\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx \]
Optimal antiderivative \[ -\frac {14 a^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d e}-\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )}{5 d e}+\frac {14 a^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^2*(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 i \, \sqrt {2} a^{2} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} a^{2} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, a^{2} \cos \left (d x + c\right ) e^{\frac {1}{2}} \sin \left (d x + c\right ) + 10 \, a^{2} \cos \left (d x + c\right ) e^{\frac {1}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{15 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]