\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {4 \left (c +d \right ) \left (11 A \left (c -17 d \right ) d -3 B \left (c^{2}-9 c d +56 d^{2}\right )\right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{1155 f}+\frac {4 a^{2} \left (c +d \right ) \left (15 c^{2}+10 c d +7 d^{2}\right ) \left (11 A \left (c -17 d \right ) d -3 B \left (c^{2}-9 c d +56 d^{2}\right )\right ) \cos \left (f x +e \right )}{3465 d^{2} f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a^{2} \left (11 A \left (c -17 d \right ) d -3 B \left (c^{2}-9 c d +56 d^{2}\right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{693 d^{2} f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a^{2} \left (3 B \left (c -4 d \right )-11 A d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4}}{99 d^{2} f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {8 a \left (5 c -d \right ) \left (c +d \right ) \left (11 A \left (c -17 d \right ) d -3 B \left (c^{2}-9 c d +56 d^{2}\right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{3465 d f}-\frac {2 a B \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4} \sqrt {a +a \sin \left (f x +e \right )}}{11 d f} \]
command
integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________