3.2852 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{\sqrt{x+2} \sqrt{x+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2}{\sqrt{x+3}}\right ),\frac{1}{4}\right )}{\sqrt{-x-3} \sqrt{-x-2}}-\frac{2 i K(4) \sqrt{x+2}}{\sqrt{-x-2}}+\frac{K\left (\frac{3}{4}\right ) \sqrt{x+3}}{\sqrt{-x-3}} \]

[Out]

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

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Rubi [A]  time = 0.0180816, antiderivative size = 52, normalized size of antiderivative = 0.57, 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} \[ -\frac{\sqrt{x+2} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{4}+\frac{3}{4}}}\right )|\frac{1}{4}\right )}{\sqrt{-x-3} \sqrt{-x-2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Mathematica [A]  time = 0.087631, size = 75, normalized size = 0.82 \[ \frac{2 i \sqrt{\frac{3}{x-1}+1} \sqrt{\frac{4}{x-1}+1} (x-1) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-1}}\right ),\frac{4}{3}\right )}{\sqrt{-3 (x-1)-12} \sqrt{-x-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[1 + 3/(-1 + x)]*Sqrt[1 + 4/(-1 + x)]*(-1 + x)*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-1 + x]], 4/3])/(Sq
rt[-12 - 3*(-1 + x)]*Sqrt[-2 - x])

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Maple [A]  time = 0.031, size = 54, normalized size = 0.6 \begin{align*}{\frac{2\,\sqrt{3}}{3\,{x}^{2}+6\,x-9}{\it EllipticF} \left ( \sqrt{-2-x},{\frac{i}{3}}\sqrt{3} \right ) \sqrt{3+x}\sqrt{1-x}\sqrt{-1+x}\sqrt{-3-x}} \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]

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

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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} + 4 \, x^{2} + 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 + 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{- 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)