Write the given expression as a single quotient in which only positive exponents and/or radicals appear.
\[
2 x\left(x^{2}+1\right)^{1 / 2}+x^{2} \cdot \frac{1}{2}\left(x^{2}+1\right)^{-1 / 2} \cdot 2 x
\]

Step 1 ：Given the expression \(2x(x^{2}+1)^{1 / 2}+x^{2} \cdot \frac{1}{2}(x^{2}+1)^{-1 / 2} \cdot 2x\)

Step 2 ：The first term is a product of 2, x and the square root of \((x^2 + 1)\). The second term is a product of \(x^2\), 1/2, the reciprocal of the square root of \((x^2 + 1)\) and 2x.

Step 3 ：Simplify the second term by cancelling out the 2 in the numerator and the 2 in the denominator. The x in the numerator can be combined with the \(x^2\) to give \(x^3\). The reciprocal of the square root of \((x^2 + 1)\) can be written as \((x^2 + 1)^{-1/2}\).

Step 4 ：Combine the two terms into a single quotient by adding the exponents of \((x^2 + 1)\) in the two terms. The exponent of \((x^2 + 1)\) in the first term is 1/2 and in the second term is -1/2. Adding these gives an exponent of 0, which means \((x^2 + 1)\) to the power of 0 is 1.

Step 5 ：So, the expression simplifies to \(2x + x^3\).

Step 6 ：However, the expression simplifies to \(x(1.0x^{2} + 2(x^{2} + 1)^{1.0})/(x^{2} + 1)^{0.5}\). This is a single quotient in which only positive exponents and/or radicals appear.

Step 7 ：\(\boxed{x(1.0x^{2} + 2(x^{2} + 1)^{1.0})/(x^{2} + 1)^{0.5}}\) is the final answer.