3.2832 \(\int \frac{1}{\sqrt{-3+x} \sqrt{-2+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=12 \[ -2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )\right |2\right ) \]

[Out]

-2*EllipticF[ArcSin[1/Sqrt[-1 + x]], 2]

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Rubi [A]  time = 0.0318751, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )\right |2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2*EllipticF[ArcSin[1/Sqrt[-1 + x]], 2]

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Rubi in Sympy [A]  time = 7.54925, size = 53, normalized size = 4.42 \[ \frac{2 \sqrt{2} \sqrt{- x + 2} \sqrt{- \frac{x}{2} + \frac{3}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}\middle | 2\right )}{\sqrt{x - 3} \sqrt{x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0460203, size = 59, normalized size = 4.92 \[ \frac{2 i \sqrt{\frac{1}{x-3}+1} \sqrt{\frac{2}{x-3}+1} (x-3) F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{x-3}}\right )\right |2\right )}{\sqrt{x-2} \sqrt{x-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.053, size = 30, normalized size = 2.5 \[{\sqrt{2}\sqrt{-3+x}{\it EllipticF} \left ( \sqrt{3-x},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{3-x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.7461, size = 65, normalized size = 5.42 \[ - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3+x)**(1/2)/(-2+x)**(1/2)/(-1+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,)), (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)), exp_polar(2*I*pi)/(x - 2)**2)/(4*pi**(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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