3.2834 \(\int \frac{1}{\sqrt{-2-x} \sqrt{-3+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]

[Out]

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

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Rubi [A]  time = 0.0721526, 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 \[ -\frac{2 \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.02323, size = 61, normalized size = 1.49 \[ \frac{2 \sqrt{2} \sqrt{- \frac{x}{2} + \frac{3}{2}} \sqrt{\frac{x}{3} + \frac{2}{3}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}\middle | - \frac{2}{3}\right )}{\sqrt{- x - 2} \sqrt{x - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.149831, size = 72, normalized size = 1.76 \[ \frac{2 i \sqrt{\frac{x-3}{x-1}} \sqrt{\frac{x-1}{x+2}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{-x-2}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{\frac{x-3}{x+2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.059, size = 76, normalized size = 1.9 \[ -{\frac{2\,\sqrt{3}}{3\,{x}^{3}-6\,{x}^{2}-15\,x+18}\sqrt{-2-x}\sqrt{-3+x}\sqrt{-1+x}\sqrt{2+x}\sqrt{1-x}\sqrt{3-x}{\it EllipticF} \left ({\frac{\sqrt{5}}{5}\sqrt{2+x}},{\frac{\sqrt{5}\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-2-x)^(1/2)*(-3+x)^(1/2)*(-1+x)^(1/2)*(2+x)^(1/2)*3^(1/2)*(1-x)^(1/2)*(3-x
)^(1/2)*EllipticF(1/5*5^(1/2)*(2+x)^(1/2),1/3*5^(1/2)*3^(1/2))/(x^3-2*x^2-5*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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