\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 a^{5} \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{3 d \,e^{5} \left (a -a \sin \left (d x +c \right )\right )^{2}}-\frac {10 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{2} \sqrt {e \cos \left (d x +c \right )}}+\frac {10 a^{3} \sqrt {e \cos \left (d x +c \right )}}{3 d \,e^{3}} \]
command
integrate((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (-i \, \sqrt {2} a^{3} \sin \left (d x + c\right ) + i \, \sqrt {2} a^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (i \, \sqrt {2} a^{3} \sin \left (d x + c\right ) - i \, \sqrt {2} a^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (3 \, a^{3} \sin \left (d x + c\right ) - 7 \, a^{3}\right )} \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (d e^{\frac {5}{2}} \sin \left (d x + c\right ) - d e^{\frac {5}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]