\[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {b^{2} \cot \left (d x +c \right ) \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}}{d}-b^{2} x \left (\cot ^{2}\left (d x +c \right )\right ) \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}-\frac {b^{2} \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}\, \tan \left (d x +c \right )}{3 d}+\frac {b^{2} \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}\, \left (\tan ^{3}\left (d x +c \right )\right )}{5 d}-\frac {b^{2} \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}\, \left (\tan ^{5}\left (d x +c \right )\right )}{7 d}+\frac {b^{2} \sqrt {\left (\tan ^{4}\left (d x +c \right )\right ) b}\, \left (\tan ^{7}\left (d x +c \right )\right )}{9 d} \]
command
integrate((tan(d*x+c)^4*b)^(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 \[ \text {Timed out} \]_______________________________________________________