3.2827 \(\int \frac{1}{\sqrt{2-x} \sqrt{3-x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=18 \[ F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|\frac{3}{4}\right ) \]

[Out]

EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], 3/4]

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Rubi [A]  time = 0.0404446, antiderivative size = 18, 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 \[ F\left (\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{3}}\right )|\frac{3}{4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[1 + x]/Sqrt[3]], 3/4]

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Rubi in Sympy [A]  time = 4.68758, size = 17, normalized size = 0.94 \[ F\left (\operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{x + 1}}{3} \right )}\middle | \frac{3}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.11947, size = 65, normalized size = 3.61 \[ -\frac{2 i \sqrt{1-\frac{3}{2-x}} \sqrt{\frac{1}{2-x}+1} (2-x) F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{2-x}}\right )\right |-3\right )}{\sqrt{-(x-3) (x+1)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.089, size = 19, normalized size = 1.1 \[{\frac{2\,\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{2}\sqrt{1+x}},{\frac{2\,\sqrt{3}}{3}} \right ) } \]

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

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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)