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

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

[Out]

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

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Rubi [A]  time = 0.0050956, 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, Rules used = {115} \[ -\sqrt{\frac{2}{3}} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )\right |-1\right ) \]

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])

Rule 115

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2-3 x} \sqrt{-x} \sqrt{2+3 x}} \, dx &=-\sqrt{\frac{2}{3}} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0065802, size = 26, normalized size = 0.96 \[ \frac{x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{9 x^2}{4}\right )}{\sqrt{-x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Hypergeometric2F1[1/4, 1/2, 5/4, (9*x^2)/4])/Sqrt[-x]

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Maple [A]  time = 0.028, size = 21, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{2}\sqrt{4+6\,x}},{\frac{\sqrt{2}}{2}} \right ) } \end{align*}

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*(4+6*x)^(1/2),1/2*2^(1/2))*3^(1/2)

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

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x} \sqrt{3 \, x + 2} \sqrt{-3 \, x + 2}}{9 \, x^{3} - 4 \, x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(-x)*sqrt(3*x + 2)*sqrt(-3*x + 2)/(9*x^3 - 4*x), x)

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Sympy [B]  time = 10.9068, size = 82, normalized size = 3.04 \begin{align*} \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}}} \end{align*}

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

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, algorithm="giac")

[Out]

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