3.2856 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1-x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0061095, 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, Rules used = {118} \[ 2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{-x-1}}\right )\right |2\right ) \]

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]

Rule 118

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(-2*Sqrt[d/
f]*EllipticF[ArcSin[Rt[-((b*e - a*f)/f), 2]/Sqrt[a + b*x]], (f*(b*c - a*d))/(d*(b*e - a*f))])/(d*Rt[-((b*e - a
*f)/f), 2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/b, 0] && GtQ[f/b, 0] && LeQ[c, (a*d)/b] && LeQ[e, (a*
f)/b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1-x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{-1-x}}\right )\right |2\right )\\ \end{align*}

Mathematica [C]  time = 0.0333234, size = 67, normalized size = 4.79 \[ \frac{2 i \sqrt{\frac{x+1}{x+3}} \sqrt{\frac{x+2}{x+3}} (x+3) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{-x-3}}\right ),2\right )}{\sqrt{-x-2} \sqrt{-x-1}} \]

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 = 0.036, size = 54, normalized size = 3.9 \begin{align*}{\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}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{-x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}{x^{3} + 6 \, x^{2} + 11 \, x + 6}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(-sqrt(-x - 1)*sqrt(-x - 2)*sqrt(-x - 3)/(x^3 + 6*x^2 + 11*x + 6), x)

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Sympy [C]  time = 7.03044, size = 66, normalized size = 4.71 \begin{align*} \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}}} \end{align*}

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_polar(-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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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