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The elliptic curve $E_1: y^2=x^3-x/25+9/125$ is isomorphic to the one $E_2: Y^2=X^3-25X+9\cdot 5^3,$ here we have $X=25x.$ Integral points on $E_1$ are integral points on $E_2.$ The latter can be computed via Sage.

E=EllipticCurve([-25,9*5^3])
E.integral_points()
[(4 : 33 : 1)]

Hence the only candidates points are $(4/25 : \pm 33/125 : 1)$ on your curve. Therefore there are no integral points.

The elliptic curve $E_1: y^2=x^3-x/25+9/125$ is isomorphic to the one $E_2: Y^2=X^3-25X+9\cdot 5^3,$ here we have $X=25x.$ Integral points on $E_1$ are integral points on $E_2.$ The latter can be computed via Sage.

E=EllipticCurve([-25,9*5^3])
E.integral_points()
[(4 : 33 : 1)]

Hence the only candidates points are $(4/25 : \pm 33/125 : 1)$ on your curve. Therefore there are no integral points.