\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {4 a^{3} \left (17 A +15 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (121 A +105 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (17 A +15 B \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {20 a^{3} \left (22 A +21 B \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{693 d}+\frac {2 a B \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a +a \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{11 d}+\frac {2 \left (11 A +15 B \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a^{3}+a^{3} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{99 d}+\frac {4 a^{3} \left (121 A +105 B \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (121 \, A + 105 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (121 \, A + 105 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (17 \, A + 15 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (17 \, A + 15 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, B a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 135 \, {\left (11 \, A + 14 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 154 \, {\left (17 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right ) + 30 \, {\left (121 \, A + 105 \, B\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B a^{3} \cos \left (d x + c\right )^{5} + {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 3 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + A a^{3} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]