\[ \int \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx \]
Optimal antiderivative \[ \frac {\left (a -3 b \right ) \left (a +b \right ) \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{8 b^{\frac {3}{2}} f}-\frac {\cos \left (f x +e \right ) \left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{4 b f}+\frac {\left (a -3 b \right ) \cos \left (f x +e \right ) \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{8 b f} \]
command
integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (2 \, \cos \left (f x + e\right )^{2} - \frac {a b f^{4} + 5 \, b^{2} f^{4}}{b^{2} f^{4}}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (a^{2} - 2 \, a b - 3 \, b^{2}\right )} \log \left ({\left | \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} + \frac {\sqrt {-b f^{2}} \cos \left (f x + e\right )}{f} \right |}\right )}{8 \, \sqrt {-b} b {\left | f \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________