\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 A \left (a +a \sec \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{3 d \sqrt {\sec \left (d x +c \right )}}+\frac {4 a^{3} \left (5 A +20 B +21 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 d}-\frac {2 \left (5 A -3 C \right ) \left (a^{2}+a^{2} \sec \left (d x +c \right )\right )^{2} \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 a d}-\frac {2 \left (5 A -5 B -9 C \right ) \left (a^{3}+a^{3} \sec \left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 d}+\frac {4 a^{3} \left (5 A -5 B -9 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (5 A +5 B +3 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (5 \, A + 5 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (5 \, A + 5 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 i \, \sqrt {2} {\left (5 \, A - 5 \, B - 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} {\left (5 \, A - 5 \, B - 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (5 \, A a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, A + 15 \, B + 18 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 3 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{15 \, d \cos \left (d x + c\right )^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C a^{3} \sec \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]