\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {64 a \,c^{4} \left (\cos ^{3}\left (f x +e \right )\right )}{105 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {16 a \,c^{3} \left (\cos ^{3}\left (f x +e \right )\right )}{35 f \sqrt {c -c \sin \left (f x +e \right )}}+\frac {2 a \,c^{2} \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {c -c \sin \left (f x +e \right )}}{7 f} \]
command
integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (525 \, a c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 35 \, a c^{2} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 63 \, a c^{2} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, a c^{2} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{420 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________