\[ \int x^3 (a+a \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {1280 a \sqrt {a +a \sin \left (f x +e \right )}}{9 f^{4}}+\frac {16 a \,x^{2} \sqrt {a +a \sin \left (f x +e \right )}}{f^{2}}+\frac {640 a x \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{9 f^{3}}-\frac {8 a \,x^{3} \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 f}+\frac {32 a x \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{9 f^{3}}-\frac {4 a \,x^{3} \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 f}-\frac {64 a \left (\sin ^{2}\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \sqrt {a +a \sin \left (f x +e \right )}}{27 f^{4}}+\frac {8 a \,x^{2} \left (\sin ^{2}\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 f^{2}} \]
command
integrate(x^3*(a+a*sin(f*x+e))^(3/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________