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