\[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {a^{\frac {5}{2}} \left (A d \left (3 c^{2}+10 c d +19 d^{2}\right )-B \left (15 c^{3}+30 c^{2} d +7 c \,d^{2}-20 d^{3}\right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right )}{4 d^{\frac {7}{2}} \left (c +d \right )^{\frac {5}{2}} f}+\frac {a \left (-A d +B c \right ) \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{2 d \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{2}}+\frac {a^{3} \left (3 A d \left (c +3 d \right )-B \left (15 c^{2}+25 c d +4 d^{2}\right )\right ) \cos \left (f x +e \right )}{4 d^{3} \left (c +d \right )^{2} f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {a^{2} \left (A d \left (c +7 d \right )-B \left (5 c^{2}+7 c d -4 d^{2}\right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{4 d^{2} \left (c +d \right )^{2} f \left (c +d \sin \left (f x +e \right )\right )} \]
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 \[ \text {Timed out} \]_______________________________________________________