\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{2 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (c -d \right ) \left (A c +7 A d +3 B c -11 B d \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} f}+\frac {d \left (3 A c -9 A d -15 B c +13 B d \right ) \cos \left (f x +e \right )}{3 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {\left (3 A -7 B \right ) d^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{6 a^{2} f} \]
command
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, \sqrt {2} {\left (A \sqrt {a} c^{2} + 3 \, B \sqrt {a} c^{2} + 6 \, A \sqrt {a} c d - 14 \, B \sqrt {a} c d - 7 \, A \sqrt {a} d^{2} + 11 \, B \sqrt {a} d^{2}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (A \sqrt {a} c^{2} + 3 \, B \sqrt {a} c^{2} + 6 \, A \sqrt {a} c d - 14 \, B \sqrt {a} c d - 7 \, A \sqrt {a} d^{2} + 11 \, B \sqrt {a} d^{2}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (A \sqrt {a} c^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} c^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, A \sqrt {a} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B \sqrt {a} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + A \sqrt {a} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, B a^{\frac {9}{2}} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________