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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = 0.07, size = 67, normalized size = 4.79 \[ \frac {2 i \sqrt {\frac {x+1}{x+3}} \sqrt {\frac {x+2}{x+3}} (x+3) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {-x-3}}\right ),2\right )}{\sqrt {-x-2} \sqrt {-x-1}} \]

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])

________________________________________________________________________________________

fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}{x^{3} + 6 \, x^{2} + 11 \, x + 6}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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)

________________________________________________________________________________________

maple [B]  time = 0.04, size = 54, normalized size = 3.86 \[ \frac {\sqrt {x +2}\, \sqrt {2}\, \sqrt {x +3}\, \sqrt {-x -2}\, \sqrt {-x -3}\, \EllipticF \left (\sqrt {-x -1}, \frac {\sqrt {2}}{2}\right )}{-x^{2}-5 x -6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x-3)^(1/2)/(-x-2)^(1/2)/(-x-1)^(1/2),x)

[Out]

1/(-x^2-5*x-6)*EllipticF((-x-1)^(1/2),1/2*2^(1/2))*(x+2)^(1/2)*2^(1/2)*(x+3)^(1/2)*(-x-2)^(1/2)*(-x-3)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {-x-1}\,\sqrt {-x-2}\,\sqrt {-x-3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((- x - 1)^(1/2)*(- x - 2)^(1/2)*(- x - 3)^(1/2)),x)

[Out]

int(1/((- x - 1)^(1/2)*(- x - 2)^(1/2)*(- x - 3)^(1/2)), x)

________________________________________________________________________________________

sympy [C]  time = 7.37, 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_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))

________________________________________________________________________________________