Not first, Not second, its third order

Calculus Level 5

If the solution of the differential equation

\[\large{\frac{d^3 y}{dx^3}+2\frac{d^2 y}{dx^2}+\frac{dy}{dx}=e^{2x}+x^2+x}\] is

\(\large{y=c_{1}+(c_{2}+c_{3}x)e^{Ax}+\frac{B}{C}e^{Dx}+\frac{E}{F}x^{G}+Hx^{I}-\frac{J}{K}x^{L}}\)

where \(c_{1},c_{2},c_{3}\) are constants.

\(A,B,C,D,E,F,G,H,I,J,K,L\) are integers and all the fractions are co prime.

Find \(A+B+C+D+E+F+G+H+I+J+K+L\)

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