\[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (11 A +9 C \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{77 b^{3} d}+\frac {2 C \left (b \cos \left (d x +c \right )\right )^{\frac {9}{2}} \sin \left (d x +c \right )}{11 b^{5} d}+\frac {10 \left (11 A +9 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {b \cos \left (d x +c \right )}}+\frac {10 \left (11 A +9 C \right ) \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{231 b d} \]
command
integrate(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(b*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (11 i \, A + 9 i \, C\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-11 i \, A - 9 i \, C\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (21 \, C \cos \left (d x + c\right )^{4} + 3 \, {\left (11 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 55 \, A + 45 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{231 \, b d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right )}}{b}, x\right ) \]