3.2836 \(\int \frac{1}{\sqrt{-1-x} \sqrt{-3+x} \sqrt{-2+x}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 \sqrt{x+1} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{1}{3}}}\right )|\frac{4}{3}\right )}{\sqrt{3} \sqrt{-x-1}} \]

[Out]

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

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Rubi [A]  time = 0.0739897, 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 \[ -\frac{2 \sqrt{x+1} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{1}{3}}}\right )|\frac{4}{3}\right )}{\sqrt{3} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.63633, size = 53, normalized size = 1.29 \[ \frac{4 i \sqrt{- \frac{x}{3} + \frac{2}{3}} \sqrt{- \frac{x}{4} + \frac{3}{4}} F\left (i \operatorname{asinh}{\left (\frac{\sqrt{- x - 1}}{2} \right )}\middle | \frac{4}{3}\right )}{\sqrt{x - 3} \sqrt{x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1-x)**(1/2)/(-3+x)**(1/2)/(-2+x)**(1/2),x)

[Out]

4*I*sqrt(-x/3 + 2/3)*sqrt(-x/4 + 3/4)*elliptic_f(I*asinh(sqrt(-x - 1)/2), 4/3)/(
sqrt(x - 3)*sqrt(x - 2))

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Mathematica [C]  time = 0.0800732, size = 72, normalized size = 1.76 \[ \frac{2 i \sqrt{\frac{x-3}{x-2}} \sqrt{\frac{x-2}{x+1}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{-x-1}}\right )|\frac{4}{3}\right )}{\sqrt{3} \sqrt{\frac{x-3}{x+1}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.059, size = 68, normalized size = 1.7 \[ -{\frac{2\,\sqrt{3}}{3\,{x}^{3}-12\,{x}^{2}+3\,x+18}\sqrt{-1-x}\sqrt{-3+x}\sqrt{-2+x}\sqrt{1+x}\sqrt{2-x}\sqrt{3-x}{\it EllipticF} \left ({\frac{1}{2}\sqrt{1+x}},{\frac{2\,\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-1-x)^(1/2)*(-3+x)^(1/2)*(-2+x)^(1/2)*(1+x)^(1/2)*3^(1/2)*(2-x)^(1/2)*(3-x
)^(1/2)*EllipticF(1/2*(1+x)^(1/2),2/3*3^(1/2))/(x^3-4*x^2+x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(x - 3)*sqrt(-x - 1)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x - 2} \sqrt{x - 3} \sqrt{-x - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(x - 3)*sqrt(-x - 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x - 1} \sqrt{x - 3} \sqrt{x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x - 2} \sqrt{x - 3} \sqrt{-x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(x - 3)*sqrt(-x - 1)),x, algorithm="giac")

[Out]

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