\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx \]
Optimal antiderivative \[ -\frac {4 a \left (c +d \right ) \left (13 A d \left (3 c^{2}-38 c d +355 d^{2}\right )-B \left (15 c^{3}-150 c^{2} d +799 c \,d^{2}-4184 d^{3}\right )\right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{15015 d f}-\frac {2 a B \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{4}}{13 d f}-\frac {4 a^{3} \left (c +d \right ) \left (15 c^{2}+10 c d +7 d^{2}\right ) \left (13 A d \left (3 c^{2}-38 c d +355 d^{2}\right )-B \left (15 c^{3}-150 c^{2} d +799 c \,d^{2}-4184 d^{3}\right )\right ) \cos \left (f x +e \right )}{45045 d^{3} f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a^{3} \left (13 A d \left (3 c^{2}-38 c d +355 d^{2}\right )-B \left (15 c^{3}-150 c^{2} d +799 c \,d^{2}-4184 d^{3}\right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{9009 d^{3} f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a^{3} \left (-39 A c d +299 A \,d^{2}+15 B \,c^{2}-75 B c d +280 B \,d^{2}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4}}{1287 d^{3} f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {8 a^{2} \left (5 c -d \right ) \left (c +d \right ) \left (13 A d \left (3 c^{2}-38 c d +355 d^{2}\right )-B \left (15 c^{3}-150 c^{2} d +799 c \,d^{2}-4184 d^{3}\right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{45045 d^{2} f}+\frac {2 a^{2} \left (-13 A d +5 B c -16 B d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4} \sqrt {a +a \sin \left (f x +e \right )}}{143 d^{2} f} \]
command
integrate((a+a*sin(f*x+e))^(5/2)*(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________