\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (-e x +2\right )^{\frac {3}{2}} \sqrt {3}}{4 e \left (e x +2\right )^{4}}-\frac {9 \arctanh \left (\frac {\sqrt {-e x +2}}{2}\right ) \sqrt {3}}{2048 e}+\frac {3 \sqrt {3}\, \sqrt {-e x +2}}{8 e \left (e x +2\right )^{3}}-\frac {3 \sqrt {3}\, \sqrt {-e x +2}}{128 e \left (e x +2\right )^{2}}-\frac {9 \sqrt {3}\, \sqrt {-e x +2}}{1024 e \left (e x +2\right )} \]
command
integrate((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(13/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3}{4096} \, \sqrt {3} {\left (\frac {4 \, {\left (3 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 44 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} + 176 \, {\left (-x e + 2\right )}^{\frac {3}{2}} - 192 \, \sqrt {-x e + 2}\right )}}{{\left (x e + 2\right )}^{4}} + 3 \, \log \left (\sqrt {-x e + 2} + 2\right ) - 3 \, \log \left (-\sqrt {-x e + 2} + 2\right )\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________