\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 b \left (7 A +5 C \right ) \left (b \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{21 d}+\frac {2 b^{2} \left (7 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {b \sec \left (d x +c \right )}}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 C \left (b \sec \left (d x +c \right )\right )^{\frac {5}{2}} \tan \left (d x +c \right )}{7 d} \]
command
integrate((b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left ({\left (7 \, A + 5 \, C\right )} b^{2} \cos \left (d x + c\right )^{2} + 3 \, C b^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{2} \sec \left (d x + c\right )^{4} + A b^{2} \sec \left (d x + c\right )^{2}\right )} \sqrt {b \sec \left (d x + c\right )}, x\right ) \]