\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (468 a^{3} b B +364 a \,b^{3} B +39 a^{4} \left (5 A +3 C \right )+78 a^{2} b^{2} \left (9 A +7 C \right )+7 b^{4} \left (13 A +11 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{195 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (77 a^{4} B +330 B \,a^{2} b^{2}+45 b^{4} B +44 a^{3} b \left (7 A +5 C \right )+20 a \,b^{3} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (3458 a^{3} b B +4004 a \,b^{3} B +192 a^{4} C +77 b^{4} \left (13 A +11 C \right )+11 a^{2} b^{2} \left (637 A +491 C \right )\right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6435 d}+\frac {2 b \left (2171 a^{2} b B +1053 b^{3} B +192 a^{3} C +2 a \,b^{2} \left (1573 A +1259 C \right )\right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9009 d}+\frac {2 \left (143 A \,b^{2}+221 a b B +48 a^{2} C +121 b^{2} C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{1287 d}+\frac {2 \left (13 b B +8 a C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{143 d}+\frac {2 C \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{13 d}+\frac {2 \left (77 a^{4} B +330 B \,a^{2} b^{2}+45 b^{4} B +44 a^{3} b \left (7 A +5 C \right )+20 a \,b^{3} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate((a+b*cos(d*x+c))^4*(A+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 (3465 \, C b^{4} \cos \left (d x + c\right )^{5} + 15015 \, B a^{4} + 8580 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 64350 \, B a^{2} b^{2} + 3900 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 8775 \, B b^{4} + 4095 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{4} + 385 \, {\left (78 \, C a^{2} b^{2} + 52 \, B a b^{3} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} + 585 \, {\left (44 \, C a^{3} b + 66 \, B a^{2} b^{2} + 4 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 9 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (117 \, C a^{4} + 468 \, B a^{3} b + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 \, B a b^{3} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 195 \, \sqrt {2} {\left (77 i \, B a^{4} + 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 330 i \, B a^{2} b^{2} + 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 45 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 \, \sqrt {2} {\left (-77 i \, B a^{4} - 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 330 i \, B a^{2} b^{2} - 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} - 45 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 468 i \, B a^{3} b - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 364 i \, B a b^{3} - 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\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 (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 468 i \, B a^{3} b + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 i \, B a b^{3} + 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\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 )}{45045 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]