\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {4 a^{3} \left (143 A +105 C \right ) \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{231 d}+\frac {8 a^{3} \left (44 A +35 C \right ) \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{385 d}+\frac {2 C \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{11 d}+\frac {4 C \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a^{2}+a^{2} \sec \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{33 a d}+\frac {2 \left (33 A +35 C \right ) \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a^{3}+a^{3} \sec \left (d x +c \right )\right ) \sin \left (d x +c \right )}{231 d}+\frac {4 a^{3} \left (7 A +5 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 d}-\frac {4 a^{3} \left (7 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (143 A +105 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate(sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (462 \, {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (9 \, A + 10 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (11 \, A + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, C a^{3} \cos \left (d x + c\right ) + 105 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d \cos \left (d x + c\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C a^{3} \sec \left (d x + c\right )^{6} + 3 \, C a^{3} \sec \left (d x + c\right )^{5} + {\left (A + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (3 \, A + C\right )} a^{3} \sec \left (d x + c\right )^{3} + 3 \, A a^{3} \sec \left (d x + c\right )^{2} + A a^{3} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]