\[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx \]
Optimal antiderivative \[ \left (a^{2} \left (A \,c^{3}-3 A c \,d^{2}-3 B \,c^{2} d +B \,d^{3}-c^{3} C +3 c C \,d^{2}\right )+b^{2} \left (c^{3} C +3 B \,c^{2} d -3 c C \,d^{2}-B \,d^{3}-A \left (c^{3}-3 c \,d^{2}\right )\right )-2 a b \left (\left (A -C \right ) d \left (3 c^{2}-d^{2}\right )+B \left (c^{3}-3 c \,d^{2}\right )\right )\right ) x +\frac {\left (2 a b \left (c^{3} C +3 B \,c^{2} d -3 c C \,d^{2}-B \,d^{3}-A \left (c^{3}-3 c \,d^{2}\right )\right )-a^{2} \left (\left (A -C \right ) d \left (3 c^{2}-d^{2}\right )+B \left (c^{3}-3 c \,d^{2}\right )\right )+b^{2} \left (\left (A -C \right ) d \left (3 c^{2}-d^{2}\right )+B \left (c^{3}-3 c \,d^{2}\right )\right )\right ) \ln \left (\cos \left (f x +e \right )\right )}{f}-\frac {d \left (2 a b \left (c^{2} C +2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-a^{2} \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )+b^{2} \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )\right ) \tan \left (f x +e \right )}{f}+\frac {\left (2 a b \left (A c -B d -c C \right )+a^{2} \left (B c +\left (A -C \right ) d \right )-b^{2} \left (B c +\left (A -C \right ) d \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}{2 f}+\frac {\left (a^{2} B -b^{2} B +2 a b \left (A -C \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{3}}{3 f}+\frac {\left (5 a^{2} C \,d^{2}-6 a b d \left (-5 B d +c C \right )+b^{2} \left (c^{2} C -3 B c d +15 \left (A -C \right ) d^{2}\right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{4}}{60 d^{3} f}-\frac {b \left (-3 b B d -a C d +b c C \right ) \tan \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{4}}{15 d^{2} f}+\frac {C \left (a +b \tan \left (f x +e \right )\right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{4}}{6 d f} \]
command
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^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 \[ \text {Timed out} \]_______________________________________________________