\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {56 a^{4} \left (A -C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a^{4} \left (4 A +5 B +4 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (5 B +8 C \right ) \left (a +a \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{15 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 C \left (a +a \cos \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{5 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (5 A +15 B +19 C \right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{5 d \sqrt {\cos \left (d x +c \right )}}+\frac {4 a^{4} \left (A -25 B -41 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{15 d}-\frac {4 \left (6 A +25 B +34 C \right ) \left (a^{4}+a^{4} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{15 d} \]
command
integrate(cos(d*x+c)^(5/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (10 i \, \sqrt {2} {\left (4 \, A + 5 \, B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 10 i \, \sqrt {2} {\left (4 \, A + 5 \, B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 42 i \, \sqrt {2} {\left (A - C\right )} a^{4} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} {\left (A - C\right )} a^{4} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3 \, A a^{4} \cos \left (d x + c\right )^{4} + 5 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, A + 20 \, B + 33 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 5 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 3 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{15 \, d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A a^{4} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]