\[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (a -5 b \right ) \left (a +b \right )^{2} \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{16 b^{\frac {3}{2}} f}+\frac {\left (a -5 b \right ) \cos \left (f x +e \right ) \left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{24 b f}-\frac {\cos \left (f x +e \right ) \left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{6 b f}+\frac {\left (a -5 b \right ) \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{16 b f} \]
command
integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (\frac {2 \, {\left (4 \, b f^{2} \cos \left (f x + e\right )^{2} - \frac {7 \, a b^{4} f^{10} + 13 \, b^{5} f^{10}}{b^{4} f^{8}}\right )} \cos \left (f x + e\right )^{2}}{f^{2}} + \frac {3 \, {\left (a^{2} b^{3} f^{8} + 12 \, a b^{4} f^{8} + 11 \, b^{5} f^{8}\right )}}{b^{4} f^{8}}\right )} \cos \left (f x + e\right )}{48 \, f} + \frac {{\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} \log \left ({\left | \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} + \frac {\sqrt {-b f^{2}} \cos \left (f x + e\right )}{f} \right |}\right )}{16 \, \sqrt {-b} b {\left | f \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________