\[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {5}{2}} \left (c -d \right ) \left (A d \left (3 c +5 d \right )-B \left (5 c^{2}+5 c d -2 d^{2}\right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right )}{d^{\frac {7}{2}} \left (c +d \right )^{\frac {3}{2}} f}+\frac {a \left (-A d +B c \right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{d \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )}-\frac {a^{3} \left (3 A d \left (3 c +d \right )-B \left (15 c^{2}-5 c d -14 d^{2}\right )\right ) \cos \left (f x +e \right )}{3 d^{3} \left (c +d \right ) f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {a^{2} \left (-3 A d +5 B c +2 B d \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 d^{2} \left (c +d \right ) f} \]
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 \[ \text {Timed out} \]_______________________________________________________