\[ \int \frac {\sqrt {\cos (c+d x)} (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 B \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate(cos(d*x+c)^(1/2)*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {i \, \sqrt {2} B {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - i \, \sqrt {2} B {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (B \sqrt {\cos \left (d x + c\right )}, x\right ) \]