\[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {4 a^{3} \left (5 A +9 B +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (35 A +21 B +13 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 C \left (a +a \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{7 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (7 B +6 C \right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{35 a d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (5 A +9 B +7 C \right ) \left (a^{3}+a^{3} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{15 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a^{3} \left (140 A +147 B +106 C \right ) \sin \left (d x +c \right )}{105 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (21 \, {\left (15 \, A + 18 \, B + 14 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 21 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 21 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 15 \, C a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{105 \, d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C a^{3} \sec \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]