Not The Brute Force Approach

first off.. i loved this problem, thanks to the creator for giving me such great experience. start like this $\begin{aligned} \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}+4= x^2-11x\\ \frac{3}{x-3}+1+\frac{5}{x-5}+1+\frac{17}{x-17}+1+\frac{19}{x-19}+1= x^2-11x\\ \frac{x}{x-3}+\frac{x}{x-5}+\frac{x}{x-17}+\frac{x}{x-19}=x(x-11)\end{aligned}$ we get x=0 as a solution. now divide by x and put x=y+11 $\begin{aligned} \frac{1}{y+8}+\frac{1}{y-8}+\frac{1}{y+6}+\frac{1}{y-6}=y\\ \frac{2y}{y^2-64}+\frac{2y}{y^2-36}=y \end{aligned}$ y=0 ->x=11 is another solution. we divide again to get $\begin{aligned} \frac{2}{y^2-64}+\frac{2}{y^2-36}=1\\ 2(y^2-36)+2(y^2-64)=(y^2-64)(y^2-36)\\ 4y^2-200=y^4-100y^2+2304\\ y^4-104y^2+2504=0\\ y^2=52\pm \sqrt{200}\\ y=\pm\sqrt{52\pm \sqrt{200}}\\ x=11\pm\sqrt{52\pm \sqrt{200}}\end{aligned}$ we get that $x=0,11,11\pm\sqrt{52\pm \sqrt{200}}$ the required $x_{largest}=11+\sqrt{52+ \sqrt{200}}$ $11+52+200=\boxed{263}$

General method should expand $blindly$ but not too difficult with Excel. { $A Brute Force Approach!$ }

$x^5 - 55 x^4 + 1106 x^3 - 9878 x^2 + 37957 x - 50171 = 0$ {An x = 0 was eliminated.}

Figures of Excel correlated with other 18 S.F. solver:

$\alpha$ = 11.00000000000000

$\beta$ = 19.13278154285057

$\gamma$ = 17.15287448078287

$\delta$ = 4.847125519217132

$\epsilon$ = 2.867218457149427

All checked to be correct to very close's proximity.

Numerical method can achieve a merit easily for 19.13278154285057 $\rightarrow$ $11$ + $\sqrt{52 + \sqrt{200}}$ .

11 + 52 + 200 = 263

Answer: $\boxed{263}$

Very easy and fast one gets the solution via Xmaxima:

solve(x^2-11*x-4-3/(x-3)-5/(x-5)-17/(x-17)-19/(x-19),x);

                          7/2                     7/2
            sqrt(208 - 5 2   )      sqrt(208 - 5 2   )
  [x = 11 - ------------------, x = ------------------ + 11, 
                    2                       2

                    7/2                     7/2
            sqrt(5 2    + 208)      sqrt(5 2    + 208)
   x = 11 - ------------------, x = ------------------ + 11, x = 11, x = 0]
                    2                       2

As third last value of the solutions' list you find the largest real solution (x=19.13278...) and after some equivalent transformation you have $x = 11 + \sqrt{ 52 + \sqrt{ 200 }}$ .