\[ \int \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right ) \left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{4 f}-\frac {3 \left (a +b \right )^{2} \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{8 f \sqrt {b}}-\frac {3 \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{8 f} \]
command
integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (2 \, b \cos \left (f x + e\right )^{2} - \frac {5 \, {\left (a b^{2} f^{4} + b^{3} f^{4}\right )}}{b^{2} f^{4}}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} + \frac {\sqrt {-b f^{2}} \cos \left (f x + e\right )}{f} \right |}\right )}{8 \, \sqrt {-b} {\left | f \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________