\[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {-e x +2}}{2}\right ) \sqrt {3}}{2 e}-\frac {\sqrt {3}\, \sqrt {-e x +2}}{e \left (e x +2\right )} \]
command
integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{4} \, \sqrt {3} {\left (\frac {4 \, \sqrt {-x e + 2}}{x e + 2} - \log \left (\sqrt {-x e + 2} + 2\right ) + \log \left (-\sqrt {-x e + 2} + 2\right )\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________