Evaluate: `int((x^4-x)^(1//4))/(x^5)\ dx`

Explanation

To evaluate the integral

\[ \int \frac{(x^4 - x)^{1/4}}{x^5} \, dx, \]

we will follow these steps:

### Step 1: Simplify the integrand
We can rewrite the integrand by factoring out \(x^4\) from the expression inside the parentheses:

\[ \int \frac{(x^4(1 - \frac{1}{x^3}))^{1/4}}{x^5} \, dx. \]

This simplifies to:

\[ \int \frac{x^{4/4}(1 - \frac{1}{x^3})^{1/4}}{x^5} \, dx = \int \frac{(1 - \frac{1}{x^3})^{1/4}}{x^{5 - 1}} \, dx = \int (1 - \frac{1}{x^3})^{1/4} \frac{1}{x} \, dx. \]

### Step 2: Substitute
Let

\[ t = 1 - \frac{1}{x^3}. \]

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

\[ \frac{dt}{dx} = \frac{3}{x^4} \implies dt = \frac{3}{x^4} \, dx \implies dx = \frac{x^4}{3} \, dt. \]

### Step 3: Express \(x\) in terms of \(t\)
From the substitution, we can express \(x^4\) in terms of \(t\):

\[ x^4 = \frac{1 - t}{1} \implies x^4 = \frac{1}{1 - t}. \]

### Step 4: Substitute in the integral
Now substituting \(dx\) and \(x\) into the integral:

\[ \int (1 - \frac{1}{x^3})^{1/4} \frac{1}{x} \cdot \frac{x^4}{3} \, dt = \frac{1}{3} \int t^{1/4} \cdot \frac{1}{x} \cdot x^4 \, dt. \]

This simplifies to:

\[ \frac{1}{3} \int t^{1/4} \cdot \frac{1}{x^3} \, dt. \]

### Step 5: Substitute back for \(x\)
Since \(x^3 = \frac{1}{1 - t}\), we have:

\[ \frac{1}{x^3} = 1 - t. \]

Thus, the integral becomes:

\[ \frac{1}{3} \int t^{1/4} (1 - t) \, dt. \]

### Step 6: Evaluate the integral
Now we can evaluate the integral:

\[ \frac{1}{3} \int t^{1/4} (1 - t) \, dt = \frac{1}{3} \left( \int t^{1/4} \, dt - \int t^{5/4} \, dt \right). \]

Calculating these integrals:

\[ \int t^{1/4} \, dt = \frac{t^{5/4}}{5/4} = \frac{4}{5} t^{5/4}, \]
\[ \int t^{5/4} \, dt = \frac{t^{9/4}}{9/4} = \frac{4}{9} t^{9/4}. \]

Thus,

\[ \frac{1}{3} \left( \frac{4}{5} t^{5/4} - \frac{4}{9} t^{9/4} \right) + C. \]

### Step 7: Substitute back for \(t\)
Substituting back \(t = 1 - \frac{1}{x^3}\):

\[ = \frac{1}{3} \left( \frac{4}{5} \left(1 - \frac{1}{x^3}\right)^{5/4} - \frac{4}{9} \left(1 - \frac{1}{x^3}\right)^{9/4} \right) + C. \]

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

\[ \frac{4}{15} \left(1 - \frac{1}{x^3}\right)^{5/4} - \frac{4}{27} \left(1 - \frac{1}{x^3}\right)^{9/4} + C. \]