\[ \int \frac {(a+a \sec (c+d x))^{5/2}}{\sqrt {\sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {5 a^{\frac {5}{2}} \arcsinh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a +a \sec \left (d x +c \right )}}\right )}{d}+\frac {a^{3} \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{d \sqrt {a +a \sec \left (d x +c \right )}}+\frac {a^{2} \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {a +a \sec \left (d x +c \right )}}{d} \]
command
integrate((a+a*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________