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