3.2850 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-2+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{x+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2}{\sqrt{x+3}}\right ),\frac{5}{4}\right )}{\sqrt{-x-3}}-\frac{i K\left (-\frac{1}{4}\right ) \sqrt{x+3}}{\sqrt{-x-3}} \]

[Out]

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

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Rubi [A]  time = 0.0099321, antiderivative size = 36, normalized size of antiderivative = 0.63, 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} \[ -\frac{\sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{4}+\frac{3}{4}}}\right )|\frac{5}{4}\right )}{\sqrt{-x-3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0934196, size = 63, normalized size = 1.11 \[ \frac{i \sqrt{\frac{x-2}{x-1}} \sqrt{\frac{x-1}{x+3}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{-x-3}}\right ),\frac{5}{4}\right )}{\sqrt{\frac{x-2}{x+3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 65, normalized size = 1.1 \begin{align*}{\frac{1}{-{x}^{3}+7\,x-6}\sqrt{-3-x}\sqrt{-2+x}\sqrt{-1+x}\sqrt{3+x}\sqrt{1-x}\sqrt{2-x}{\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/(-3-x)^(1/2)/(-2+x)^(1/2)/(-1+x)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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