(Non) Context Free language?
It helps to think of $L$ as having two parts.


First part: $L_1$ is all strings not in the form $a^i b^j c^k$. We can write this with the regular expression $\neg [a^* b^* c^*]$, so it's regular.


(I've been informed this notation isn't universal, so to spell it out: it's the complement of the regular language $a^* b^* c^*$, and the complement of a regular language is regular.)

Second part: $L_2$ is all strings in the form $a^i b^j c^k$ such that the given conditions on $i, j, k$ are not fulfilled.


The given conditions are that $i, j, k$ are all different. So how can those conditions be violated? Either $i=j$, or $j=k$, or $i=k$. We can call these three options $L_{2a}$, $L_{2b}$, and $L_{2c}$ if you want.

Let's take just one of them as an example; the others work the same way. $L_{2a}$ is all strings of the form $a^i b^i c^k$. Can you write a regular expression for this? How about a CFG? (The answers are no, and yes: in fact, this is one of the classic examples for what a CFG can do that a regular expression can't.)

Now, since $L_{2a}$, $L_{2b}$, and $L_{2c}$ are all context-free, their union ($L_2$) is also context-free. And since $L_1$ is regular, it's also context-free. So $L$, as the union of two context-free languages, is context-free.