Simplifying algebra fraction expression The below equation is after taking the derivative for a calculus problem, but when I  try to simplify I don't see how. My calculator shows that
$${5(\sqrt{1+x^2}-x) \over \sqrt{1+x^2}(1+(x-\sqrt{1+x^2})^2)}$$ simplifies to 
$${5 \over 2(x^2+1)}$$
and no matter how much I rearrange the equation I just don't see how anything cancels. What am I missing?

$$\sqrt{1+x^2}(1+(x-\sqrt{1+x^2})^2) \\ =\sqrt{1+x^2}+x^2\sqrt{1+x^2}+\sqrt{1+x^2}(1+x^2)-2x(\sqrt{1+x^2})^2 \\ =2\sqrt{1+x^2}+2x^2\sqrt{1+x^2}-2x(1+x^2) \\ = (2+2x^2)\sqrt{1+x^2}-2x(1+x^2) \\ = 2(1+x^2)\left[\sqrt{1+x^2}-x\right]$$
Then cancel from the numerator and denominator to get your result.
