\[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 b^{7}}{9 f \left (b \sec \left (f x +e \right )\right )^{\frac {9}{2}}}-\frac {6 b^{5}}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}+\frac {6 b^{3}}{f \sqrt {b \sec \left (f x +e \right )}} \]
command
integrate((b*sec(f*x+e))^(5/2)*sin(f*x+e)^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{4} - 27 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{2} + 135 \, \sqrt {b \cos \left (f x + e\right )} b^{4} + \frac {15 \, b^{5}}{\sqrt {b \cos \left (f x + e\right )} \cos \left (f x + e\right )}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{45 \, b^{2} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \sin \left (f x + e\right )^{7}\,{d x} \]________________________________________________________________________________________