\[ \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {384 \left (-e x +2\right )^{\frac {5}{2}} \sqrt {3}}{5 e}+\frac {288 \left (-e x +2\right )^{\frac {7}{2}} \sqrt {3}}{7 e}-\frac {8 \left (-e x +2\right )^{\frac {9}{2}} \sqrt {3}}{e}+\frac {6 \left (-e x +2\right )^{\frac {11}{2}} \sqrt {3}}{11 e} \]
command
integrate((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{1155} \, \sqrt {3} {\left (11088 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - {\left ({\left (315 \, {\left (x e - 2\right )}^{5} \sqrt {-x e + 2} + 3080 \, {\left (x e - 2\right )}^{4} \sqrt {-x e + 2} + 11880 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 22176 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 18480 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-4\right )} + 27008 \, e^{\left (-4\right )}\right )} e^{4} - 44 \, {\left ({\left (35 \, {\left (x e - 2\right )}^{4} \sqrt {-x e + 2} + 270 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 756 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 840 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-3\right )} - 832 \, e^{\left (-3\right )}\right )} e^{3} - 55440 \, {\left (-x e + 2\right )}^{\frac {3}{2}} + 88704\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}\,{d x} \]________________________________________________________________________________________