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