3.2831 \(\int \frac{1}{\sqrt{1-x} \sqrt{2-x} \sqrt{3-x}} \, dx\)

Optimal. Leaf size=14 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{3-x}}\right )\right |2\right ) \]

[Out]

2*EllipticF[ArcSin[1/Sqrt[3 - x]], 2]

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Rubi [A]  time = 0.043583, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{3-x}}\right )\right |2\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*EllipticF[ArcSin[1/Sqrt[3 - x]], 2]

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Rubi in Sympy [A]  time = 6.7179, size = 19, normalized size = 1.36 \[ \sqrt{2} i F\left (i \operatorname{asinh}{\left (\sqrt{- x + 1} \right )}\middle | \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*I*elliptic_f(I*asinh(sqrt(-x + 1)), 1/2)

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Mathematica [C]  time = 0.0471969, size = 67, normalized size = 4.79 \[ \frac{2 i \sqrt{\frac{x-3}{x-1}} \sqrt{\frac{x-2}{x-1}} (x-1) F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{1-x}}\right )\right |2\right )}{\sqrt{2-x} \sqrt{3-x}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[(-3 + x)/(-1 + x)]*Sqrt[(-2 + x)/(-1 + x)]*(-1 + x)*EllipticF[I*ArcS
inh[1/Sqrt[1 - x]], 2])/(Sqrt[2 - x]*Sqrt[3 - x])

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Maple [B]  time = 0.059, size = 55, normalized size = 3.9 \[ -{\frac{\sqrt{2}}{2\,{x}^{2}-6\,x+4}{\it EllipticF} \left ( \sqrt{3-x},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-2+x}\sqrt{-2+2\,x}\sqrt{2-x}\sqrt{2-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.7028, size = 66, normalized size = 4.71 \[ \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), exp_pol
ar(-2*I*pi)/(x - 2)**2)/(4*pi**(3/2)) - meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,))
, ((0, 1/2, 0), (-1/4, 1/4, 1/4)), (x - 2)**(-2))/(4*pi**(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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