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