\[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b}{7 a f \left (a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {b \sec \left (f x +e \right )}}-\frac {4 b}{7 a^{3} f \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {b \sec \left (f x +e \right )}}-\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (2 f x +2 e \right )}{2}}\, \EllipticF \left (\cos \left (e +\frac {\pi }{4}+f x \right ), \sqrt {2}\right ) \sqrt {b \sec \left (f x +e \right )}\, \left (\sqrt {\sin }\left (2 f x +2 e \right )\right )}{7 \sin \left (e +\frac {\pi }{4}+f x \right ) a^{4} f \sqrt {a \sin \left (f x +e \right )}} \]
command
integrate((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ), -1\right ) + 2 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ), -1\right ) - {\left (2 \, \cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{7 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} - 2 \, a^{5} f \cos \left (f x + e\right )^{2} + a^{5} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right )}}{{\left (a^{5} \cos \left (f x + e\right )^{4} - 2 \, a^{5} \cos \left (f x + e\right )^{2} + a^{5}\right )} \sin \left (f x + e\right )}, x\right ) \]