Connect `int x^(m-1)(a+bx^(n))^(p)dx` with `int x^(m-n-1)(a+bx^(n))^(p)dx` and evaluate `int(x^(8)dx)/((1-x^(3))^(1//3))`.

Answer

To solve the integral \( \int \frac{x^8}{(1-x^3)^{1/3}} \, dx \), we will first rewrite it in a form that allows us to connect it with the given integral \( \int x^{m-1} (a + b x^n)^p \, dx \).

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{x^8}{(1-x^3)^{1/3}} \, dx \]
This can be rewritten as:
\[ \int x^8 (1-x^3)^{-1/3} \, dx \]

### Step 2: Identify Values of \( m \) and \( n \)
In the expression \( x^{m-1} (1 - x^3)^{p} \), we can identify:
- \( m - 1 = 8 \) which gives \( m = 9 \)
- The term \( (1 - x^3)^{-1/3} \) indicates that \( n = 3 \) and \( p = -\frac{1}{3} \).

### Step 3: Connect the Integrals
Now, we can connect this integral with the form:
\[ \int x^{m-n-1} (a + b x^n)^p \, dx \]
Here, we have:
- \( m - n - 1 = 9 - 3 - 1 = 5 \)
- Thus, we can express our integral as:
\[ \int x^5 (1 - x^3)^{-1/3} \, dx \]

### Step 4: Substitution
Next, we will use the substitution \( t = 1 - x^3 \). Then, we have:
\[ dt = -3x^2 \, dx \quad \Rightarrow \quad dx = -\frac{dt}{3x^2} \]
From \( t = 1 - x^3 \), we can express \( x^3 = 1 - t \) and thus \( x = (1 - t)^{1/3} \). Therefore, \( x^2 = (1 - t)^{2/3} \).

### Step 5: Substitute in the Integral
Substituting these into the integral gives:
\[ \int x^5 (1 - x^3)^{-1/3} \, dx = \int (1 - t)^{5/3} t^{-1/3} \left(-\frac{dt}{3(1-t)^{2/3}}\right) \]
This simplifies to:
\[ -\frac{1}{3} \int (1 - t)^{5/3} t^{-1/3} (1 - t)^{-2/3} \, dt = -\frac{1}{3} \int (1 - t)^{1} t^{-1/3} \, dt \]

### Step 6: Evaluate the Integral
Now we can evaluate:
\[ -\frac{1}{3} \int (1 - t) t^{-1/3} \, dt \]
Using the formula for the integral of \( x^m(1-x)^n \):
\[ \int x^m(1-x)^n \, dx = \frac{m!n!}{(m+n+1)!} \]
Here, \( m = -\frac{1}{3} \) and \( n = 1 \).

### Step 7: Final Result
After performing the integration and simplifying, we find:
\[ -\frac{1}{3} \left( \frac{1! \cdot \frac{2}{3}!}{(\frac{2}{3}+1)!} \right) + C \]

### Conclusion
Thus, the final answer for the integral \( \int \frac{x^8}{(1-x^3)^{1/3}} \, dx \) is:
\[ -\frac{1}{3} \left( \frac{1! \cdot \frac{2}{3}!}{(\frac{2}{3}+1)!} \right) + C \]