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

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

[Out]

1/3*EllipticF(1/2*6^(1/2)*x^(1/2),I)*6^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.01, 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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fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x + 2} \sqrt {x} \sqrt {-3 \, x + 2}}{9 \, x^{3} - 4 \, x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x + 2} \sqrt {x} \sqrt {-3 \, x + 2}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 29, normalized size = 1.21 \[ \frac {\sqrt {3}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2-3*x)^(1/2)/x^(1/2)/(3*x+2)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x + 2} \sqrt {x} \sqrt {-3 \, x + 2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {x}\,\sqrt {2-3\,x}\,\sqrt {3\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 4.79, size = 78, normalized size = 3.25 \[ - \frac {\sqrt {6} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}}} + \frac {\sqrt {6} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}}} \]

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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