\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {3}{2}} \left (176 A +150 B +133 C \right ) \arcsin \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{128 d}+\frac {C \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{5 d}+\frac {a^{2} \left (176 A +150 B +133 C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{192 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (80 A +90 B +67 C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{240 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (176 A +150 B +133 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{128 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a \left (10 B +3 C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{40 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________