\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {7 \left (33 A -17 B \right ) \sin \left (d x +c \right )}{30 a^{3} d \sec \left (d x +c \right )^{\frac {3}{2}}}-\frac {\left (A -B \right ) \sin \left (d x +c \right )}{5 d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a +a \sec \left (d x +c \right )\right )^{3}}-\frac {\left (12 A -7 B \right ) \sin \left (d x +c \right )}{15 a d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a +a \sec \left (d x +c \right )\right )^{2}}-\frac {3 \left (21 A -11 B \right ) \sin \left (d x +c \right )}{10 d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a^{3}+a^{3} \sec \left (d x +c \right )\right )}-\frac {\left (21 A -11 B \right ) \sin \left (d x +c \right )}{2 a^{3} d \sqrt {\sec \left (d x +c \right )}}+\frac {7 \left (33 A -17 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d}-\frac {\left (21 A -11 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d} \]
command
integrate((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, {\left (\sqrt {2} {\left (-21 i \, A + 11 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-21 i \, A + 11 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-21 i \, A + 11 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-21 i \, A + 11 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, {\left (\sqrt {2} {\left (21 i \, A - 11 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (21 i \, A - 11 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (21 i \, A - 11 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (21 i \, A - 11 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-33 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-33 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-33 i \, A + 17 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-33 i \, A + 17 i \, B\right )}\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 \, {\left (\sqrt {2} {\left (33 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (33 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (33 i \, A - 17 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (33 i \, A - 17 i \, B\right )}\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 (12 \, A \cos \left (d x + c\right )^{5} - 4 \, {\left (6 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{4} - 3 \, {\left (147 \, A - 79 \, B\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (357 \, A - 188 \, B\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left (21 \, A - 11 \, B\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{6} + 3 \, a^{3} \sec \left (d x + c\right )^{5} + 3 \, a^{3} \sec \left (d x + c\right )^{4} + a^{3} \sec \left (d x + c\right )^{3}}, x\right ) \]