\[ \int (b \sec (e+f x))^{5/2} \sin ^4(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}} \left (\sin ^{3}\left (f x +e \right )\right )}{3 f}+\frac {4 b^{3} \sin \left (f x +e \right )}{3 f \sqrt {b \sec \left (f x +e \right )}}-\frac {8 b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f} \]
command
integrate((b*sec(f*x+e))^(5/2)*sin(f*x+e)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (-2 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - {\left (b^{2} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{3 \, f \cos \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \, b^{2} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {b \sec \left (f x + e\right )} \sec \left (f x + e\right )^{2}, x\right ) \]