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