3.2841 \(\int \frac {1}{\sqrt {1+x} \sqrt {2+x} \sqrt {3+x}} \, dx\)

Optimal. Leaf size=12 \[ -2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {x+3}}\right ),2\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {118} \[ -2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {x+3}}\right )\right |2\right ) \]

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]

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

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Mathematica [C]  time = 0.12, size = 55, normalized size = 4.58 \[ \frac {2 i \sqrt {\frac {1}{x+1}+1} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x+1}}\right ),2\right )}{\sqrt {\frac {x+2}{x+3}} \sqrt {\frac {x+3}{x+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 3} \sqrt {x + 2} \sqrt {x + 1}}{x^{3} + 6 \, x^{2} + 11 \, x + 6}, x\right ) \]

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, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(1/2)/(2+x)^(1/2)/(3+x)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.04, size = 44, normalized size = 3.67 \[ -\frac {\sqrt {2}\, \left (x +3\right ) \sqrt {-x -1}\, \sqrt {x +1}\, \EllipticF \left (\sqrt {-x -1}, \frac {\sqrt {2}}{2}\right )}{x^{2}+4 x +3} \]

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]

-EllipticF((-x-1)^(1/2),1/2*2^(1/2))*2^(1/2)*(x+3)*(-x-1)^(1/2)*(x+1)^(1/2)/(x^2+4*x+3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(1/2)/(2+x)^(1/2)/(3+x)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {x+1}\,\sqrt {x+2}\,\sqrt {x+3}} \,d x \]

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)

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sympy [C]  time = 7.36, 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/(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,)), (x + 2)**(-2))/(4*pi**(3/2)) + meije
rg(((-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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