\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {16 a^{2} \left (3 A +2 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{2} \left (7 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a^{2} \left (21 A +19 C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{105 d}+\frac {2 C \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +a \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{9 d}+\frac {8 C \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{63 d}+\frac {4 a^{2} \left (7 A +5 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 d} \]
command
integrate((a+a*cos(d*x+c))^2*(A+C*cos(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
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 84 i \, \sqrt {2} {\left (3 \, A + 2 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 i \, \sqrt {2} {\left (3 \, A + 2 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, C a^{2} \cos \left (d x + c\right )^{3} + 90 \, C a^{2} \cos \left (d x + c\right )^{2} + 7 \, {\left (9 \, A + 16 \, C\right )} a^{2} \cos \left (d x + c\right ) + 30 \, {\left (7 \, A + 5 \, C\right )} a^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} + {\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]