\[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {5}{2}} \left (19 A +20 B +8 C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{4 d}-\frac {a^{3} \left (27 A -12 B -56 C \right ) \sin \left (d x +c \right )}{12 d \sqrt {a +a \cos \left (d x +c \right )}}-\frac {a^{2} \left (21 A +12 B -8 C \right ) \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{12 d}+\frac {a \left (5 A +4 B \right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \tan \left (d x +c \right )}{4 d}+\frac {A \left (a +a \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
command
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,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 \[ \text {Timed out} \]_____________________________________________________