\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (9 a^{2} A +7 A \,b^{2}+14 a b B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {10 \left (9 b^{2} B +11 a \left (2 A b +B a \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (9 a^{2} A +7 A \,b^{2}+14 a b B \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {2 \left (9 b^{2} B +11 a \left (2 A b +B a \right )\right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77 d}+\frac {2 b \left (11 A b +13 B a \right ) \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99 d}+\frac {2 b B \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{11 d}+\frac {10 \left (9 b^{2} B +11 a \left (2 A b +B a \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate(cos(d*x+c)^(5/2)*(a+b*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
\[ \frac {2 \, {\left (315 \, B b^{2} \cos \left (d x + c\right )^{4} + 385 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} + 825 \, B a^{2} + 1650 \, A a b + 675 \, B b^{2} + 45 \, {\left (11 \, B a^{2} + 22 \, A a b + 9 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, A a^{2} + 14 \, B a b + 7 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 75 \, \sqrt {2} {\left (11 i \, B a^{2} + 22 i \, A a b + 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 75 \, \sqrt {2} {\left (-11 i \, B a^{2} - 22 i \, A a b - 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-9 i \, A a^{2} - 14 i \, B a b - 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (9 i \, A a^{2} + 14 i \, B a b + 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{5} + A a^{2} \cos \left (d x + c\right )^{2} + {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]