3.2837 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-1-x} \sqrt{-2+x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 \sqrt{x+1} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{3}{5}}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x-3} \sqrt{-x-1}} \]

[Out]

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

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Rubi [A]  time = 0.118975, 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 \[ -\frac{2 \sqrt{x+1} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{3}{5}}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x-3} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.63543, size = 65, normalized size = 1.14 \[ - \frac{2 \sqrt{2} \sqrt{- \frac{x}{3} + \frac{2}{3}} \sqrt{\frac{x}{2} + \frac{3}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- x - 1}}{2} \right )}\middle | - \frac{2}{3}\right )}{\sqrt{- x - 3} \sqrt{x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0534599, size = 75, normalized size = 1.32 \[ \frac{2 i \sqrt{\frac{3}{x-2}+1} \sqrt{\frac{5}{x-2}+1} (x-2) F\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-2}}\right )|\frac{5}{3}\right )}{\sqrt{-3 (x-2)-15} \sqrt{-x-1}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[1 + 3/(-2 + x)]*Sqrt[1 + 5/(-2 + x)]*(-2 + x)*EllipticF[I*ArcSinh[Sq
rt[3]/Sqrt[-2 + x]], 5/3])/(Sqrt[-15 - 3*(-2 + x)]*Sqrt[-1 - x])

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Maple [C]  time = 0.054, size = 57, normalized size = 1. \[{\frac{2\,\sqrt{3}}{3\,{x}^{2}+3\,x-18}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2-2\,x}},{\frac{i}{3}}\sqrt{3}\sqrt{2} \right ) \sqrt{3+x}\sqrt{2-x}\sqrt{-2+x}\sqrt{-3-x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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