\[ \int \frac {1}{(2+e x)^{9/2} \sqrt [4]{12-3 e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+4\right )^{\frac {3}{4}} 3^{\frac {3}{4}}}{45 e \left (e x +2\right )^{\frac {9}{2}}}-\frac {\left (-e^{2} x^{2}+4\right )^{\frac {3}{4}} 3^{\frac {3}{4}}}{165 e \left (e x +2\right )^{\frac {7}{2}}}-\frac {2 \left (-e^{2} x^{2}+4\right )^{\frac {3}{4}} 3^{\frac {3}{4}}}{1155 e \left (e x +2\right )^{\frac {5}{2}}}-\frac {2 \left (-e^{2} x^{2}+4\right )^{\frac {3}{4}} 3^{\frac {3}{4}}}{3465 e \left (e x +2\right )^{\frac {3}{2}}} \]
command
integrate(1/(e*x+2)^(9/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{221760} \cdot 3^{\frac {3}{4}} {\left (77 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {15}{4}} + 315 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {11}{4}} + 495 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {7}{4}} + 385 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {3}{4}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {9}{2}}}\,{d x} \]________________________________________________________________________________________