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