To solve the equation \(3x + 1 = \sqrt{8x + 1} + \sqrt{2x - 1}\), we need to isolate the variables and eliminate the square roots. Here is a step-by-step solution:

1. **Isolate one of the square roots:**
\[
3x + 1 - \sqrt{2x - 1} = \sqrt{8x + 1}
\]

2. **Square both sides to eliminate the square root on the right:**
\[
(3x + 1 - \sqrt{2x - 1})^2 = (\sqrt{8x + 1})^2
\]
\[
(3x + 1 - \sqrt{2x - 1})^2 = 8x + 1
\]

3. **Expand the left side:**
\[
(3x + 1)^2 - 2(3x + 1)\sqrt{2x - 1} + (\sqrt{2x - 1})^2 = 8x + 1
\]
\[
9x^2 + 6x + 1 - 2(3x + 1)\sqrt{2x - 1} + 2x - 1 = 8x + 1
\]
\[
9x^2 + 8x + 1 - 2(3x + 1)\sqrt{2x - 1} = 8x + 1
\]

4. **Simplify the equation:**
\[
9x^2 + 8x + 1 - 2(3x + 1)\sqrt{2x - 1} = 8x + 1
\]
\[
9x^2 - 2(3x + 1)\sqrt{2x - 1} = 0
\]

5. **Isolate the square root term:**
\[
9x^2 = 2(3x + 1)\sqrt{2x - 1}
\]

6. **Square both sides again to eliminate the remaining square root:**
\[
(9x^2)^2 = [2(3x + 1)\sqrt{2x - 1}]^2
\]
\[
81x^4 = 4(3x + 1)^2(2x - 1)
\]

7. **Expand the right side:**
\[
81x^4 = 4(9x^2 + 6x + 1)(2x - 1)
\]
\[
81x^4 = 4(18x^3 - 9x^2 + 12x^2 - 6x + 2x - 1)
\]
\[
81x^4 = 4(18x^3 + 3x^2 - 4x - 1)
\]
\[
81x^4 = 72x^3 + 12x^2 - 16x - 4
\]

8. **Rearrange the equation:**
\[
81x^4 - 72x^3 - 12x^2 + 16x + 4 = 0
\]

This is a quartic equation, which can be quite complex to solve algebraically. However, we can check for possible rational roots using the Rational Root Theorem or use numerical methods to approximate the solutions.

Let's check for possible rational roots. The Rational Root Theorem suggests that any rational solution, \( \frac{p}{q} \), is a factor of the constant term (4) divided by a factor of the leading coefficient (81). Possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{1}{9}, \pm \frac{2}{9}, \pm \frac{4}{9}, \pm \frac{1}{27}, \pm \frac{2}{27}, \pm \frac{4}{27}, \pm \frac{1}{81}, \pm \frac{2}{81}, \pm \frac{4}{81} \).

Testing these values, we find that \( x = 1 \) is a solution:

\[
3(1) + 1 = \sqrt{8(1) + 1} + \sqrt{2(1) - 1}
\]
\[
4 = \sqrt{9} + \sqrt{1}
\]
\[
4 = 3 + 1
\]
\[
4 = 4
\]

Thus, \( x = 1 \) is a solution. Further solutions may exist, but they would require more advanced techniques or numerical methods to find.