3.2829 \(\int \frac{1}{\sqrt{1-x} \sqrt{3-x} \sqrt{2+x}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )|\frac{3}{5}\right )}{\sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.0431427, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )|\frac{3}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.73332, size = 26, normalized size = 1.04 \[ - \sqrt{2} F\left (\operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{- x + 1}}{3} \right )}\middle | - \frac{3}{2}\right ) \]

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]

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

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Mathematica [B]  time = 0.0964608, size = 68, normalized size = 2.72 \[ -\frac{2 \sqrt{\frac{x-3}{x-1}} (x-1) \sqrt{\frac{x+2}{x-1}} F\left (\sin ^{-1}\left (\frac{\sqrt{3}}{\sqrt{1-x}}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-x^2+x+6}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(-3 + x)/(-1 + x)]*(-1 + x)*Sqrt[(2 + x)/(-1 + x)]*EllipticF[ArcSin[Sqr
t[3]/Sqrt[1 - x]], -2/3])/(Sqrt[3]*Sqrt[6 + x - x^2])

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Maple [A]  time = 0.118, size = 25, normalized size = 1. \[{\frac{2\,\sqrt{3}}{3}{\it EllipticF} \left ({\frac{\sqrt{5}}{5}\sqrt{2+x}},{\frac{\sqrt{5}\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)