\[ \int \frac {(2+e x)^{3/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {5 \,3^{\frac {3}{4}} \left (-e x +2\right )^{\frac {3}{4}} \left (e x +2\right )^{\frac {1}{4}}}{6 e}-\frac {3^{\frac {3}{4}} \left (-e x +2\right )^{\frac {3}{4}} \left (e x +2\right )^{\frac {5}{4}}}{6 e}-\frac {5 \arctan \left (-1+\frac {\left (-e x +2\right )^{\frac {1}{4}} \sqrt {2}}{\left (e x +2\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{6 e}-\frac {5 \arctan \left (1+\frac {\left (-e x +2\right )^{\frac {1}{4}} \sqrt {2}}{\left (e x +2\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{6 e}-\frac {5 \ln \left (\sqrt {3}-\frac {\left (-e x +2\right )^{\frac {1}{4}} \sqrt {6}}{\left (e x +2\right )^{\frac {1}{4}}}+\frac {\sqrt {3}\, \sqrt {-e x +2}}{\sqrt {e x +2}}\right ) 3^{\frac {3}{4}} \sqrt {2}}{12 e}+\frac {5 \ln \left (\sqrt {3}+\frac {\left (-e x +2\right )^{\frac {1}{4}} \sqrt {6}}{\left (e x +2\right )^{\frac {1}{4}}}+\frac {\sqrt {3}\, \sqrt {-e x +2}}{\sqrt {e x +2}}\right ) 3^{\frac {3}{4}} \sqrt {2}}{12 e} \]
command
integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{24} \cdot 3^{\frac {3}{4}} {\left ({\left (x e + 2\right )}^{2} {\left (5 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {7}{4}} + 9 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {3}{4}}\right )} + 20 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + 20 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - 10 \, \sqrt {2} \log \left (\sqrt {2} {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{x e + 2} - 1} + 1\right ) + 10 \, \sqrt {2} \log \left (-\sqrt {2} {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{x e + 2} - 1} + 1\right )\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (e x + 2\right )}^{\frac {3}{2}}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}\,{d x} \]________________________________________________________________________________________