`int (x^(2))/((x^(2) +1) (x^(2)-2) (x^(2) + 3))dx`

Answer

To solve the integral

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

we will follow these steps:

### Step 1: Substitution
Let \( x^2 = t \). Then, \( dx = \frac{1}{2\sqrt{t}} dt \) or \( dx = \frac{1}{2\sqrt{x^2}} dt = \frac{1}{2\sqrt{t}} dt \).

Substituting in the integral, we have:

\[ \int \frac{t}{(t + 1)(t - 2)(t + 3)} \cdot \frac{1}{2\sqrt{t}} dt = \frac{1}{2} \int \frac{t^{1/2}}{(t + 1)(t - 2)(t + 3)} dt. \]

### Step 2: Partial Fraction Decomposition
Next, we perform partial fraction decomposition on

\[ \frac{t}{(t + 1)(t - 2)(t + 3)}. \]

We can express this as:

\[ \frac{t}{(t + 1)(t - 2)(t + 3)} = \frac{A}{t + 1} + \frac{B}{t - 2} + \frac{C}{t + 3}. \]

Multiplying through by the denominator \((t + 1)(t - 2)(t + 3)\) gives:

\[ t = A(t - 2)(t + 3) + B(t + 1)(t + 3) + C(t + 1)(t - 2). \]

### Step 3: Solve for Coefficients
To find \(A\), \(B\), and \(C\), we can substitute convenient values for \(t\):

1. Let \(t = -1\):
\[ -1 = A(-3)(2) \implies A = \frac{1}{6}. \]

2. Let \(t = 2\):
\[ 2 = B(3)(5) \implies B = \frac{2}{15}. \]

3. Let \(t = -3\):
\[ -3 = C(-2)(-4) \implies C = \frac{3}{10}. \]

### Step 4: Rewrite the Integral
Now we can rewrite the integral as:

\[ \frac{1}{2} \int \left( \frac{1/6}{t + 1} + \frac{2/15}{t - 2} + \frac{3/10}{t + 3} \right) dt. \]

### Step 5: Integrate Each Term
Now we integrate each term separately:

1. \(\int \frac{1}{t + 1} dt = \ln |t + 1|\).
2. \(\int \frac{1}{t - 2} dt = \ln |t - 2|\).
3. \(\int \frac{1}{t + 3} dt = \ln |t + 3|\).

Thus, we have:

\[ \frac{1}{2} \left( \frac{1}{6} \ln |t + 1| + \frac{2}{15} \ln |t - 2| + \frac{3}{10} \ln |t + 3| \right) + C. \]

### Step 6: Substitute Back
Substituting back \(t = x^2\):

\[ = \frac{1}{12} \ln |x^2 + 1| + \frac{1}{15} \ln |x^2 - 2| + \frac{1}{20} \ln |x^2 + 3| + C. \]

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

\[ \int \frac{x^2}{(x^2 + 1)(x^2 - 2)(x^2 + 3)} \, dx = \frac{1}{12} \ln |x^2 + 1| + \frac{1}{15} \ln |x^2 - 2| + \frac{1}{20} \ln |x^2 + 3| + C. \]