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

Optimal. Leaf size=24 \[ \sqrt{\frac{2}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right ),-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.004738, antiderivative size = 24, 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, 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 = 0.0071765, size = 23, normalized size = 0.96 \[ \sqrt{x} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{9 x^2}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 29, normalized size = 1.2 \begin{align*}{\frac{\sqrt{3}}{3}{\it EllipticF} \left ({\frac{1}{2}\sqrt{4+6\,x}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-x}{\frac{1}{\sqrt{x}}}} \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)*(-x)^(1/2)/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x + 2} \sqrt{x} \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(3*x + 2)*sqrt(x)*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{3 \, x + 2} \sqrt{x} \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(3*x + 2)*sqrt(x)*sqrt(-3*x + 2)/(9*x^3 - 4*x), x)

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Sympy [B]  time = 5.6352, size = 78, normalized size = 3.25 \begin{align*} - \frac{\sqrt{6}{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}{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)*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)*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{3 \, x + 2} \sqrt{x} \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(3*x + 2)*sqrt(x)*sqrt(-3*x + 2)), x)