\[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} A \sin \left (d x +c \right )}{7 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 a \left (2 A b +B a \right ) \sin \left (d x +c \right )}{5 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (5 a^{2} A +7 A \,b^{2}+14 a b B \right ) \sin \left (d x +c \right )}{21 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (6 A a b +3 B \,a^{2}+5 b^{2} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (5 a^{2} A +7 A \,b^{2}+14 a b B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (5 i \, A a^{2} + 14 i \, B a b + 7 i \, A b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, A a^{2} - 14 i \, B a b - 7 i \, A b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{2} - 6 i \, A a b - 5 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{2} + 6 i \, A a b + 5 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, A a^{2} \cos \left (d x + c\right )^{3} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (5 \, A a^{2} + 14 \, B a b + 7 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B b^{2} \sec \left (d x + c\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {7}{2}}}, x\right ) \]