3.2872 \(\int \frac{1}{\sqrt{(6-x) (-2+x)} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

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Rubi [A]  time = 0.0866572, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[(6 - x)*(-2 + x)]*Sqrt[-1 + x]),x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

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Rubi in Sympy [A]  time = 13.0673, size = 66, normalized size = 2.64 \[ - \frac{8 \sqrt{\frac{x}{5} - \frac{1}{5}} \sqrt{- \frac{x^{2}}{16} + \frac{x}{2} - \frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- \frac{x}{2} + 3}}{2} \right )}\middle | \frac{4}{5}\right )}{\sqrt{x - 1} \sqrt{- x^{2} + 8 x - 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((6-x)*(-2+x))**(1/2)/(-1+x)**(1/2),x)

[Out]

-8*sqrt(x/5 - 1/5)*sqrt(-x**2/16 + x/2 - 3/4)*elliptic_f(asin(sqrt(2)*sqrt(-x/2
+ 3)/2), 4/5)/(sqrt(x - 1)*sqrt(-x**2 + 8*x - 12))

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Mathematica [C]  time = 0.0225956, size = 74, normalized size = 2.96 \[ \frac{i \sqrt{\frac{4}{x-6}+1} \sqrt{\frac{5}{x-6}+1} (x-6)^{3/2} F\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-6}}\right )|\frac{5}{4}\right )}{\sqrt{-(x-6) (x-2)} \sqrt{x-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[(6 - x)*(-2 + x)]*Sqrt[-1 + x]),x]

[Out]

(I*Sqrt[1 + 4/(-6 + x)]*Sqrt[1 + 5/(-6 + x)]*(-6 + x)^(3/2)*EllipticF[I*ArcSinh[
2/Sqrt[-6 + x]], 5/4])/(Sqrt[-((-6 + x)*(-2 + x))]*Sqrt[-1 + x])

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Maple [B]  time = 0.025, size = 43, normalized size = 1.7 \[ -{\frac{2\,\sqrt{5}}{5}\sqrt{-2+x}\sqrt{6-x}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ){\frac{1}{\sqrt{- \left ( x-6 \right ) \left ( -2+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((6-x)*(-2+x))^(1/2)/(-1+x)^(1/2),x)

[Out]

-2/5*5^(1/2)*(-2+x)^(1/2)*(6-x)^(1/2)*EllipticF(1/2*(6-x)^(1/2),2/5*5^(1/2))/(-(
x-6)*(-2+x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x - 2\right )}{\left (x - 6\right )}} \sqrt{x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 6\right ) \left (x - 2\right )} \sqrt{x - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((6-x)*(-2+x))**(1/2)/(-1+x)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 6)*(x - 2))*sqrt(x - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x - 2\right )}{\left (x - 6\right )}} \sqrt{x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-(x - 2)*(x - 6))*sqrt(x - 1)), x)