\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {4 a b \left (9 A +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a b \left (9 A +7 C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {2 \left (4 a^{2} C +b^{2} \left (11 A +9 C \right )\right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77 d}+\frac {8 a b C \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99 d}+\frac {2 C \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{11 d}+\frac {2 \left (11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^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
\[ \frac {462 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 462 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (315 \, C b^{2} \cos \left (d x + c\right )^{4} + 770 \, C a b \cos \left (d x + c\right )^{3} + 154 \, {\left (9 \, A + 7 \, C\right )} a b \cos \left (d x + c\right ) + 165 \, {\left (7 \, A + 5 \, C\right )} a^{2} + 75 \, {\left (11 \, A + 9 \, C\right )} b^{2} + 45 \, {\left (11 \, C a^{2} + {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + 2 \, C a b \cos \left (d x + c\right )^{4} + 2 \, A a b \cos \left (d x + c\right )^{2} + A a^{2} \cos \left (d x + c\right ) + {\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]