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