\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (27 a^{2} b B +7 b^{3} B +9 a^{3} C +21 C a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (77 a^{3} B +165 B a \,b^{2}+165 a^{2} b C +45 C \,b^{3}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (27 a^{2} b B +7 b^{3} B +9 a^{3} C +21 C a \,b^{2}\right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {2 b \left (33 a b B +26 a^{2} C +9 b^{2} C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77 d}+\frac {2 b^{2} \left (11 b B +15 a C \right ) \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99 d}+\frac {2 b C \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{11 d}+\frac {2 \left (77 a^{3} B +165 B a \,b^{2}+165 a^{2} b C +45 C \,b^{3}\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+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
\[ \frac {2 \, {\left (315 \, C b^{3} \cos \left (d x + c\right )^{4} + 1155 \, B a^{3} + 2475 \, C a^{2} b + 2475 \, B a b^{2} + 675 \, C b^{3} + 385 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 135 \, {\left (11 \, C a^{2} b + 11 \, B a b^{2} + 3 \, C b^{3}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, C a^{3} + 27 \, B a^{2} b + 21 \, C a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (77 i \, B a^{3} + 165 i \, C a^{2} b + 165 i \, B a b^{2} + 45 i \, C b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-77 i \, B a^{3} - 165 i \, C a^{2} b - 165 i \, B a b^{2} - 45 i \, C b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-9 i \, C a^{3} - 27 i \, B a^{2} b - 21 i \, C a b^{2} - 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (9 i \, C a^{3} + 27 i \, B a^{2} b + 21 i \, C a b^{2} + 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} + B a^{3} \cos \left (d x + c\right ) + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]