\[ \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a \cos \left (f x +e \right )}{5 \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {8 a \cos \left (f x +e \right )}{15 \left (c +d \right )^{2} f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {16 a \cos \left (f x +e \right )}{15 \left (c +d \right )^{3} f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________