\[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \, dx \]
Optimal antiderivative \[ \frac {10 \sin \left (d x +c \right )}{21 a d e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{7 d e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (d x +c \right )\right )}+\frac {10 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a d \,e^{2} \sqrt {e \cos \left (d x +c \right )}} \]
command
integrate(1/(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
\[ -\frac {5 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - i \, \sqrt {2} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )}}{21 \, {\left (a d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a e^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]