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

Optimal. Leaf size=27 \[ -\sqrt{\frac{2}{3}} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )\right |-1\right ) \]

[Out]

-(Sqrt[2/3]*EllipticF[ArcSin[Sqrt[3/2]*Sqrt[-x]], -1])

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Rubi [A]  time = 0.0371795, antiderivative size = 27, 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 \[ -\sqrt{\frac{2}{3}} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2/3]*EllipticF[ArcSin[Sqrt[3/2]*Sqrt[-x]], -1])

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Rubi in Sympy [A]  time = 3.764, size = 26, normalized size = 0.96 \[ - \frac{\sqrt{6} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{- x}}{2} \right )}\middle | -1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(6)*elliptic_f(asin(sqrt(6)*sqrt(-x)/2), -1)/3

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Mathematica [A]  time = 0.0509301, size = 50, normalized size = 1.85 \[ \frac{\sqrt{6-\frac{8}{3 x^2}} \sqrt{-x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{4-9 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[6 - 8/(3*x^2)]*Sqrt[-x^2]*EllipticF[ArcSin[Sqrt[2/3]/Sqrt[x]], -1])/Sqrt[4
- 9*x^2]

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Maple [A]  time = 0.046, size = 24, normalized size = 0.9 \[{\frac{\sqrt{3}}{3}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{2+3\,x}},{\frac{\sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 34.4515, size = 82, normalized size = 3.04 \[ \frac{\sqrt{6} i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{4 e^{- 2 i \pi }}{9 x^{2}}} \right )}}{24 \pi ^{\frac{3}{2}}} - \frac{\sqrt{6} i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{4}{9 x^{2}}} \right )}}{24 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)
), 4*exp_polar(-2*I*pi)/(9*x**2))/(24*pi**(3/2)) - sqrt(6)*I*meijerg(((-1/4, 0,
1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), 4/(9*x**2))/(24*pi**(3/2
))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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