\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^5(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right )}{f}+\frac {\left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{3 f}-\frac {\left (a +2 b \right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{5 b^{2} f}+\frac {\left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {7}{2}}}{7 b^{2} f}+\frac {a \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{f} \]
command
integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^5,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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________