3.2855 \(\int \frac{1}{\sqrt{-2-x} \sqrt{-1-x} \sqrt{-3+x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 \sqrt{x+1} \sqrt{x+2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{2}{5}}}\right ),\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x-2} \sqrt{-x-1}} \]

[Out]

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

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Rubi [A]  time = 0.0189782, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {121, 118} \[ -\frac{2 \sqrt{x+1} \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{2}{5}}}\right )|\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x-2} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0498898, size = 69, normalized size = 1.21 \[ \frac{i \sqrt{\frac{4}{x-3}+1} \sqrt{\frac{5}{x-3}+1} (x-3) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-3}}\right ),\frac{5}{4}\right )}{\sqrt{-x-2} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.03, size = 46, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}-x-6}{\it EllipticF} \left ( \sqrt{-1-x},{\frac{i}{2}} \right ) \sqrt{2+x}\sqrt{3-x}\sqrt{-3+x}\sqrt{-2-x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticF((-1-x)^(1/2),1/2*I)*(2+x)^(1/2)*(3-x)^(1/2)*(-3+x)^(1/2)*(-2-x)^(1/2)/(x^2-x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x - 3)*sqrt(-x - 1)*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 - 3} \sqrt{-x - 1} \sqrt{-x - 2}}{x^{3} - 7 \, x - 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-x - 2)*sqrt(-x - 1)*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 - 1} \sqrt{-x - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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