\displaystyle \int \frac{8 x^3+6 x^2+ 100 x + 75}{x^4+25 x^2}\, dx

We are tasked with evaluating the integral:

\[
\int \frac{8x^3 + 6x^2 + 100x + 75}{x^4 + 25x^2} \, dx
\]

### Step 1: Simplifying the denominator
The denominator is \(x^4 + 25x^2\). This can be factored as:

\[
x^4 + 25x^2 = x^2(x^2 + 25)
\]

Thus, the integral becomes:

\[
\int \frac{8x^3 + 6x^2 + 100x + 75}{x^2(x^2 + 25)} \, dx
\]

### Step 2: Breaking the fraction into simpler parts
We now split the expression into terms that are easier to handle:

\[
\frac{8x^3 + 6x^2 + 100x + 75}{x^2(x^2 + 25)} = \frac{8x^3}{x^2(x^2 + 25)} + \frac{6x^2}{x^2(x^2 + 25)} + \frac{100x}{x^2(x^2 + 25)} + \frac{75}{x^2(x^2 + 25)}
\]

We can simplify each term:

1. \(\frac{8x^3}{x^2(x^2 + 25)} = \frac{8x}{x^2 + 25}\)
2. \(\frac{6x^2}{x^2(x^2 + 25)} = \frac{6}{x^2 + 25}\)
3. \(\frac{100x}{x^2(x^2 + 25)} = \frac{100x}{x^2(x^2 + 25)}\) (this one stays as is)
4. \(\frac{75}{x^2(x^2 + 25)} = \frac{75}{x^2(x^2 + 25)}\) (this one also stays as is)

Thus, the integral becomes:

\[
\int \left( \frac{8x}{x^2 + 25} + \frac{6}{x^2 + 25} + \frac{100x}{x^2(x^2 + 25)} + \frac{75}{x^2(x^2 + 25)} \right) dx
\]

### Step 3: Solving each part

#### 1. Integral of \(\frac{8x}{x^2 + 25}\)
This can be solved using the substitution \(u = x^2 + 25\), so \(du = 2x dx\):

\[
\int \frac{8x}{x^2 + 25} \, dx = 4 \ln|x^2 + 25|
\]

#### 2. Integral of \(\frac{6}{x^2 + 25}\)
This is a standard arctangent integral:

\[
\int \frac{6}{x^2 + 25} \, dx = \frac{6}{5} \arctan\left(\frac{x}{5}\right)
\]

#### 3. Integral of \(\frac{100x}{x^2(x^2 + 25)}\)
This term requires partial fraction decomposition. We can express it as:

\[
\frac{100x}{x^2(x^2 + 25)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 25}
\]

We'll solve that below.

#### 4. Integral of \(\frac{75}{x^2(x^2 + 25)}\)
Similarly, this term will need partial fractions.

Let me compute the partial fractions and finish the solution.After performing partial fraction decomposition, we obtain:

\[
\frac{100x}{x^2(x^2 + 25)} = \frac{4}{x} - \frac{4x}{x^2 + 25}
\]
\[
\frac{75}{x^2(x^2 + 25)} = \frac{3}{x^2} - \frac{3}{x^2 + 25}
\]

### Step 4: Revising the Integral

The integral now becomes:

\[
\int \left( \frac{8x}{x^2 + 25} + \frac{6}{x^2 + 25} + \left( \frac{4}{x} - \frac{4x}{x^2 + 25} \right) + \left( \frac{3}{x^2} - \frac{3}{x^2 + 25} \right) \right) dx
\]

Simplifying:

\[
= \int \left( \frac{8x - 4x}{x^2 + 25} + \frac{6 - 3}{x^2 + 25} + \frac{4}{x} + \frac{3}{x^2} \right) dx
\]

\[
= \int \left( \frac{4x}{x^2 + 25} + \frac{3}{x^2 + 25} + \frac{4}{x} + \frac{3}{x^2} \right) dx
\]

### Step 5: Solving the integrals

1. \(\int \frac{4x}{x^2 + 25} \, dx = 2 \ln|x^2 + 25|\)
2. \(\int \frac{3}{x^2 + 25} \, dx = \frac{3}{5} \arctan\left(\frac{x}{5}\right)\)
3. \(\int \frac{4}{x} \, dx = 4 \ln|x|\)
4. \(\int \frac{3}{x^2} \, dx = -\frac{3}{x}\)

### Final Answer:
The full integral is:

\[
2 \ln|x^2 + 25| + \frac{3}{5} \arctan\left(\frac{x}{5}\right) + 4 \ln|x| - \frac{3}{x} + C
\]

Would you like further details or clarifications on any part of this process?

Here are 5 related questions for practice:
1. How do you factor a quartic expression like \(x^4 + 25x^2\)?
2. What are the standard forms for integrals involving \(\frac{1}{x^2 + a^2}\)?
3. How does substitution simplify the integration process for rational functions?
4. What is partial fraction decomposition, and when is it useful?
5. How do you handle integrals involving both \(x^2\) and arctangent functions?

**Tip:** Always check if a denominator can be factored or simplified to make integrals more manageable.

This tutorial solves the integral \(\int \frac{8x^3 + 6x^2 + 100x + 75}{x^4 + 25x^2} \, dx\) by applying techniques such as partial fraction decomposition and the arctangent formula. Ideal for university students studying advanced calculus.

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