3.2828 \(\int \frac{1}{\sqrt{1-x} \sqrt{2+x} \sqrt{3+x}} \, dx\)

Optimal. Leaf size=18 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )\right |-3\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], -3]

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Rubi [A]  time = 0.0375513, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+2}}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], -3]

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Rubi in Sympy [A]  time = 4.07538, size = 19, normalized size = 1.06 \[ - F\left (\operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{- x + 1}}{3} \right )}\middle | \frac{3}{4}\right ) \]

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]

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

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Mathematica [C]  time = 0.151251, size = 78, normalized size = 4.33 \[ -\frac{2 i \sqrt{-(x-1) (x+2)} \sqrt{x+3} F\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-1}}\right )|\frac{4}{3}\right )}{\sqrt{\frac{9}{x-1}+3} (x-1)^{3/2} \sqrt{\frac{x+3}{x-1}}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[-((-1 + x)*(2 + x))]*Sqrt[3 + x]*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-
1 + x]], 4/3])/(Sqrt[3 + 9/(-1 + x)]*(-1 + x)^(3/2)*Sqrt[(3 + x)/(-1 + x)])

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Maple [A]  time = 0.093, size = 32, normalized size = 1.8 \[ -{\frac{2\,\sqrt{3}}{3}\sqrt{-2-x}{\it EllipticF} \left ( \sqrt{-2-x},{\frac{i}{3}}\sqrt{3} \right ){\frac{1}{\sqrt{2+x}}}} \]

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]

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