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

Optimal. Leaf size=1 \[ 0 \]

[Out]

0

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Rubi [C]  time = 3.07177, antiderivative size = 579, normalized size of antiderivative = 579., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 (1-a) \sqrt{x} \sqrt{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2} \tan ^{-1}\left (\frac{\sqrt{-a^2+2 a-1} \sqrt{x}}{\sqrt{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}}\right )}{a \sqrt{-a^2+2 a-1} \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac{\sqrt [4]{(2-a) a} \left (-a+\sqrt{(2-a) a}+2\right ) \sqrt{x} \left (\frac{x}{\sqrt{(2-a) a}}+1\right ) \sqrt{\frac{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac{x}{\sqrt{(2-a) a}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt [4]{(2-a) a}}\right )|\frac{1}{4} \left (\frac{-a^2+2 a+1}{\sqrt{(2-a) a}}+2\right )\right )}{\left (a+\sqrt{(2-a) a}\right ) \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac{\left (\sqrt{2-a}-\sqrt{a}\right ) (1-a) \sqrt [4]{(2-a) a} \sqrt{x} \left (\frac{x}{\sqrt{(2-a) a}}+1\right ) \sqrt{\frac{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac{x}{\sqrt{(2-a) a}}+1\right )^2}} \Pi \left (\frac{\left (\sqrt{2-a}+\sqrt{a}\right )^2}{4 \sqrt{(2-a) a}};2 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt [4]{(2-a) a}}\right )|\frac{1}{4} \left (\frac{-a^2+2 a+1}{\sqrt{(2-a) a}}+2\right )\right )}{\left (\sqrt{2-a}+\sqrt{a}\right ) a \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(1 - a)*Sqrt[x]*Sqrt[(2 - a)*a - (1 + 2*a - a^2)*x + x^2]*ArcTan[(Sqrt[-1 + 2
*a - a^2]*Sqrt[x])/Sqrt[(2 - a)*a - (1 + 2*a - a^2)*x + x^2]])/(a*Sqrt[-1 + 2*a
- a^2]*Sqrt[(2 - a)*a*x - (1 + 2*a - a^2)*x^2 + x^3]) + (((2 - a)*a)^(1/4)*(2 -
a + Sqrt[(2 - a)*a])*Sqrt[x]*(1 + x/Sqrt[(2 - a)*a])*Sqrt[((2 - a)*a - (1 + 2*a
- a^2)*x + x^2)/((2 - a)*a*(1 + x/Sqrt[(2 - a)*a])^2)]*EllipticF[2*ArcTan[Sqrt[x
]/((2 - a)*a)^(1/4)], (2 + (1 + 2*a - a^2)/Sqrt[(2 - a)*a])/4])/((a + Sqrt[(2 -
a)*a])*Sqrt[(2 - a)*a*x - (1 + 2*a - a^2)*x^2 + x^3]) + ((Sqrt[2 - a] - Sqrt[a])
*(1 - a)*((2 - a)*a)^(1/4)*Sqrt[x]*(1 + x/Sqrt[(2 - a)*a])*Sqrt[((2 - a)*a - (1
+ 2*a - a^2)*x + x^2)/((2 - a)*a*(1 + x/Sqrt[(2 - a)*a])^2)]*EllipticPi[(Sqrt[2
- a] + Sqrt[a])^2/(4*Sqrt[(2 - a)*a]), 2*ArcTan[Sqrt[x]/((2 - a)*a)^(1/4)], (2 +
(1 + 2*a - a^2)/Sqrt[(2 - a)*a])/4])/((Sqrt[2 - a] + Sqrt[a])*a*Sqrt[(2 - a)*a*
x - (1 + 2*a - a^2)*x^2 + 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((-2+a+x)/(-a+x)/((-a+2)*a*x+(a**2-2*a-1)*x**2+x**3)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.464059, size = 100, normalized size = 100. \[ -\frac{2 i \sqrt{\frac{1}{x-1}+1} (x-1)^{3/2} \sqrt{\frac{(a-1)^2}{x-1}+1} \left (F\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|(a-1)^2\right )-2 \Pi \left (1-a;i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|(a-1)^2\right )\right )}{\sqrt{(x-1) x \left (a^2-2 a+x\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + (-1 + a)^2/(-1 + x)]*(-1 + x)^(3/2)*(El
lipticF[I*ArcSinh[1/Sqrt[-1 + x]], (-1 + a)^2] - 2*EllipticPi[1 - a, I*ArcSinh[1
/Sqrt[-1 + x]], (-1 + a)^2]))/Sqrt[(-1 + x)*x*(-2*a + a^2 + x)]

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Maple [C]  time = 0.063, size = 317, normalized size = 317. \[ 2\,{\frac{{a}^{2}-2\,a}{\sqrt{{a}^{2}{x}^{2}-{a}^{2}x-2\,a{x}^{2}+{x}^{3}+2\,ax-{x}^{2}}}\sqrt{{\frac{{a}^{2}-2\,a+x}{{a}^{2}-2\,a}}}\sqrt{{\frac{-1+x}{-{a}^{2}+2\,a-1}}}\sqrt{{\frac{x}{-{a}^{2}+2\,a}}}{\it EllipticF} \left ( \sqrt{{\frac{{a}^{2}-2\,a+x}{{a}^{2}-2\,a}}},\sqrt{{\frac{-{a}^{2}+2\,a}{-{a}^{2}+2\,a-1}}} \right ) }+2\,{\frac{ \left ( 2\,a-2 \right ) \left ({a}^{2}-2\,a \right ) }{\sqrt{{a}^{2}{x}^{2}-{a}^{2}x-2\,a{x}^{2}+{x}^{3}+2\,ax-{x}^{2}} \left ( -{a}^{2}+a \right ) }\sqrt{{\frac{{a}^{2}-2\,a+x}{{a}^{2}-2\,a}}}\sqrt{{\frac{-1+x}{-{a}^{2}+2\,a-1}}}\sqrt{{\frac{x}{-{a}^{2}+2\,a}}}{\it EllipticPi} \left ( \sqrt{{\frac{{a}^{2}-2\,a+x}{{a}^{2}-2\,a}}},{\frac{-{a}^{2}+2\,a}{-{a}^{2}+a}},\sqrt{{\frac{-{a}^{2}+2\,a}{-{a}^{2}+2\,a-1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.263826, size = 95, normalized size = 95. \[ \frac{\log \left (-\frac{a^{2} - 2 \,{\left (a^{2} - a\right )} x - x^{2} + 2 \, \sqrt{{\left (a^{2} - 2 \, a - 1\right )} x^{2} + x^{3} -{\left (a^{2} - 2 \, a\right )} x} a}{a^{2} - 2 \, a x + x^{2}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + x - 2)/(sqrt(x*(x - 1)*(a**2 - 2*a + x))*(-a + x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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