`int(2x^(12)+5x^(9))/((x^(5)+x^(3)+1)^(3))dx=`

Options

A. `(x^(2)+2x)/((x^(5)+x^(3)+1)^(2))+c`
B. `(x^(10))/(2(x^(5)+x^()+1)^(2))+c`
C. `ln|x^(5)+x^(3)+1|+sqrt(2x^(7)+5x^(4))+c`
D. none of these

Answer

To solve the integral

\[ \int \frac{2x^{12} + 5x^{9}}{(x^{5} + x^{3} + 1)^{3}} \, dx, \]

we can follow these steps:

### Step 1: Simplify the Denominator
We start by factoring out \(x^5\) from the denominator:

\[ x^{5} + x^{3} + 1 = x^{5}\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right). \]

Thus, we rewrite the integral as:

\[ \int \frac{2x^{12} + 5x^{9}}{\left(x^{5}\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right)\right)^{3}} \, dx. \]

### Step 2: Rewrite the Integral
Taking \(x^5\) out of the cube in the denominator gives us:

\[ = \int \frac{2x^{12} + 5x^{9}}{x^{15}\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right)^{3}} \, dx. \]

### Step 3: Divide Each Term
Now we divide each term in the numerator by \(x^{15}\):

\[ = \int \left( \frac{2}{x^{3}} + \frac{5}{x^{6}} \right) \cdot \frac{1}{\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right)^{3}} \, dx. \]

### Step 4: Substitution
Let

\[ t = 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}. \]

Now we need to find \(dt\):

\[ dt = \left(-\frac{2}{x^{3}} - \frac{5}{x^{6}}\right)dx. \]

Thus, we can express \(dx\) in terms of \(dt\):

\[ dx = -\frac{1}{\left(-\frac{2}{x^{3}} - \frac{5}{x^{6}}\right)} dt. \]

### Step 5: Substitute in the Integral
Substituting \(t\) and \(dx\) into the integral gives:

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

### Step 6: Integrate
Now we integrate:

\[ = -\left(-\frac{1}{2t^{2}}\right) + C = \frac{1}{2t^{2}} + C. \]

### Step 7: Substitute Back
Substituting back for \(t\):

\[ = \frac{1}{2\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right)^{2}} + C. \]

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

\[ \int \frac{2x^{12} + 5x^{9}}{(x^{5} + x^{3} + 1)^{3}} \, dx = \frac{1}{2\left(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}\right)^{2}} + C. \]