3.2845 \(\int \frac{1}{\sqrt{1-x} \sqrt{2+x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=18 \[ 2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right ),-3\right ) \]

[Out]

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

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Rubi [A]  time = 0.0047029, antiderivative size = 18, 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} \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.10253, size = 78, normalized size = 4.33 \[ -\frac{2 i \sqrt{-(x-1) (x+2)} \sqrt{x+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-1}}\right ),\frac{4}{3}\right )}{\sqrt{\frac{9}{x-1}+3} (x-1)^{3/2} \sqrt{\frac{x+3}{x-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[-((-1 + x)*(2 + x))]*Sqrt[3 + x]*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-1 + x]], 4/3])/(Sqrt[3 + 9/(-1
+ x)]*(-1 + x)^(3/2)*Sqrt[(3 + x)/(-1 + x)])

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Maple [A]  time = 0.036, size = 32, normalized size = 1.8 \begin{align*} -{\frac{2\,\sqrt{3}}{3}\sqrt{-2-x}{\it EllipticF} \left ( \sqrt{-2-x},{\frac{i}{3}}\sqrt{3} \right ){\frac{1}{\sqrt{2+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]

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

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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} + 4 \, x^{2} + 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 + 4*x^2 + x - 6), x)

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

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