\[ \int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {3 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a \left (\sec ^{2}\left (d x +c \right )\right )}}\right )}{8 d}+\frac {a \left (a \left (\sec ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \tan \left (d x +c \right )}{4 d}+\frac {3 a^{2} \sqrt {a \left (\sec ^{2}\left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{8 d} \]
command
integrate((a+a*tan(d*x+c)^2)^(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________