\[ \int (a \sin (e+f x))^{5/2} (b \tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {24 a^{2} b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \sqrt {a \sin \left (f x +e \right )}}{5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {\cos \left (f x +e \right )}\, \sqrt {b \tan \left (f x +e \right )}}-\frac {2 b \left (a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {b \tan \left (f x +e \right )}}{5 f}+\frac {12 a^{2} b \sqrt {a \sin \left (f x +e \right )}\, \sqrt {b \tan \left (f x +e \right )}}{5 f} \]
command
integrate((a*sin(f*x+e))^(5/2)*(b*tan(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-a b} a^{2} b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 6 \, \sqrt {2} \sqrt {-a b} a^{2} b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (a^{2} b \cos \left (f x + e\right )^{2} + 5 \, a^{2} b\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}\right )}}{5 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (a^{2} b \cos \left (f x + e\right )^{2} - a^{2} b\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} \tan \left (f x + e\right ), x\right ) \]