It was during the solving of this puzzle on Wolfram Alpha that I decided to subscribe to Wolfram Alpha Pro.

Wolfram Alpha gives the answer as sqrt(109), at x = 300/763 - 3*sqrt(4441)/763 or (300 - 3sqrt(4441))/763 ~= .131163191378042.

The derivative of the function given in the puzzle was found to be f'(x) = (70 x)/(sqrt(1 - x^2) sqrt(149 - 140 sqrt(1 - x^2))) - 21/sqrt(58 - 42 x), which has a zero at the above mentioned value of x.

This is from Wolfram's text version of the step-by-step solution.