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Maxima can not compute the limit but expanding into series and integrating term by term helps:

sage: maxima('powerseries(x^2*exp(-x)/(exp(-x)+1),exp(-x),0)')
x^2*%e^-x*'sum((-1)^i1*%e^-(i1*x),i1,0,inf)
sage: var('k x');
sage: assume(k+1>0);
sage: sum(integrate(x^2*exp(-x)*(-1)^k*exp(-(k*x)),x,0,oo),k,0,oo)
3/2*zeta(3)