\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx \]
Optimal antiderivative \[ -\frac {22 a^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d e}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (d x +c \right )\right )^{2}}{7 d e}-\frac {22 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{3}+a^{3} \sin \left (d x +c \right )\right )}{35 d e}+\frac {22 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^3*(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {231 i \, \sqrt {2} a^{3} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} a^{3} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 63 \, a^{3} \cos \left (d x + c\right ) e^{\frac {1}{2}} \sin \left (d x + c\right ) - 140 \, a^{3} \cos \left (d x + c\right ) e^{\frac {1}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{105 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]