\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2}{3 b f \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x +e \right )}}-\frac {8 \sqrt {d \sec \left (f x +e \right )}}{3 b \,d^{2} f \sqrt {b \tan \left (f x +e \right )}} \]
command
integrate(1/(d*sec(f*x+e))**(3/2)/(b*tan(f*x+e))**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {8 \tan ^{3}{\left (e + f x \right )}}{3 f \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}} - \frac {2 \tan {\left (e + f x \right )}}{f \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}} & \text {for}\: f \neq 0 \\\frac {x}{\left (b \tan {\left (e \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e \right )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________