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

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

[Out]

-2*EllipticF[ArcSin[1/Sqrt[3 + x]], 2]

________________________________________________________________________________________

Rubi [A]  time = 0.0035878, 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, 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*}

Mathematica [C]  time = 0.0873702, size = 55, normalized size = 4.58 \[ \frac{2 i \sqrt{\frac{1}{x+1}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{x+1}}\right ),2\right )}{\sqrt{\frac{x+2}{x+3}} \sqrt{\frac{x+3}{x+1}}} \]

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)])

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 30, normalized size = 2.5 \begin{align*}{\sqrt{2}\sqrt{1+x}{\it EllipticF} \left ( \sqrt{-1-x},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{-1-x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x + 3} \sqrt{x + 2} \sqrt{x + 1}}{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/(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)

________________________________________________________________________________________

Sympy [C]  time = 6.8763, size = 65, normalized size = 5.42 \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{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}}} \end{align*}

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))

________________________________________________________________________________________

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

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)