\[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {C \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{3 d}+\frac {B x \sqrt {b \cos \left (d x +c \right )}}{2 \sqrt {\cos \left (d x +c \right )}}+\frac {\left (3 A +2 C \right ) \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{3 d \sqrt {\cos \left (d x +c \right )}}+\frac {B \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {b \cos \left (d x +c \right )}}{2 d} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2)*(b*cos(d*x+c))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} 0 & \text {for}\: c = - d x + \frac {\pi }{2} \vee c = - d x + \frac {3 \pi }{2} \\x \sqrt {b \cos {\left (c \right )}} \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \sqrt {\cos {\left (c \right )}} & \text {for}\: d = 0 \\\frac {A \sqrt {b \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{d \sqrt {\cos {\left (c + d x \right )}}} + \frac {B x \sqrt {b \cos {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{2 \sqrt {\cos {\left (c + d x \right )}}} + \frac {B x \sqrt {b \cos {\left (c + d x \right )}} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}{2} + \frac {B \sqrt {b \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )} \sqrt {\cos {\left (c + d x \right )}}}{2 d} + \frac {2 C \sqrt {b \cos {\left (c + d x \right )}} \sin ^{3}{\left (c + d x \right )}}{3 d \sqrt {\cos {\left (c + d x \right )}}} + \frac {C \sqrt {b \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}{d} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________