\[ \int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx \]
Optimal antiderivative \[ \frac {2 b B \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{5 d}+\frac {2 A \,b^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {b \cos \left (d x +c \right )}}+\frac {2 A \,b^{2} \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{3 d}+\frac {6 b^{2} B \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {b \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} A b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} A b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 9 i \, \sqrt {2} B b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 9 i \, \sqrt {2} B b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, B b^{2} \cos \left (d x + c\right ) + 5 \, A b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{3} + A b^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right ), x\right ) \]