Solve $x^5=\frac{133x-78}{133-78x}$
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.
Another obvious solution is $x=-1$. So we can write the equation as, $$(x^2-1)(78x^4+ax^3+bx^2+cx+78)=0$$ and comparing coefficients, we can easily find $a=c=-133$, $b=78$.
Now divide this (second polynomial) by $x^2$ and substitute $t=x+\frac1x$ to get a quadratic in $t$. It's easy to solve then.