\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5 b f \left (d \sec \left (f x +e \right )\right )^{\frac {5}{2}}} \]
command
integrate((b*tan(f*x+e))**(3/2)/(d*sec(f*x+e))**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {2 \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}}{5 f \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}} & \text {for}\: f \neq 0 \\\frac {x \left (b \tan {\left (e \right )}\right )^{\frac {3}{2}}}{\left (d \sec {\left (e \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________