Answer
$\displaystyle \frac{2x^{3}+2x+x^{3}}{(x^{2}+1)^{1/2}}$
Work Step by Step
$2x(x^{2}+1)^{1/2}+x^{2}\displaystyle \cdot\frac{1}{2}(x^{2}+1)^{-1/2}\cdot 2x$ =
... rewrite the second term moving $x^{2}+1$ into the denominator
$=2x(x^{2}+1)^{1/2}+\displaystyle \frac{x^{3}}{(x^{2}+1)^{1/2}}$
... rewrite, with the first term being a fraction with the common denominator
$=\displaystyle \frac{2x(x^{2}+1)^{1/2}\cdot(x^{2}+1)^{1/2}+x^{3}}{(x^{2}+1)^{1/2}}$
... apply $a^{m}\cdot a^{n}=a^{m+n}$ to the first term of the numerator
$=\displaystyle \frac{2x(x^{2}+1)^{1/2+1/2}+x^{3}}{(x^{2}+1)^{1/2}}$
$=\displaystyle \frac{2x(x^{2}+1)^{1}+x^{3}}{(x^{2}+1)^{1/2}}$
... distribute,
$=\displaystyle \frac{2x^{3}+2x+x^{3}}{(x^{2}+1)^{1/2}}$
... all exponents are positive
