3.2847 \(\int \frac{1}{\sqrt{1-x} \sqrt{2-x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+3}}{2}\right ),\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/Sqrt[5]

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Rubi [A]  time = 0.0054182, antiderivative size = 23, 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} \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+3}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/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{2-x} \sqrt{3+x}} \, dx &=\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{3+x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [C]  time = 0.0809231, size = 65, normalized size = 2.83 \[ -\frac{2 i \sqrt{1-\frac{4}{1-x}} \sqrt{\frac{1}{1-x}+1} (1-x) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{1-x}}\right ),-4\right )}{\sqrt{-(x-2) (x+3)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 16, normalized size = 0.7 \begin{align*}{\it EllipticF} \left ({\frac{1}{5}\sqrt{15+5\,x}},{\frac{\sqrt{5}}{2}} \right ) \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]

EllipticF(1/5*(15+5*x)^(1/2),1/2*5^(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} - 7 \, 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 - 7*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{2 - x} \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(2 - x)*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)