+2*a)*x**3)**(1/2),x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (2 \, a - 1\right )} x - a}{\sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}{\left (a - x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="giac")

[Out]

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