Solve $x^5=\frac{133x-78}{133-78x}$ Solve $$x^5=\dfrac{133x-78}{133-78x}$$ We have $D:133-78x\ne0,x\ne\dfrac{133}{78}$. In $x\in D$ the given equation is equivalent to $$x^5(133-78x)=133x-78\\78x^6-133x^5+133x-78=0$$ As $x=0$ is not a solution, we can divide both sides by $x^3\ne0$ to get $$78x^3-133x^2+\dfrac{133}{x^2}-\dfrac{78}{x^3}=0\\78\left(x^3-\dfrac{1}{x^3}\right)-133\left(x^2-\dfrac{1}{x^2}\right)=0$$ I am not able to come up with a subtle substitution. It would have been easier if we had $x^2\color{red}{+}\dfrac{1}{x^2}.$
$x=1$ is an "obvious" solution. What else? Thank you!

Multiplying to a common denominator gives
$$
0=78x^6 - 133x^5 + 133x - 78=(13x^2 + 6x + 13)(3x - 2)(2x - 3)(x + 1)(x - 1).
$$
The factorisation arises by the Rational Root Theorem.
