\[ \int \left (b \tan ^4(e+f x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {b^{2} \cot \left (f x +e \right ) \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}}{f}-b^{2} x \left (\cot ^{2}\left (f x +e \right )\right ) \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}-\frac {b^{2} \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )}{3 f}+\frac {b^{2} \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}\, \left (\tan ^{3}\left (f x +e \right )\right )}{5 f}-\frac {b^{2} \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}\, \left (\tan ^{5}\left (f x +e \right )\right )}{7 f}+\frac {b^{2} \sqrt {b \left (\tan ^{4}\left (f x +e \right )\right )}\, \left (\tan ^{7}\left (f x +e \right )\right )}{9 f} \]
command
integrate((b*tan(f*x+e)^4)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________