\[ \int \frac {1}{(d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {b \tan \left (f x +e \right )}}{5 b f \left (d \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {8 \sqrt {b \tan \left (f x +e \right )}}{5 b \,d^{2} f \sqrt {d \sec \left (f x +e \right )}} \]
command
integrate(1/(d*sec(f*x+e))**(5/2)/(b*tan(f*x+e))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {8 \tan ^{3}{\left (e + f x \right )}}{5 f \sqrt {b \tan {\left (e + f x \right )}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}} + \frac {2 \tan {\left (e + f x \right )}}{f \sqrt {b \tan {\left (e + f x \right )}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}} & \text {for}\: f \neq 0 \\\frac {x}{\sqrt {b \tan {\left (e \right )}} \left (d \sec {\left (e \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________