\[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx \]
Optimal antiderivative \[ \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{14 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {15}{2}}}+\frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{56 c f \left (c -c \sin \left (f x +e \right )\right )^{\frac {13}{2}}}+\frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{280 c^{2} f \left (c -c \sin \left (f x +e \right )\right )^{\frac {11}{2}}}+\frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{2240 c^{3} f \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}} \]
command
integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (35 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 84 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 20 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{2240 \, c^{8} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________