\[ \int \frac {1}{(c \sec (a+b x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sin \left (b x +a \right )}{5 b c \left (c \sec \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {6 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,c^{2} \sqrt {\cos \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}} \]
command
integrate(1/(c*sec(b*x+a))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, \sqrt {\frac {c}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) + 3 i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 3 i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c \sec \left (b x + a\right )}}{c^{3} \sec \left (b x + a\right )^{3}}, x\right ) \]