\[ \int \frac {(e \cos (c+d x))^{5/2}}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 a d}+\frac {2 e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a d \sqrt {\cos \left (d x +c \right )}} \]
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
\[ \frac {2 \, \cos \left (d x + c\right )^{\frac {3}{2}} e^{\frac {5}{2}} + 3 i \, \sqrt {2} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3 \, a d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} e^{2} \cos \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}, x\right ) \]