Evaluate: `int(dx)/(x^2(1+x^5)^(4/5))`

Explanation

To evaluate the integral

\[ \int \frac{dx}{x^2 (1 + x^5)^{4/5}}, \]

we will follow a systematic approach.

### Step 1: Rewrite the integral
We start with the integral:

\[ \int \frac{dx}{x^2 (1 + x^5)^{4/5}}. \]

### Step 2: Substitute \( t = 1 + x^5 \)
Let us make the substitution \( t = 1 + x^5 \). Then, we differentiate \( t \) with respect to \( x \):

\[ dt = 5x^4 \, dx \implies dx = \frac{dt}{5x^4}. \]

### Step 3: Express \( x^2 \) in terms of \( t \)
From the substitution \( t = 1 + x^5 \), we can express \( x^5 \) as \( x^5 = t - 1 \). Therefore,

\[ x = (t - 1)^{1/5}. \]

Now, we can find \( x^2 \):

\[ x^2 = \left((t - 1)^{1/5}\right)^2 = (t - 1)^{2/5}. \]

### Step 4: Substitute \( dx \) and \( x^2 \) into the integral
Now we substitute \( dx \) and \( x^2 \) into the integral:

\[ \int \frac{1}{(t - 1)^{2/5} (t)^{4/5}} \cdot \frac{dt}{5x^4}. \]

Next, we need to express \( x^4 \):

\[ x^4 = \left((t - 1)^{1/5}\right)^4 = (t - 1)^{4/5}. \]

Thus, \( dx = \frac{dt}{5(t - 1)^{4/5}} \).

### Step 5: Substitute everything into the integral
Now substituting everything back into the integral gives us:

\[ \int \frac{1}{(t - 1)^{2/5} (t)^{4/5}} \cdot \frac{dt}{5(t - 1)^{4/5}} = \frac{1}{5} \int \frac{dt}{(t - 1)^{6/5} t^{4/5}}. \]

### Step 6: Simplify the integral
Now we can simplify the integral:

\[ \frac{1}{5} \int (t - 1)^{-6/5} t^{-4/5} \, dt. \]

### Step 7: Use the formula for integration
Using the formula for integration of the form \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \):

\[ = \frac{1}{5} \cdot \frac{(t - 1)^{-1/5}}{-1/5} + C = -\frac{1}{5} (t - 1)^{-1/5} + C. \]

### Step 8: Substitute back \( t = 1 + x^5 \)
Now substituting back \( t = 1 + x^5 \):

\[ = -\frac{1}{5} (1 + x^5 - 1)^{-1/5} + C = -\frac{1}{5} (x^5)^{-1/5} + C = -\frac{1}{5} \frac{1}{x} + C. \]

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

\[ -\frac{1}{5x} + C. \]