I need to obtain an analytical solution to an equation of the following form: $$ (x-a)(x-b)(x-c)=d(x-e)(x-f), $$ where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable.
Of course, the equation can be reduced to a "simple" equation for the roots of a 3rd-order polynomial, and its solution is provided by the Cardano formula (and other similar methods), but the result is too complex...
So the question arises: does any simpler method exist that makes use of the known roots of the "constituent polynomials" (left-hand and right-hand side of my equation)?
P.S. It is possible to try the Cardano method with the polynomial coefficients expressed through the roots $a$...$c$,$e$,$f$ and then to simplify the resulting gargantuan formulae using a CAS (computer algebra system) hoping that the CAS will cope with it... But I am not so sure that the CAS is powerful enough in the dark art of formulae simplification.