\[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 e^{3}}{11 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {11}{2}}}-\frac {2 e^{3} \cos \left (d x +c \right )}{11 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {11}{2}}}-\frac {2 e^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{11 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {11}{2}}}-\frac {4 e}{7 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {16 e \cos \left (d x +c \right )}{77 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {4 \cos \left (d x +c \right )}{231 a^{2} d e \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{231 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d \,e^{2} \sqrt {e \sin \left (d x +c \right )}} \]
command
integrate(1/(a+a*sec(d*x+c))^2/(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (\sqrt {-i} {\left (\sqrt {2} \cos \left (d x + c\right )^{4} + 2 \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 \, \sqrt {2} \cos \left (d x + c\right ) - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {i} {\left (\sqrt {2} \cos \left (d x + c\right )^{4} + 2 \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 \, \sqrt {2} \cos \left (d x + c\right ) - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (2 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + 47 \, \cos \left (d x + c\right ) + 24\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{231 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} e^{\frac {5}{2}} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 2 \, a^{2} d \cos \left (d x + c\right ) e^{\frac {5}{2}} - a^{2} d e^{\frac {5}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (a^{2} e^{3} \cos \left (d x + c\right )^{2} - a^{2} e^{3} + {\left (a^{2} e^{3} \cos \left (d x + c\right )^{2} - a^{2} e^{3}\right )} \sec \left (d x + c\right )^{2} + 2 \, {\left (a^{2} e^{3} \cos \left (d x + c\right )^{2} - a^{2} e^{3}\right )} \sec \left (d x + c\right )\right )} \sin \left (d x + c\right )}, x\right ) \]