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