3.2838 \(\int \frac{1}{\sqrt{-2-x} \sqrt{-1-x} \sqrt{-3+x}} \, dx\)

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

[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 = 0.12141, 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 \[ -\frac{2 \sqrt{x+1} \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{2}{5}}}\right )|\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x-2} \sqrt{-x-1}} \]

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])

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Rubi in Sympy [A]  time = 9.04088, size = 49, normalized size = 0.86 \[ - \frac{2 \sqrt{- \frac{x}{4} + \frac{3}{4}} \sqrt{x + 2} F\left (\operatorname{asin}{\left (\sqrt{- x - 1} \right )}\middle | - \frac{1}{4}\right )}{\sqrt{- x - 2} \sqrt{x - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0568539, size = 69, normalized size = 1.21 \[ \frac{i \sqrt{\frac{4}{x-3}+1} \sqrt{\frac{5}{x-3}+1} (x-3) F\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-3}}\right )|\frac{5}{4}\right )}{\sqrt{-x-2} \sqrt{-x-1}} \]

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 = 0.053, size = 46, normalized size = 0.8 \[{\frac{1}{{x}^{2}-x-6}{\it EllipticF} \left ( \sqrt{-1-x},{\frac{i}{2}} \right ) \sqrt{2+x}\sqrt{3-x}\sqrt{-3+x}\sqrt{-2-x}} \]

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. \[ \int \frac{1}{\sqrt{x - 3} \sqrt{-x - 1} \sqrt{-x - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 3)*sqrt(-x - 1)*sqrt(-x - 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. \[{\rm integral}\left (\frac{1}{\sqrt{x - 3} \sqrt{-x - 1} \sqrt{-x - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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