\[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {5 a^{\frac {5}{2}} C \arcsinh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a +a \sec \left (d x +c \right )}}\right )}{d}+\frac {2 A \left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{5 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a A \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{3 d \sqrt {\sec \left (d x +c \right )}}+\frac {a^{3} \left (64 A +15 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 d \sqrt {a +a \sec \left (d x +c \right )}}-\frac {a^{2} \left (16 A -15 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {a +a \sec \left (d x +c \right )}}{15 d} \]
command
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2),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 \[ \text {Timed out} \]_____________________________________________________