Evaluate the definite integral
\[
\int_{1}^{5} \frac{10 x^{2}+5}{\sqrt{x}} d x
\]
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Step 1 ：The given integral is a definite integral with limits from 1 to 5. The integrand is a rational function where the numerator is a polynomial and the denominator is a square root function.

Step 2 ：To solve this integral, we can use the power rule for integration, which states that the integral of x^n dx is (1/(n+1))x^(n+1). However, before we can apply this rule, we need to rewrite the integrand in a form that allows us to use the power rule.

Step 3 ：We can do this by splitting the fraction into two separate fractions and simplifying the exponents of x.

Step 4 ：Let's simplify the function f = (10*x**2 + 5)/sqrt(x) to f_simplified = 5*(2*x**2 + 1)/sqrt(x).

Step 5 ：Now, we can compute the definite integral of the simplified function. The result is a numerical value, which is the area under the curve of the function from x=1 to x=5.

Step 6 ：The definite integral of the given function from 1 to 5 is \(\boxed{-14 + 110\sqrt{5}}\).