Evaluate `int(sqrt(1+root(3)(x)))/(root(3)(x^(2)))dx`.

Answer

To evaluate the integral

\[ \int \frac{\sqrt{1+\sqrt[3]{x}}}{\sqrt[3]{x^2}} \, dx, \]

we will use substitution. Let's go through the solution step by step.

### Step 1: Substitution
Let

\[ t = 1 + \sqrt[3]{x}. \]

Then, we differentiate both sides with respect to \(x\):

\[ \frac{dt}{dx} = \frac{1}{3} x^{-\frac{2}{3}}. \]

This implies:

\[ dx = 3 x^{\frac{2}{3}} dt. \]

### Step 2: Express \(x\) in terms of \(t\)
From our substitution, we have:

\[ \sqrt[3]{x} = t - 1 \quad \Rightarrow \quad x = (t - 1)^3. \]

### Step 3: Substitute \(x\) and \(dx\) into the integral
Now, we need to express the integral in terms of \(t\):

\[ \sqrt[3]{x^2} = \sqrt[3]{((t - 1)^3)^2} = (t - 1)^2. \]

Thus, the integral becomes:

\[ \int \frac{\sqrt{t}}{(t - 1)^2} \cdot 3(t - 1)^2 dt. \]

### Step 4: Simplify the integral
The \((t - 1)^2\) in the numerator and denominator cancels out:

\[ 3 \int \sqrt{t} \, dt. \]

### Step 5: Evaluate the integral
Now, we can evaluate the integral:

\[ 3 \int t^{\frac{1}{2}} \, dt = 3 \cdot \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + C = 2t^{\frac{3}{2}} + C. \]

### Step 6: Substitute back for \(t\)
Recalling our substitution \(t = 1 + \sqrt[3]{x}\), we substitute back:

\[ 2(1 + \sqrt[3]{x})^{\frac{3}{2}} + C. \]

### Final Answer
Thus, the final answer is:

\[ 2(1 + \sqrt[3]{x})^{\frac{3}{2}} + C. \]

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