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