$$S = \frac{3}{x} + \frac{5}{x^2} + \frac{7}{x^3}$$
To add these fractions, we need a common denominator. The least common denominator is $x^3$.
We rewrite each fraction with the common denominator $x^3$:
$$\frac{3}{x} = \frac{3x^2}{x^3}$$
$$\frac{5}{x^2} = \frac{5x}{x^3}$$
$$\frac{7}{x^3} = \frac{7}{x^3}$$
Now we can add the fractions:
$$S = \frac{3x^2}{x^3} + \frac{5x}{x^3} + \frac{7}{x^3} = \frac{3x^2 + 5x + 7}{x^3}$$
The numerator $3x^2 + 5x + 7$ is a quadratic expression, and it cannot be factored easily. Therefore, the sum in simplest form is
$$\frac{3x^2 + 5x + 7}{x^3}$$
Final Answer: The final answer is $\boxed{\frac{3x^2+5x+7}{x^3}}$