\[ \int (b \sec (e+f x))^{5/2} \sin ^3(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}+\frac {2 b^{3}}{f \sqrt {b \sec \left (f x +e \right )}} \]
command
integrate((b*sec(f*x+e))^(5/2)*sin(f*x+e)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, \sqrt {b \cos \left (f x + e\right )} b + \frac {b^{2}}{\sqrt {b \cos \left (f x + e\right )} \cos \left (f x + e\right )}\right )} b \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \sin \left (f x + e\right )^{3}\,{d x} \]________________________________________________________________________________________