3.2830 \(\int \frac{1}{\sqrt{1-x} \sqrt{2-x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+3}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/Sqrt[5]

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Rubi [A]  time = 0.0415293, antiderivative size = 23, 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 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{x+3}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/Sqrt[5]

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Rubi in Sympy [A]  time = 4.76603, size = 15, normalized size = 0.65 \[ - 2 F\left (\operatorname{asin}{\left (\frac{\sqrt{- x + 1}}{2} \right )}\middle | -4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.115298, size = 65, normalized size = 2.83 \[ -\frac{2 i \sqrt{1-\frac{4}{1-x}} \sqrt{\frac{1}{1-x}+1} (1-x) F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{1-x}}\right )\right |-4\right )}{\sqrt{-(x-2) (x+3)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.095, size = 17, normalized size = 0.7 \[{\it EllipticF} \left ({\frac{\sqrt{5}}{5}\sqrt{3+x}},{\frac{\sqrt{5}}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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