\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {2 a \left (7 A +5 B \right ) \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{21 d}+\frac {2 a \left (A +B \right ) \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}+\frac {2 a B \left (\sec ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d}+\frac {6 a \left (A +B \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 d}-\frac {6 a \left (A +B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (7 A +5 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate(sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))*(A+B*sec(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} {\left (7 \, A + 5 \, B\right )} a \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (7 \, A + 5 \, B\right )} a \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 63 i \, \sqrt {2} {\left (A + B\right )} a \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 i \, \sqrt {2} {\left (A + B\right )} a \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {2 \, {\left (63 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 5 \, B\right )} a \cos \left (d x + c\right )^{2} + 21 \, {\left (A + B\right )} a \cos \left (d x + c\right ) + 15 \, B a\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B a \sec \left (d x + c\right )^{4} + {\left (A + B\right )} a \sec \left (d x + c\right )^{3} + A a \sec \left (d x + c\right )^{2}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]