\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx \]
Optimal antiderivative \[ -\frac {22 a^{4} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{9 d e}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (d x +c \right )\right )^{3}}{9 d e}-\frac {10 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{2}}{21 d e}-\frac {22 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a^{4}+a^{4} \sin \left (d x +c \right )\right )}{21 d e}+\frac {22 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {231 i \, \sqrt {2} a^{4} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} a^{4} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (36 \, a^{4} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 168 \, a^{4} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 7 \, {\left (a^{4} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 13 \, a^{4} \cos \left (d x + c\right ) e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{63 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]