\[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (7 A +9 C \right ) \sin \left (d x +c \right )}{45 b^{3} d \left (b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 \left (7 A +9 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {b \sec \left (d x +c \right )}}+\frac {2 A \tan \left (d x +c \right )}{9 d \left (b \sec \left (d x +c \right )\right )^{\frac {9}{2}}} \]
command
integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, A \cos \left (d x + c\right )^{4} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45 \, b^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right )}}{b^{5} \sec \left (d x + c\right )^{5}}, x\right ) \]