\[ \int x^3 \sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {x^{3} \sec \left (f x +e \right ) \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a -a \sin \left (f x +e \right )}}{2 c f}-\frac {6 c \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{f^{4}}+\frac {3 c \,x^{2} \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{f^{2}}+\frac {3 c x \sec \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{8 f^{3}}-\frac {3 c \,x^{3} \sec \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{4 f}-\frac {3 c \sin \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{8 f^{4}}+\frac {3 c \,x^{2} \sin \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{4 f^{2}}-\frac {6 c x \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}\, \tan \left (f x +e \right )}{f^{3}}-\frac {3 c x \sin \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}\, \tan \left (f x +e \right )}{4 f^{3}} \]
command
integrate(x^3*(c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________