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