`int (8x ^(43)+13 x ^(38))/((x ^(13)+x ^(5) +1)^(4))dx =`

Options

A. `(x ^(39))/(3 (x ^(13) +x ^(5) +1)^(3))+C`
B. `(x ^(39))/( (x ^(13) -x ^(5) +1)^(3))+C`
C. `(x ^(39))/( 5(x ^(13) -x ^(5) +1)^(5))+C`
D. None of these

Answer

To solve the integral

\[ \int \frac{8x^{43} + 13x^{38}}{(x^{13} + x^{5} + 1)^{4}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
We can factor out \(x^{13}\) from the denominator. This gives us:

\[ \int \frac{8x^{43} + 13x^{38}}{(x^{13}(1 + x^{-8} + x^{-13}))^{4}} \, dx. \]

### Step 2: Rewrite the Denominator
The denominator can be rewritten as:

\[ x^{52} (1 + x^{-8} + x^{-13})^{4}. \]

Thus, we can rewrite the integral as:

\[ \int \frac{8x^{43} + 13x^{38}}{x^{52}(1 + x^{-8} + x^{-13})^{4}} \, dx. \]

### Step 3: Simplify the Numerator
Now, we simplify the numerator:

\[ 8x^{43} + 13x^{38} = 8x^{43 - 52} + 13x^{38 - 52} = 8x^{-9} + 13x^{-14}. \]

### Step 4: Substitute into the Integral
Now, substituting this back into the integral gives:

\[ \int \frac{8x^{-9} + 13x^{-14}}{(1 + x^{-8} + x^{-13})^{4}} \, dx. \]

### Step 5: Use Substitution
Let

\[ t = 1 + x^{-8} + x^{-13}. \]

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

\[ \frac{dt}{dx} = -8x^{-9} - 13x^{-14}. \]

Thus,

\[ dt = (-8x^{-9} - 13x^{-14}) \, dx. \]

This implies:

\[ dx = \frac{dt}{-8x^{-9} - 13x^{-14}}. \]

### Step 6: Substitute \(dx\) into the Integral
Substituting \(dx\) into the integral gives:

\[ \int \frac{-dt}{t^{4}}. \]

### Step 7: Integrate
Now we can integrate:

\[ -\int t^{-4} \, dt = \frac{1}{3} t^{-3} + C. \]

### Step 8: Back Substitute for \(t\)
Substituting back for \(t\):

\[ = \frac{1}{3} (1 + x^{-8} + x^{-13})^{-3} + C. \]

### Step 9: Simplify the Expression
We can express this as:

\[ = \frac{1}{3} \frac{1}{(1 + x^{-8} + x^{-13})^{3}} + C. \]

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

\[ \frac{1}{3(1 + x^{-8} + x^{-13})^{3}} + C. \]