\[ \int (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx \]
Optimal antiderivative \[ \frac {2 A \,b^{5} \sin \left (d x +c \right )}{7 d \left (b \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {2 b^{3} \left (5 A +7 C \right ) \sin \left (d x +c \right )}{21 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 b^{2} \left (5 A +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {b \cos \left (d x +c \right )}} \]
command
integrate((b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left ({\left (5 \, A + 7 \, C\right )} b \cos \left (d x + c\right )^{2} + 3 \, A b\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{6}, x\right ) \]