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