\[ \int (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}-\frac {2 a \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {3 i \, \sqrt {2} \sqrt {-i} a e^{\frac {1}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} \sqrt {i} a e^{\frac {1}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, b e^{\frac {1}{2}} \sin \left (d x + c\right )^{\frac {3}{2}}}{3 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]