Proving that L is not regular using closure properties
Let $ L_1 = \{\ ab^jc^j\ |\ j \geq 0\ \} $ and $ L_2 = \{\ a^ib^jc^k\ |\ i, j, k \geq 0 \ and\ i \neq 1\ \} $. Then 

$$ 
L = L_1 \cup L_2.
$$

If we take 
$$ \begin{align}
L\ \cap\ ab^*c^* &amp;= (L_1\cup L_2) \cap ab^*c^* \\
 &amp;= (L_1\cap ab^*c^*)\ \cup\ (L_2 \cap ab^*c^*)\\
\end{align}
$$

Since $L_1\cap ab^*c^* = L_1$ and  $L_2 \cap ab^*c^* = \emptyset$, we have

$$
L\ \cap ab^*c^* = L_1
$$

Now, $ab^*c^*$ is regular, and $L_1$ is not regular (can be shown using the Pumping Lemma). If $L$ was regular, by the closure properties, $L_1$ would be to, which is a contradiction. Hence, $L$ is not regular.