3.2839 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1-x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0440658, 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 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{-x-1}}\right )\right |2\right ) \]

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 = 12.7607, size = 60, normalized size = 4.29 \[ - \frac{2 \sqrt{2} \sqrt{\frac{x}{2} + \frac{3}{2}} \sqrt{x + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- x - 1}}{2} \right )}\middle | 2\right )}{\sqrt{- x - 3} \sqrt{- x - 2}} \]

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

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Mathematica [C]  time = 0.0438489, size = 67, normalized size = 4.79 \[ \frac{2 i \sqrt{\frac{x+1}{x+3}} \sqrt{\frac{x+2}{x+3}} (x+3) F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{-x-3}}\right )\right |2\right )}{\sqrt{-x-2} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[(1 + x)/(3 + x)]*Sqrt[(2 + x)/(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 = 0.057, size = 54, normalized size = 3.9 \[{\frac{\sqrt{2}}{-{x}^{2}-5\,x-6}{\it EllipticF} \left ( \sqrt{-1-x},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2+x}\sqrt{3+x}\sqrt{-2-x}\sqrt{-3-x}} \]

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/(-x^2-5*x-6)*EllipticF((-1-x)^(1/2),1/2*2^(1/2))*(2+x)^(1/2)*2^(1/2)*(3+x)^(1/
2)*(-2-x)^(1/2)*(-3-x)^(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 = 12.3631, 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/(-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,)), 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. \[ \int \frac{1}{\sqrt{-x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}\,{d x} \]

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)