\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 a \left (3 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (A +3 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a A \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}+\frac {2 a A \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 d} \]
command
integrate(cos(d*x+c)^(5/2)*(a+a*sec(d*x+c))*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} {\left (A + 3 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (A + 3 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 i \, \sqrt {2} {\left (3 \, A + 5 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} {\left (3 \, A + 5 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, A a \cos \left (d x + c\right ) + 5 \, A a\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C a \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + C a \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A a \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]