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