\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {4 a^{2} \left (9 A +8 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{2} \left (6 A +5 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{2} \left (9 A +8 B \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {2 a^{2} \left (9 A +11 B \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{63 d}+\frac {2 B \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{9 d}+\frac {4 a^{2} \left (6 A +5 B \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (6 \, A + 5 \, B\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (6 \, A + 5 \, B\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (9 \, A + 8 \, B\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (9 \, A + 8 \, B\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, B a^{2} \cos \left (d x + c\right )^{3} + 45 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 14 \, {\left (9 \, A + 8 \, B\right )} a^{2} \cos \left (d x + c\right ) + 30 \, {\left (6 \, A + 5 \, B\right )} a^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B a^{2} \cos \left (d x + c\right )^{4} + {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} + A a^{2} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]