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