3.2843 \(\int \frac{1}{\sqrt{2-x} \sqrt{1+x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=24 \[ \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right ),-\frac{3}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0058621, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {119} \[ \sqrt{2} F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|-\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 119

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

Rubi steps

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

Mathematica [B]  time = 0.0752608, size = 67, normalized size = 2.79 \[ -\frac{2 (x+3) \sqrt{1-\frac{5}{x+3}} \sqrt{1-\frac{2}{x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5}}{\sqrt{x+3}}\right ),\frac{2}{5}\right )}{\sqrt{-5 (x+3)^2+35 (x+3)-50}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + x)*Sqrt[1 - 5/(3 + x)]*Sqrt[1 - 2/(3 + x)]*EllipticF[ArcSin[Sqrt[5]/Sqrt[3 + x]], 2/5])/Sqrt[-50 + 35
*(3 + x) - 5*(3 + x)^2]

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Maple [A]  time = 0.044, size = 34, normalized size = 1.4 \begin{align*} -{\frac{2\,\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2-2\,x}},{\frac{i}{3}}\sqrt{6} \right ) \sqrt{-1-x}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x + 3} \sqrt{x + 1} \sqrt{-x + 2}}{x^{3} + 2 \, x^{2} - 5 \, x - 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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