Problem 5. (10 points) Calculate
\[
\int \frac{x^{2}+2}{\left(x^{2}-2 x+5\right)(x+3)} d x
\]
by using partial fractions. (You need to show your calculation process.)

Step 1 ：Decompose the integrand into simpler fractions. The denominator of the integrand is a product of a quadratic and a linear term. Therefore, the integrand can be decomposed into the form \(\frac{A}{x+3} + \frac{Bx+C}{x^2-2x+5}\), where A, B, and C are constants to be determined.

Step 2 ：Set up the equation \(x^2 + 2 = A*(x^2 - 2*x + 5) + (x + 3)*(B*x + C)\) and solve for A, B, and C. The solution is \(A = \frac{11}{20}\), \(B = \frac{9}{20}\), and \(C = -\frac{1}{4}\).

Step 3 ：Substitute the values of A, B, and C back into the decomposed form of the integrand to get \(\frac{9x}{20} - \frac{1}{4}{x^2 - 2x + 5} + \frac{11}{20(x + 3)}\).

Step 4 ：Integrate each term separately to get \(\frac{11}{20}\ln|x + 3| + \frac{9}{40}\ln|x^2 - 2x + 5| + \frac{1}{10}\arctan\left(\frac{x}{2} - \frac{1}{2}\right)\).

Step 5 ：Final Answer: The integral of the given function is \(\boxed{\frac{11}{20}\ln|x + 3| + \frac{9}{40}\ln|x^2 - 2x + 5| + \frac{1}{10}\arctan\left(\frac{x}{2} - \frac{1}{2}\right) + C}\), where C is the constant of integration.