`int x^8/(1+x^3)^(1//3)dx`

Answer

To solve the integral \( \int \frac{x^8}{(1+x^3)^{1/3}} \, dx \), we will follow these steps:

### Step 1: Substitution
Let \( u = 1 + x^3 \). Then, we differentiate to find \( du \):
\[ du = 3x^2 \, dx \implies dx = \frac{du}{3x^2} \]

### Step 2: Express \( x^2 \) in terms of \( u \)
From the substitution \( u = 1 + x^3 \), we can express \( x^3 \) as:
\[ x^3 = u - 1 \implies x = (u - 1)^{1/3} \]
Now, we can find \( x^2 \):
\[ x^2 = ((u - 1)^{1/3})^2 = (u - 1)^{2/3} \]

### Step 3: Substitute \( x^8 \) and \( dx \) in the integral
Now, we need to express \( x^8 \) in terms of \( u \):
\[ x^8 = ((u - 1)^{1/3})^8 = (u - 1)^{8/3} \]
Substituting these into the integral gives:
\[ \int \frac{(u - 1)^{8/3}}{u^{1/3}} \cdot \frac{du}{3(u - 1)^{2/3}} = \int \frac{(u - 1)^{8/3}}{u^{1/3} \cdot 3(u - 1)^{2/3}} \, du \]
This simplifies to:
\[ \int \frac{(u - 1)^{6/3}}{3u^{1/3}} \, du = \frac{1}{3} \int \frac{(u - 1)^2}{u^{1/3}} \, du \]

### Step 4: Expand the integrand
Now, we expand \( (u - 1)^2 \):
\[ (u - 1)^2 = u^2 - 2u + 1 \]
Thus, the integral becomes:
\[ \frac{1}{3} \int \left( \frac{u^2}{u^{1/3}} - \frac{2u}{u^{1/3}} + \frac{1}{u^{1/3}} \right) \, du = \frac{1}{3} \int \left( u^{5/3} - 2u^{2/3} + u^{-1/3} \right) \, du \]

### Step 5: Integrate term by term
Now we can integrate each term:
\[ \int u^{5/3} \, du = \frac{u^{8/3}}{8/3} = \frac{3}{8} u^{8/3} \]
\[ \int -2u^{2/3} \, du = -2 \cdot \frac{u^{5/3}}{5/3} = -\frac{6}{5} u^{5/3} \]
\[ \int u^{-1/3} \, du = \frac{u^{2/3}}{2/3} = \frac{3}{2} u^{2/3} \]

Combining these results, we have:
\[ \frac{1}{3} \left( \frac{3}{8} u^{8/3} - \frac{6}{5} u^{5/3} + \frac{3}{2} u^{2/3} \right) + C \]

### Step 6: Substitute back for \( u \)
Now substitute back \( u = 1 + x^3 \):
\[ = \frac{1}{8} (1 + x^3)^{8/3} - \frac{2}{5} (1 + x^3)^{5/3} + \frac{1}{2} (1 + x^3)^{2/3} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{x^8}{(1+x^3)^{1/3}} \, dx = \frac{1}{8} (1 + x^3)^{8/3} - \frac{2}{5} (1 + x^3)^{5/3} + \frac{1}{2} (1 + x^3)^{2/3} + C \]