$\int \frac{x^{3}+x}{x-1} d x$

Step 1 ：Perform polynomial long division to simplify the integrand: \(\frac{x^3 + x}{x - 1} = x^2 + x + 2 + \frac{2}{x - 1}\)

Step 2 ：Integrate each part separately: \(\int (x^2 + x + 2) dx + \int \frac{2}{x - 1} dx\)

Step 3 ：Integrate the first part: \(\frac{x^3}{3} + \frac{x^2}{2} + 2x\)

Step 4 ：Integrate the second part: \(2\ln|x - 1|\)

Step 5 ：Combine the results: \(\boxed{\frac{x^3}{3} + \frac{x^2}{2} + 2x + 2\ln|x - 1| + C}\)