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