\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \left (13 A b +9 B a \right ) \sin \left (d x +c \right )}{63 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 a \left (7 a^{2} A +22 A \,b^{2}+27 a b B \right ) \sin \left (d x +c \right )}{45 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a A \left (a +b \sec \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{9 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (15 A \,a^{2} b +7 A \,b^{3}+5 a^{3} B +21 B a \,b^{2}\right ) \sin \left (d x +c \right )}{21 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (7 a^{3} A +27 A a \,b^{2}+27 a^{2} b B +15 b^{3} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (15 A \,a^{2} b +7 A \,b^{3}+5 a^{3} B +21 B a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, \sqrt {2} {\left (5 i \, B a^{3} + 15 i \, A a^{2} b + 21 i \, B a b^{2} + 7 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{3} - 15 i \, A a^{2} b - 21 i \, B a b^{2} - 7 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-7 i \, A a^{3} - 27 i \, B a^{2} b - 27 i \, A a b^{2} - 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (7 i \, A a^{3} + 27 i \, B a^{2} b + 27 i \, A a b^{2} + 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, A a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, A a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{3} + 15 \, A a^{2} b + 21 \, B a b^{2} + 7 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]