3.2844 \(\int \frac{1}{\sqrt{2-x} \sqrt{3-x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=18 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right ),\frac{3}{4}\right ) \]

[Out]

EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], 3/4]

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Rubi [A]  time = 0.0058277, antiderivative size = 18, 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} \[ F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|\frac{3}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], 3/4]

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

Mathematica [C]  time = 0.0846531, size = 65, normalized size = 3.61 \[ -\frac{2 i \sqrt{1-\frac{3}{2-x}} \sqrt{\frac{1}{2-x}+1} (2-x) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{2-x}}\right ),-3\right )}{\sqrt{-(x-3) (x+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 19, normalized size = 1.1 \begin{align*}{\frac{2\,\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{2}\sqrt{1+x}},{\frac{2\,\sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + 1)*sqrt(-x + 3)*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 + 1} \sqrt{-x + 3} \sqrt{-x + 2}}{x^{3} - 4 \, x^{2} + x + 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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