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