3.9.59 \(\int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx\) [859]

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]

-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 = 27, normalized size of antiderivative = 1.00, 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 = {116} \begin {gather*} -\sqrt {\frac {2}{3}} F\left (\left .\text {ArcSin}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )\right |-1\right ) \end {gather*}

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 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 26, normalized size = 0.96 \begin {gather*} \frac {x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {9 x^2}{4}\right )}{\sqrt {-x}} \end {gather*}

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 = 0.11, size = 21, normalized size = 0.78

method	result	size
default	\(\frac {\EllipticF \left (\frac {\sqrt {4+6 x}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {3}}{3}\)	\(21\)
elliptic	\(\frac {\sqrt {x \left (9 x^{2}-4\right )}\, \sqrt {4+6 x}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {4+6 x}}{2}, \frac {\sqrt {2}}{2}\right )}{6 \sqrt {-x}\, \sqrt {2+3 x}\, \sqrt {9 x^{3}-4 x}}\)	\(64\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 6, normalized size = 0.22 \begin {gather*} \frac {2}{3} \, {\rm weierstrassPInverse}\left (\frac {16}{9}, 0, x\right ) \end {gather*}

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]

2/3*weierstrassPInverse(16/9, 0, x)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (22) = 44\).
time = 12.84, size = 82, normalized size = 3.04 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {-x}\,\sqrt {2-3\,x}\,\sqrt {3\,x+2}} \,d x \end {gather*}

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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