\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (9 A b +9 B a +7 b C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (7 a A +5 b B +5 a C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (9 A b +9 B a +7 b C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {2 \left (b B +a C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d}+\frac {2 b C \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9 d}+\frac {2 \left (7 a A +5 b B +5 a C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (35 \, C b \cos \left (d x + c\right )^{3} + 45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, A + 5 \, C\right )} a + 75 \, B b + 7 \, {\left (9 \, B a + {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (i \, {\left (7 \, A + 5 \, C\right )} a + 5 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 5 \, C\right )} a - 5 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-9 i \, B a - i \, {\left (9 \, A + 7 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (9 i \, B a + i \, {\left (9 \, A + 7 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right ) + {\left (B a + A b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]