\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \left (19 A +21 C \right ) \sin \left (d x +c \right )}{105 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 A \left (a +a \sec \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{9 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {8 A \left (a^{2}+a^{2} \sec \left (d x +c \right )\right ) \sin \left (d x +c \right )}{63 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a^{2} \left (5 A +7 C \right ) \sin \left (d x +c \right )}{21 d \sqrt {\sec \left (d x +c \right )}}+\frac {16 a^{2} \left (2 A +3 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{2} \left (5 A +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+a*sec(d*x+c))^2*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 84 i \, \sqrt {2} {\left (2 \, A + 3 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 i \, \sqrt {2} {\left (2 \, A + 3 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (35 \, A a^{2} \cos \left (d x + c\right )^{4} + 90 \, A a^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (16 \, A + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 30 \, {\left (5 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C a^{2} \sec \left (d x + c\right )^{4} + 2 \, C a^{2} \sec \left (d x + c\right )^{3} + {\left (A + C\right )} a^{2} \sec \left (d x + c\right )^{2} + 2 \, A a^{2} \sec \left (d x + c\right ) + A a^{2}}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]