Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.

Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in exactly $4$ different ways.
For example, $5229225$ is a palindromic number and it can be expressed in exactly $4$ different ways:

${2285}^2+{20}^3$
${2223}^2+{66}^3$
${1810}^2+{125}^3$
${1197}^2+{156}^3$

Find the sum of the five smallest such palindromic numbers.