Apply roster method

$\large{1^3}$ = $\large{1}$

$\large{2^3}$ = $\large{8}$

$\large{3^3}$ = $\large{27}$

$\large{4^3}$ = $\large{64}$

$\large{5^3}$ = $\large{125}$

$\large{6^3}$ = $\large{216}$

$\large{7^3}$ = $\large{343}$

The near $\large{X^3}$ is $\large{6^3}$ = $\large{216}$ .

Thus $\large{1+2+3+4+5+6}$ = $\large{21}$

Thank's from shakib nazmus because post this interest problem.

\color \red {x \in Z_{+} }

$x^3 \leq 216$

$x \leq \sqrt [3]{216}$

\color \red {x \leq 6}

$1+2+3+4+5+6 = 21$

So, as always, $\boxed {21}$ .