\[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \sin \left (d x +c \right )}{7 d \,e^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\frac {4 a^{2}}{7}+\frac {4 a^{2} \sin \left (d x +c \right )}{7}}{d e \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {2 a^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{4} \sqrt {e \cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^2/(e*cos(d*x+c))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{2} \sin \left (d x + c\right ) + 2 i \, \sqrt {2} a^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{2} \sin \left (d x + c\right ) - 2 i \, \sqrt {2} a^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt {\cos \left (d x + c\right )}}{7 \, {\left (d \cos \left (d x + c\right )^{2} e^{\frac {9}{2}} + 2 \, d e^{\frac {9}{2}} \sin \left (d x + c\right ) - 2 \, d e^{\frac {9}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{5} \cos \left (d x + c\right )^{5}}, x\right ) \]