\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {4 d \left (c +d \right ) \left (-9 A d +B c -8 B d \right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{105 a f}+\frac {4 a \left (c +d \right ) \left (-9 A d +B c -8 B d \right ) \left (15 c^{2}+10 c d +7 d^{2}\right ) \cos \left (f x +e \right )}{315 d f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a \left (-9 A d +B c -8 B d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{63 d f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a B \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4}}{9 d f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {8 \left (5 c -d \right ) \left (c +d \right ) \left (-9 A d +B c -8 B d \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{315 f} \]
command
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3*(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________