Evaluate
`int (x^(2)+5x-1)/(sqrt(x))dx`

Answer

To evaluate the integral

\[ I = \int \frac{x^2 + 5x - 1}{\sqrt{x}} \, dx, \]

we will first rewrite the integrand in a more manageable form.

### Step 1: Rewrite the integrand
We can express \(\sqrt{x}\) as \(x^{1/2}\). Thus, we can rewrite the integral as:

\[ I = \int \frac{x^2 + 5x - 1}{x^{1/2}} \, dx. \]

Now, we can split the fraction:

\[ I = \int \left( \frac{x^2}{x^{1/2}} + \frac{5x}{x^{1/2}} - \frac{1}{x^{1/2}} \right) \, dx. \]

### Step 2: Simplify each term
This simplifies to:

\[ I = \int \left( x^{2 - 1/2} + 5x^{1 - 1/2} - x^{-1/2} \right) \, dx, \]

which can be further simplified to:

\[ I = \int \left( x^{3/2} + 5x^{1/2} - x^{-1/2} \right) \, dx. \]

### Step 3: Integrate each term
Now, we can integrate each term separately:

1. For \(x^{3/2}\):

\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}. \]

2. For \(5x^{1/2}\):

\[ \int 5x^{1/2} \, dx = 5 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} = 5 \cdot \frac{x^{3/2}}{3/2} = \frac{10}{3} x^{3/2}. \]

3. For \(-x^{-1/2}\):

\[ \int -x^{-1/2} \, dx = -\frac{x^{-1/2 + 1}}{-1/2 + 1} = -\frac{x^{1/2}}{1/2} = -2x^{1/2}. \]

### Step 4: Combine the results
Now we combine all the integrated terms:

\[ I = \frac{2}{5} x^{5/2} + \frac{10}{3} x^{3/2} - 2x^{1/2} + C, \]

where \(C\) is the constant of integration.

### Final Answer
Thus, the evaluated integral is:

\[ I = \frac{2}{5} x^{5/2} + \frac{10}{3} x^{3/2} - 2x^{1/2} + C. \]