\[ \int (a+a \cos (c+d x))^{5/2} \sec ^2(c+d x) \, dx \]
Optimal antiderivative \[ \frac {5 a^{\frac {5}{2}} \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{d}+\frac {a^{3} \sin \left (d x +c \right )}{d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \sqrt {a +a \cos \left (d x +c \right )}\, \tan \left (d x +c \right )}{d} \]
command
integrate((a+a*cos(d*x+c))^(5/2)*sec(d*x+c)^2,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________