\[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 b^{7}}{11 f \left (b \sec \left (f x +e \right )\right )^{\frac {11}{2}}}-\frac {6 b^{5}}{7 f \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {2 b^{3}}{f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 b \sqrt {b \sec \left (f x +e \right )}}{f} \]
command
integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (7 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 33 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3} + 77 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right ) + \frac {77 \, b^{6}}{\sqrt {b \cos \left (f x + e\right )}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{77 \, b^{4} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{7}\,{d x} \]________________________________________________________________________________________