3.2871 \(\int \frac{1}{\sqrt{6-x} \sqrt{-2+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=16 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x-2}}{2}\right )\right |-4\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[-2 + x]/2], -4]

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Rubi [A]  time = 0.0368924, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x-2}}{2}\right )\right |-4\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*EllipticF[ArcSin[Sqrt[-2 + x]/2], -4]

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Rubi in Sympy [A]  time = 7.89702, size = 53, normalized size = 3.31 \[ \frac{2 \sqrt{5} \sqrt{- x + 2} \sqrt{- \frac{x}{5} + \frac{6}{5}} F\left (\operatorname{asin}{\left (\frac{\sqrt{5} \sqrt{x - 1}}{5} \right )}\middle | 5\right )}{\sqrt{- x + 6} \sqrt{x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*sqrt(-x + 2)*sqrt(-x/5 + 6/5)*elliptic_f(asin(sqrt(5)*sqrt(x - 1)/5),
5)/(sqrt(-x + 6)*sqrt(x - 2))

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Mathematica [C]  time = 0.0565711, size = 74, normalized size = 4.62 \[ \frac{i \sqrt{\frac{4}{x-6}+1} \sqrt{\frac{5}{x-6}+1} (x-6)^{3/2} F\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-6}}\right )|\frac{5}{4}\right )}{\sqrt{-(x-6) (x-2)} \sqrt{x-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[6 - x]*Sqrt[-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 [A]  time = 0.119, size = 21, normalized size = 1.3 \[ -{\frac{2\,\sqrt{5}}{5}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x - 1} \sqrt{x - 2} \sqrt{-x + 6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x - 1)*sqrt(x - 2)*sqrt(-x + 6)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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