3.615 \(\int \frac{1}{\sqrt{(2-x) x \left (4-2 x+x^2\right )}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

[Out]

-(EllipticF[ArcSin[1 - x], -1/3]/Sqrt[3])

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Rubi [A]  time = 0.0360589, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]

[Out]

-(EllipticF[ArcSin[1 - x], -1/3]/Sqrt[3])

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Rubi in Sympy [A]  time = 3.48265, size = 15, normalized size = 0.88 \[ \frac{\sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((2-x)*x*(x**2-2*x+4))**(1/2),x)

[Out]

sqrt(3)*elliptic_f(asin(x - 1), -1/3)/3

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Mathematica [C]  time = 0.314301, size = 100, normalized size = 5.88 \[ -\frac{\sqrt [3]{-1} (x-2)^2 \sqrt{\frac{x \left (x+i \sqrt{3}-1\right )}{(x-2)^2}} \sqrt{\frac{-\sqrt [3]{-1} x+x-2}{x-2}} F\left (\sin ^{-1}\left (\sqrt{-\frac{(-1)^{2/3} x}{x-2}}\right )|(-1)^{2/3}\right )}{\sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]

[Out]

-(((-1)^(1/3)*(-2 + x)^2*Sqrt[(x*(-1 + I*Sqrt[3] + x))/(-2 + x)^2]*Sqrt[(-2 + x
- (-1)^(1/3)*x)/(-2 + x)]*EllipticF[ArcSin[Sqrt[-(((-1)^(2/3)*x)/(-2 + x))]], (-
1)^(2/3)])/Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])

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Maple [B]  time = 0.039, size = 200, normalized size = 11.8 \[ 2\,{\frac{ \left ( -1+i\sqrt{3} \right ) \left ( x-2 \right ) ^{2}}{ \left ( -i\sqrt{3}-1 \right ) \sqrt{-x \left ( x-2 \right ) \left ( x-i\sqrt{3}-1 \right ) \left ( x-1+i\sqrt{3} \right ) }}\sqrt{{\frac{ \left ( -i\sqrt{3}-1 \right ) x}{ \left ( 1-i\sqrt{3} \right ) \left ( x-2 \right ) }}}\sqrt{{\frac{x-i\sqrt{3}-1}{ \left ( i\sqrt{3}+1 \right ) \left ( x-2 \right ) }}}\sqrt{{\frac{x-1+i\sqrt{3}}{ \left ( 1-i\sqrt{3} \right ) \left ( x-2 \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( -i\sqrt{3}-1 \right ) x}{ \left ( 1-i\sqrt{3} \right ) \left ( x-2 \right ) }}},\sqrt{{\frac{ \left ( 1-i\sqrt{3} \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( -i\sqrt{3}-1 \right ) \left ( i\sqrt{3}+1 \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((2-x)*x*(x^2-2*x+4))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 - 2*x + 4)*(x - 2)*x),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-(x^2 - 2*x + 4)*(x - 2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 - 2*x + 4)*(x - 2)*x),x, algorithm="fricas")

[Out]

integral(1/sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x \left (- x + 2\right ) \left (x^{2} - 2 x + 4\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2-x)*x*(x**2-2*x+4))**(1/2),x)

[Out]

Integral(1/sqrt(x*(-x + 2)*(x**2 - 2*x + 4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 - 2*x + 4)*(x - 2)*x),x, algorithm="giac")

[Out]

integrate(1/sqrt(-(x^2 - 2*x + 4)*(x - 2)*x), x)