\[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {8 a^{4} \left (24 A +19 B +16 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a^{4} \left (187 A +132 B +113 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{4} \left (913 A +803 B +667 C \right ) \sin \left (d x +c \right )}{1155 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a \left (11 B +8 C \right ) \left (a +a \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{99 d \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 C \left (a +a \cos \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{11 d \cos \left (d x +c \right )^{\frac {11}{2}}}+\frac {2 \left (33 A +55 B +43 C \right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{231 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 \left (891 A +946 B +769 C \right ) \left (a^{4}+a^{4} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{3465 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {8 a^{4} \left (24 A +19 B +16 C \right ) \sin \left (d x +c \right )}{15 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 462 i \, \sqrt {2} {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 462 i \, \sqrt {2} {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (924 \, {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (517 \, A + 528 \, B + 452 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 77 \, {\left (36 \, A + 61 \, B + 64 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 44 \, B + 75 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 385 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 315 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d \cos \left (d x + c\right )^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C a^{4} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \sec \left (d x + c\right ) + A a^{4}}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]