\[ \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx \]
Optimal antiderivative \[ -\frac {a^{2} \left (9 A -B \right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{84 f}-\frac {a \left (9 A -B \right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{72 f}-\frac {B \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{9 f}-\frac {a^{4} \left (9 A -B \right ) \cos \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{315 f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {a^{3} \left (9 A -B \right ) \cos \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}} \sqrt {a +a \sin \left (f x +e \right )}}{126 f} \]
command
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________