Efficient algorithm to translate/shift polynomials

Question

I have a polynomial $P(x)$, and given some constant $d$, I need to find the polynomial $P(x+d)$. For example, if $P(x)=x^2$ and $d=1$, then the result would be $P(x+1)=(x+1)^2=x^2+2x+1$ (with the coefficients stored in an array/vector).

I know the algorithm with $O(n^2)$ time complexity, which is based on Horner's method: $$a_0+(x+d)(a_1+(x+d)(a_2+...+(x+d)a_n))$$ However, it is not efficient enough for my needs. Is there an algorithm that is more efficient for solving this problem?

P.S. The coefficients and $d$ will always be integers, if that helps.