\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (11 A d \left (c^{2}-10 c d +73 d^{2}\right )-B \left (5 c^{3}-40 c^{2} d +169 c \,d^{2}-710 d^{3}\right )\right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{1155 d f}-\frac {2 a B \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{3}}{11 d f}-\frac {2 a^{3} \left (15 c^{2}+10 c d +7 d^{2}\right ) \left (11 A d \left (c^{2}-10 c d +73 d^{2}\right )-B \left (5 c^{3}-40 c^{2} d +169 c \,d^{2}-710 d^{3}\right )\right ) \cos \left (f x +e \right )}{3465 d^{3} f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a^{3} \left (11 A \left (3 c -19 d \right ) d -B \left (15 c^{2}-65 c d +194 d^{2}\right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{693 d^{3} f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {4 a^{2} \left (5 c -d \right ) \left (11 A d \left (c^{2}-10 c d +73 d^{2}\right )-B \left (5 c^{3}-40 c^{2} d +169 c \,d^{2}-710 d^{3}\right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{3465 d^{2} f}+\frac {2 a^{2} \left (-11 A d +5 B c -14 B d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3} \sqrt {a +a \sin \left (f x +e \right )}}{99 d^{2} f} \]
command
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________