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