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