\(\int \frac {(2+e x)^{5/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\) [921]

Optimal result

Integrand size = 24, antiderivative size = 45 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {8}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e} \]

[Out]

8/9/e*3^(1/2)/(-e*x+2)^(1/2)+2/9*3^(1/2)*(-e*x+2)^(1/2)/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45} \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {8}{3 \sqrt {3} e \sqrt {2-e x}} \]

[In]

Int[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

8/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (2*Sqrt[2 - e*x])/(3*Sqrt[3]*e)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 {2+e x}{(6-3 e x)^{3/2}} \, dx \\ & = \int \left (\frac {4}{(6-3 e x)^{3/2}}-\frac {1}{3 \sqrt {6-3 e x}}\right ) \, dx \\ & = \frac {8}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 (-6+e x) \sqrt {4-e^2 x^2}}{3 e (-2+e x) \sqrt {6+3 e x}} \]

[In]

Integrate[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(2*(-6 + e*x)*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 = 35, normalized size of antiderivative = 0.78

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (e x -6\right ) \left (e x +2\right )^{\frac {3}{2}}}{e \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}\)	\(35\)
default	\(\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x -6\right )}{9 \sqrt {e x +2}\, \left (e x -2\right ) e}\)	\(37\)
risch	\(-\frac {2 \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}+\frac {8 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}\)	\(111\)

[In]

int((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2*(e*x-2)*(e*x-6)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 6\right )}}{9 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]

[In]

integrate((e*x+2)^(5/2)/(-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*x - 6)/(e^3*x^2 - 4*e)

Sympy [F]

\[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \left (\int \frac {4 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \]

[In]

integrate((e*x+2)**(5/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

sqrt(3)*(Integral(4*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(4*
e*x*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(e**2*x**2*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 = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.44 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 i \, \sqrt {3} {\left (e x - 6\right )}}{9 \, \sqrt {e x - 2} e} \]

[In]

integrate((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

2/9*I*sqrt(3)*(e*x - 6)/(sqrt(e*x - 2)*e)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {3} \sqrt {-e x + 2}}{9 \, e} - \frac {8 \, \sqrt {3}}{9 \, e} + \frac {8 \, \sqrt {3}}{9 \, \sqrt {-e x + 2} e} \]

[In]

integrate((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*sqrt(-e*x + 2)/e - 8/9*sqrt(3)/e + 8/9*sqrt(3)/(sqrt(-e*x + 2)*e)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4\,\sqrt {e\,x+2}}{3\,e^3}-\frac {2\,x\,\sqrt {e\,x+2}}{9\,e^2}\right )\,\sqrt {12-3\,e^2\,x^2}}{\frac {4}{e^2}-x^2} \]

[In]

int((e*x + 2)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x)

[Out]

(((4*(e*x + 2)^(1/2))/(3*e^3) - (2*x*(e*x + 2)^(1/2))/(9*e^2))*(12 - 3*e^2*x^2)^(1/2))/(4/e^2 - x^2)