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