\(\int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx\) [860]

Optimal result

Integrand size = 26, antiderivative size = 36 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\frac {\sqrt {\frac {2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {3}{2}} \sqrt {e x}}{\sqrt {e}}\right ),-1\right )}{\sqrt {e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {117} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\frac {\sqrt {\frac {2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {3}{2}} \sqrt {e x}}{\sqrt {e}}\right ),-1\right )}{\sqrt {e}} \]

[In]

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

[Out]

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

Rule 117

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] && (PosQ[-b/d] || NegQ[-b/f])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {2}{3}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {e x}}{\sqrt {e}}\right )\right |-1\right )}{\sqrt {e}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {9 x^2}{4}\right )}{\sqrt {e x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\sqrt {x}\, \sqrt {3}\, F\left (\frac {\sqrt {4-6 x}}{2}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {e x}}\)	\(29\)
elliptic	\(-\frac {\sqrt {-e x \left (9 x^{2}-4\right )}\, \sqrt {4-6 x}\, \sqrt {6}\, \sqrt {x}\, F\left (\frac {\sqrt {4-6 x}}{2}, \frac {\sqrt {2}}{2}\right )}{6 \sqrt {e x}\, \sqrt {2-3 x}\, \sqrt {-9 e \,x^{3}+4 e x}}\)	\(69\)

[In]

int(1/(2-3*x)^(1/2)/(e*x)^(1/2)/(2+3*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*x^(1/2)*3^(1/2)*EllipticF(1/2*(4-6*x)^(1/2),1/2*2^(1/2))/(e*x)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=-\frac {2 \, \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {16}{9}, 0, x\right )}{3 \, e} \]

[In]

integrate(1/(2-3*x)^(1/2)/(e*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(-e)*weierstrassPInverse(16/9, 0, x)/e

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (31) = 62\).

Time = 10.97 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.44 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=- \frac {\sqrt {6} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}} \sqrt {e}} + \frac {\sqrt {6} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}} \sqrt {e}} \]

[In]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\int { \frac {1}{\sqrt {e x} \sqrt {3 \, x + 2} \sqrt {-3 \, x + 2}} \,d x } \]

[In]

integrate(1/(2-3*x)^(1/2)/(e*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\int { \frac {1}{\sqrt {e x} \sqrt {3 \, x + 2} \sqrt {-3 \, x + 2}} \,d x } \]

[In]

integrate(1/(2-3*x)^(1/2)/(e*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {e x} \sqrt {2+3 x}} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {3\,x+2}\,\sqrt {e\,x}} \,d x \]

[In]

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

[Out]

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