Perform the multiplication or division and simplify. \[ \frac{\frac{x^{3}}{x+5}}{\frac{x}{x^{2}+10 x+25}} \]

To simplify the expression \[ \frac{\frac{x^{3}}{x+5}}{\frac{x}{x^{2}+10 x+25}} \] we start by rewriting the division of fractions as multiplication by the reciprocal: \[ \frac{x^{3}}{x+5} \div \frac{x}{x^{2}+10 x+25} = \frac{x^{3}}{x+5} \cdot \frac{x^{2}+10x+25}{x} \] Next, we can factor the quadratic expression in the denominator: \[ x^{2} + 10x + 25 = (x+5)^{2} \] Now substituting this back into our expression, we get: \[ \frac{x^{3}}{x+5} \cdot \frac{(x+5)^{2}}{x} \] This expression can be rewritten as: \[ \frac{x^{3} \cdot (x+5)^{2}}{x(x+5)} \] Now we can simplify by canceling out one \(x+5\) from the numerator and the denominator: \[ = \frac{x^{3} \cdot (x+5)}{x} \] Next, we can also cancel one \(x\) from the \(x^{3}\): \[ = x^{2} (x+5) \] Now, we can distribute the \(x^{2}\): \[ = x^{3} + 5x^{2} \] Thus, the simplified expression is: \[ \boxed{x^{3} + 5x^{2}} \]