\[ \int (a \sin (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f \sqrt {b \tan \left (f x +e \right )}}+\frac {4 a^{2} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {b \tan \left (f x +e \right )}}{3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {a \sin \left (f x +e \right )}} \]
command
integrate((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
\[ -\frac {2 \, {\left (\sqrt {a \sin \left (f x + e\right )} a \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) - \sqrt {2} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - \sqrt {2} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{3 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} a \sin \left (f x + e\right ), x\right ) \]