\[ \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}+\frac {2 a e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{5 d}+\frac {6 a \,e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 i \, \sqrt {2} a e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} a e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, a \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 7 \, a \cos \left (d x + c\right ) e^{\frac {5}{2}} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{35 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a e^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]