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