\[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {5 a C \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{12 d \sqrt {\cos \left (d x +c \right )}}+\frac {C \left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{3 d \sqrt {\cos \left (d x +c \right )}}+\frac {5 a^{\frac {5}{2}} \left (8 A +5 C \right ) \arcsinh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{8 d}+\frac {a^{3} \left (24 A -49 C \right ) \sin \left (d x +c \right )}{24 d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +a \sec \left (d x +c \right )}}+\frac {a^{2} \left (24 A +31 C \right ) \sin \left (d x +c \right ) \sqrt {a +a \sec \left (d x +c \right )}}{24 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)*cos(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 \[ \text {Timed out} \]_____________________________________________________