\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{6 a f}+\frac {c^{2} \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{15 a f \sqrt {c -c \sin \left (f x +e \right )}}+\frac {2 c \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c -c \sin \left (f x +e \right )}}{15 a f} \]
command
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, {\left (10 \, a^{2} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 24 \, a^{2} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, a^{2} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{15 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________