3.2846 \(\int \frac{1}{\sqrt{1-x} \sqrt{3-x} \sqrt{2+x}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right ),\frac{3}{5}\right )}{\sqrt{5}} \]

[Out]

(2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], 3/5])/Sqrt[5]

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Rubi [A]  time = 0.0059751, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {119} \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )|\frac{3}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], 3/5])/Sqrt[5]

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{3-x} \sqrt{2+x}} \, dx &=\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{2+x}}{\sqrt{3}}\right )|\frac{3}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [B]  time = 0.0651457, size = 68, normalized size = 2.72 \[ -\frac{2 \sqrt{\frac{x-3}{x-1}} (x-1) \sqrt{\frac{x+2}{x-1}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3}}{\sqrt{1-x}}\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-x^2+x+6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(-3 + x)/(-1 + x)]*(-1 + x)*Sqrt[(2 + x)/(-1 + x)]*EllipticF[ArcSin[Sqrt[3]/Sqrt[1 - x]], -2/3])/(Sqr
t[3]*Sqrt[6 + x - x^2])

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Maple [A]  time = 0.038, size = 21, normalized size = 0.8 \begin{align*}{\frac{2\,\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{5}\sqrt{10+5\,x}},{\frac{\sqrt{15}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*EllipticF(1/5*(10+5*x)^(1/2),1/3*15^(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 + 2} \sqrt{-x + 3} \sqrt{-x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + 2)*sqrt(-x + 3)*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 + 2} \sqrt{-x + 3} \sqrt{-x + 1}}{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/(1-x)^(1/2)/(3-x)^(1/2)/(2+x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x + 2)*sqrt(-x + 3)*sqrt(-x + 1)/(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{1 - x} \sqrt{3 - x} \sqrt{x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(1 - x)*sqrt(3 - x)*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 + 3} \sqrt{-x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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