\(\int \frac {(2+e x)^{3/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\) [922]

Optimal result

Integrand size = 24, antiderivative size = 22 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]

[Out]

2/9/e*3^(1/2)/(-e*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 32} \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]

[In]

Int[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

2/(3*Sqrt[3]*e*Sqrt[2 - e*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(6-3 e x)^{3/2}} \, dx \\ & = \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {4-e^2 x^2}}{3 e (-2+e x) \sqrt {6+3 e x}} \]

[In]

Integrate[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[4 - e^2*x^2])/(3*e*(-2 + e*x)*Sqrt[6 + 3*e*x])

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36

method	result	size
gosper	\(-\frac {2 \left (e x -2\right ) \left (e x +2\right )^{\frac {3}{2}}}{e \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}\)	\(30\)
default	\(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}}{9 \sqrt {e x +2}\, \left (e x -2\right ) e}\)	\(32\)

[In]

int((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2*(e*x-2)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).

Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

-2/9*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^3*x^2 - 4*e)

Sympy [F]

\[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \]

[In]

integrate((e*x+2)**(3/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

sqrt(3)*(Integral(2*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(e*
x*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x))/9

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 i \, \sqrt {3}}{9 \, \sqrt {e x - 2} e} \]

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

-2/9*I*sqrt(3)/(sqrt(e*x - 2)*e)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {3}}{9 \, e} + \frac {2 \, \sqrt {3}}{9 \, \sqrt {-e x + 2} e} \]

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")

[Out]

-1/9*sqrt(3)/e + 2/9*sqrt(3)/(sqrt(-e*x + 2)*e)

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2\,\sqrt {e\,x+2}}{3\,e\,\sqrt {12-3\,e^2\,x^2}} \]

[In]

int((e*x + 2)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x)

[Out]

(2*(e*x + 2)^(1/2))/(3*e*(12 - 3*e^2*x^2)^(1/2))