3.2851 \(\int \frac{1}{\sqrt{-2-x} \sqrt{-3+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 \sqrt{x+2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right ),\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]

[Out]

(-2*Sqrt[2 + x]*EllipticF[ArcSin[1/Sqrt[2/3 + x/3]], 5/3])/(Sqrt[3]*Sqrt[-2 - x])

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Rubi [A]  time = 0.0107378, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {121, 118} \[ -\frac{2 \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + x]*EllipticF[ArcSin[1/Sqrt[2/3 + x/3]], 5/3])/(Sqrt[3]*Sqrt[-2 - x])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 118

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(-2*Sqrt[d/
f]*EllipticF[ArcSin[Rt[-((b*e - a*f)/f), 2]/Sqrt[a + b*x]], (f*(b*c - a*d))/(d*(b*e - a*f))])/(d*Rt[-((b*e - a
*f)/f), 2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/b, 0] && GtQ[f/b, 0] && LeQ[c, (a*d)/b] && LeQ[e, (a*
f)/b]

Rubi steps

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

Mathematica [C]  time = 0.102355, size = 72, normalized size = 1.76 \[ \frac{2 i \sqrt{\frac{x-3}{x-1}} \sqrt{\frac{x-1}{x+2}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{-x-2}}\right ),\frac{5}{3}\right )}{\sqrt{3} \sqrt{\frac{x-3}{x+2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 72, normalized size = 1.8 \begin{align*} -{\frac{2\,\sqrt{3}}{3\,{x}^{3}-6\,{x}^{2}-15\,x+18}\sqrt{-2-x}\sqrt{-3+x}\sqrt{-1+x}\sqrt{2+x}\sqrt{1-x}\sqrt{3-x}{\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/(-2-x)^(1/2)/(-3+x)^(1/2)/(-1+x)^(1/2),x)

[Out]

-2/3*(-2-x)^(1/2)*(-3+x)^(1/2)*(-1+x)^(1/2)*(2+x)^(1/2)*3^(1/2)*(1-x)^(1/2)*(3-x)^(1/2)*EllipticF(1/5*(10+5*x)
^(1/2),1/3*15^(1/2))/(x^3-2*x^2-5*x+6)

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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} - 2 \, x^{2} - 5 \, 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 - 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{- x - 2} \sqrt{x - 3} \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(-x - 2)*sqrt(x - 3)*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)