How to solve equation: $ \frac{x+1}{9x^3-4x} + \frac{3x}{27x^3 - 12x + 8} - \frac{1}{(3x-2)^2}=0 $ How to solve this equation?
$$
\frac{x+1}{9x^3-4x} + \frac{3x}{27x^3 - 12x + 8} - \frac{1}{(3x-2)^2}=0
$$
I try 
$$
\frac{81x^5 - 81x^4 - 90 x^3 + 36 x^2 + 16x -16}{x(3x-2)^2(3x+2)(27x^3 - 12x + 8)}=0
$$
And then 
$$
81x^5 - 81x^4 - 90 x^3 + 36 x^2 + 16x -16 =0  
$$
here  $( x \neq 0; x \neq \frac{2}{3}; x \neq - \frac{2}{3}; 27x^3 - 12x + 8 \neq 0 )$.
Is there some simple (and different) way to solve it?

This graph helps to show what's happening. 

It's not an answer but a companion to Claude's, in that it helps illustrate the turning points he identified and shows the fact that there's one solution. Note the scales are different, I had to stretch the Y axis. 
