3.2842 \(\int \frac{1}{\sqrt{3-x} \sqrt{1+x} \sqrt{2+x}} \, dx\)

Optimal. Leaf size=16 \[ 2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{2}\right ),-4\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[1 + x]/2], -4]

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Rubi [A]  time = 0.004844, antiderivative size = 16, 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} \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )\right |-4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[Sqrt[1 + x]/2], -4]

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{3-x} \sqrt{1+x} \sqrt{2+x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{1+x}}{2}\right )\right |-4\right )\\ \end{align*}

Mathematica [C]  time = 0.0858415, size = 74, normalized size = 4.62 \[ \frac{i \sqrt{\frac{4}{x-3}+1} \sqrt{\frac{5}{x-3}+1} (x-3)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-3}}\right ),\frac{5}{4}\right )}{\sqrt{-(x-3) (x+1)} \sqrt{x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[1 + 4/(-3 + x)]*Sqrt[1 + 5/(-3 + x)]*(-3 + x)^(3/2)*EllipticF[I*ArcSinh[2/Sqrt[-3 + x]], 5/4])/(Sqrt[-
((-3 + x)*(1 + x))]*Sqrt[2 + x])

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Maple [A]  time = 0.036, size = 25, normalized size = 1.6 \begin{align*} -{{\it EllipticF} \left ( \sqrt{-1-x},{\frac{i}{2}} \right ) \sqrt{-1-x}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x + 2} \sqrt{x + 1} \sqrt{-x + 3}}{x^{3} - 7 \, x - 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(x + 2)*sqrt(x + 1)*sqrt(-x + 3)/(x^3 - 7*x - 6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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