I've heard from Zhi-Wei Sun that he recently considered this question. In a post a few days ago to OEIS Least integer b>2n+1 such that the numbers written as [1,3,...,2n-1,2n+1] and [2n+1,2n-1,...,3,1] in base b are both prime. He gives the first of what he conjectures are infinitely many bases (for each fixed $n$) with the named property. Other fairly specific conjectures concerning Galois groups, reducibility over $\mathbb{Z}_p$ and the like can be found on that page. Any one of the conjectures would imply that $1+3x+5x^2+\cdots+(2n+1)x^{n}$ is always irreducible over the integers. A similar post a few days earlier than that concerns $1+2x+\cdots+(n+1)x^{n}$ which he would also conjecture is always irreducible over the integers.

Of course there are integer arithmetic progressions such that $f(a,b,n)=\sum_0^n(a+bk)x^k$ does factor (with $a \ne 0$ of course). At least there is $1+x+x^2+\cdots+x^n$ which is irreducible when and only when $n+1$ is prime. A fairly simple minded search over small parameters turns up

- $-n+(2-n)x+(4-n)x^2+\cdots+nx^n$ which has $(x-1)$ as a factor (and $(x+1)$ for even $n$) but no other factors up to $n=42$
- After some manipulation, the integer quadratic examples can written $s(t-2s)+(t^2-s^2)x+t(2t-s)x^2$ with linear factor $(s+tx)$
- One can first work over $\mathbb{Q}$ , stipulate a factor $x-r=x-t/s$ , set $c_{n-1}=1$ and then solve for $c_0,\cdots,c_{n-2}$ such that $(x-t/s)(c_0+c_1x+\cdots+c_{n-1}x^{n-1})=f(a,b,n)$ for $a=c_0-rc_1$ and $b=r+1-c_{n-2}$. The solutions will have $c_i$ rational functions in $r$ with denominator $n+(n-1)r+\cdots+r^{n-1}$. Then one can scale to integer examples.
- Perhaps there are nice solutions which are reducible but without a linear factor.

**later** Here is the solution for degree $5$ from which the pattern becomes clear. Thanks to Joro and Peter for seeing what I did not. The coefficients below are in arithmetic progression with difference $b=-(s^5+s^4t+s^3t^2+s^2t^3+st^4+t^5).$ It is not immediate, but also is not too hard to check that $(s-tx)$ is a factor as $x=\frac{s}{t}$ is a root.

$$\left(5{s}^{5}+4{s}^{4}t+3{s}^{3}{t}^{2}+2{s}^{2}{t}^{3}+s{t}^{4}+0t^5\right)+\left( 4{s}^{5}+3{s}^{4}t+2{s}^{3}{t}^{2}+{s}^{2}{t}^{3}+0st^4-{t}^{5} \right) x$$ $$\ \ +\left( 3{s}^{5}+2{s}^{4}t+{s}^{3}{t}^{2}+0s^2t^3-s{t}^{4}-2{t}^{5}\right){x}^{2} +\left( 2\{s}^{5}+{s}^{4}t+0s^3t^2-{s}^{2}{t}^{3}-2s{t}^{4}-3{t}^{5}
\right) {x}^{3}$$ $$\ \ \ \ +\left( {s}^{5}+0s^4t-{s}^{3}{t}^{2}-2{s}^{2}{t}^{3}-3s{t}^{4}-4{t}^{5
} \right) {x}^{4}+\left(0s^5-{s}^{4}t-2{s}^{3}{t}^{2}-3{s}^{2}{t}^{3}-4s{t}^{4}-5{
t}^{5} \right) {x}^{5} $$