\[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {a \left (8 B +5 C \right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{24 d \sqrt {\sec \left (d x +c \right )}}+\frac {C \left (a +a \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{4 d \sqrt {\sec \left (d x +c \right )}}+\frac {a^{3} \left (432 A +392 B +299 C \right ) \sin \left (d x +c \right )}{192 d \sqrt {a +a \cos \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}}+\frac {a^{2} \left (16 A +24 B +17 C \right ) \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{32 d \sqrt {\sec \left (d x +c \right )}}+\frac {a^{\frac {5}{2}} \left (304 A +200 B +163 C \right ) \arcsin \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{64 d} \]
command
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^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 \[ \text {Timed out} \]_____________________________________________________