\[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a A \sin \left (d x +c \right )}{11 d \sec \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 \left (A b +B a \right ) \sin \left (d x +c \right )}{9 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (9 a A +11 b B +11 a C \right ) \sin \left (d x +c \right )}{77 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (7 A b +7 B a +9 C b \right ) \sin \left (d x +c \right )}{45 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {10 \left (9 a A +11 b B +11 a C \right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (7 A b +7 B a +9 C b \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {10 \left (9 a A +11 b B +11 a C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {75 \, \sqrt {2} {\left (i \, {\left (9 \, A + 11 \, C\right )} a + 11 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 75 \, \sqrt {2} {\left (-i \, {\left (9 \, A + 11 \, C\right )} a - 11 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, B a - i \, {\left (7 \, A + 9 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a + i \, {\left (7 \, A + 9 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, A a \cos \left (d x + c\right )^{5} + 385 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a + 11 \, B b\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, B a + {\left (7 \, A + 9 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 75 \, {\left ({\left (9 \, A + 11 \, C\right )} a + 11 \, B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b \sec \left (d x + c\right )^{3} + {\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]