\[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {-e x +2}}{2}\right ) \sqrt {3}}{96 e}+\frac {\sqrt {3}}{48 e \sqrt {-e x +2}}-\frac {\sqrt {3}}{36 e \left (e x +2\right ) \sqrt {-e x +2}} \]
command
integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{192} \, \sqrt {3} e^{\left (-1\right )} \log \left (\sqrt {-x e + 2} + 2\right ) + \frac {1}{192} \, \sqrt {3} e^{\left (-1\right )} \log \left (-\sqrt {-x e + 2} + 2\right ) - \frac {\sqrt {3} {\left (3 \, x e + 2\right )} e^{\left (-1\right )}}{144 \, {\left ({\left (-x e + 2\right )}^{\frac {3}{2}} - 4 \, \sqrt {-x e + 2}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________