3.2825 \(\int \frac{1}{\sqrt{3-x} \sqrt{1+x} \sqrt{2+x}} \, dx\)

Optimal. Leaf size=16 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )\right |-4\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[1 + x]/2], -4]

_______________________________________________________________________________________

Rubi [A]  time = 0.0379097, antiderivative size = 16, 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 \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )\right |-4\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*EllipticF[ArcSin[Sqrt[1 + x]/2], -4]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.0236, size = 14, normalized size = 0.88 \[ 2 F\left (\operatorname{asin}{\left (\frac{\sqrt{x + 1}}{2} \right )}\middle | -4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*elliptic_f(asin(sqrt(x + 1)/2), -4)

_______________________________________________________________________________________

Mathematica [C]  time = 0.121041, size = 74, normalized size = 4.62 \[ \frac{i \sqrt{\frac{4}{x-3}+1} \sqrt{\frac{5}{x-3}+1} (x-3)^{3/2} F\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-3}}\right )|\frac{5}{4}\right )}{\sqrt{-(x-3) (x+1)} \sqrt{x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(I*Sqrt[1 + 4/(-3 + x)]*Sqrt[1 + 5/(-3 + x)]*(-3 + x)^(3/2)*EllipticF[I*ArcSinh[
2/Sqrt[-3 + x]], 5/4])/(Sqrt[-((-3 + x)*(1 + x))]*Sqrt[2 + x])

_______________________________________________________________________________________

Maple [A]  time = 0.089, size = 25, normalized size = 1.6 \[ -{1{\it EllipticF} \left ( \sqrt{-1-x},{\frac{i}{2}} \right ) \sqrt{-1-x}{\frac{1}{\sqrt{1+x}}}} \]

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]

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

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)