\[ \int \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (A +C \right ) \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}}-\frac {C \left (\sin ^{3}\left (d x +c \right )\right ) \sqrt {b \cos \left (d x +c \right )}}{3 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________