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

Answer

To solve the integral

\[ \int \frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Expression
First, we will rewrite the integral in a more manageable form. We can factor out \(\sqrt{x}\) from both the numerator and the denominator.

The numerator becomes:
\[ (\sqrt{x}+1)(x^2-\sqrt{x}) = \sqrt{x}x^2 - x + x^2 - \sqrt{x} = x^{5/2} - \sqrt{x} + x^2 - \sqrt{x}. \]

The denominator becomes:
\[ x\sqrt{x}+x+\sqrt{x} = x^{3/2} + x + \sqrt{x}. \]

### Step 2: Factor Out Common Terms
Now, we can factor out \(\sqrt{x}\) from the numerator and denominator:
\[ \int \frac{\sqrt{x}(\sqrt{x}+1)(x^{3/2}-1)}{\sqrt{x}(x^{3/2}+x+\sqrt{x})} \, dx. \]

This simplifies to:
\[ \int \frac{(\sqrt{x}+1)(x^{3/2}-1)}{x^{3/2}+x+\sqrt{x}} \, dx. \]

### Step 3: Recognize the Form
Notice that \(x^{3/2}-1\) can be factored as a difference of cubes:
\[ x^{3/2}-1 = (\sqrt{x}-1)(x+\sqrt{x}+1). \]

### Step 4: Substitute and Simplify
Now we can substitute this back into the integral:
\[ \int \frac{(\sqrt{x}+1)(\sqrt{x}-1)(x+\sqrt{x}+1)}{x^{3/2}+x+\sqrt{x}} \, dx. \]

The \(x+\sqrt{x}+1\) terms in the numerator and denominator cancel out:
\[ \int (\sqrt{x}+1)(\sqrt{x}-1) \, dx. \]

### Step 5: Expand and Integrate
Now we can expand the remaining expression:
\[ \int (\sqrt{x}^2 - 1) \, dx = \int (x - 1) \, dx. \]

### Step 6: Perform the Integration
Now we integrate:
\[ \int (x - 1) \, dx = \frac{x^2}{2} - x + C. \]

### Final Result
Thus, the final result of the integral is:
\[ \frac{x^2}{2} - x + C. \]