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 follow these steps:

### Step 1: Rewrite the Integral
First, we can rewrite the integral in terms of exponents:

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

### Step 2: Substitution
Let \( t = 1 + \sqrt[3]{x} \). Then, we differentiate \( t \):

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

Now, we need to express \( x \) in terms of \( t \):

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

### Step 3: Substitute in the Integral
Substituting \( x \) and \( dx \) into the integral:

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

### Step 4: Evaluate the Integral
Now we can evaluate the integral:

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

### Step 5: Substitute Back
Now we substitute back \( t = 1 + \sqrt[3]{x} \):

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

### Step 6: Add the Constant of Integration
Finally, we add the constant of integration \( C \):

\[ \int \frac{\sqrt{1 + \sqrt[3]{x}}}{\sqrt[3]{x^2}} \, dx = 2(1 + \sqrt[3]{x})^{3/2} + C. \]

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

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

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