To solve the integral
\[ \int dy = \int \left( y + \sqrt{\frac{y^2 - 8x}{4x}} \right) dx, \]
we will first separate the variables.
From the left side, we simply integrate with respect to \(y\):
\[ \int dy = y + C_1, \]
where \(C_1\) is an arbitrary constant of integration.
Now, let's focus on the right side:
\[ \int \left( y + \sqrt{\frac{y^2 - 8x}{4x}} \right) dx. \]
We can split this into two separate integrals:
\[ \int y , dx + \int \sqrt{\frac{y^2 - 8x}{4x}} , dx. \]
The term \(y\) is treated as a constant when integrating with respect to \(x\). So
\[ \int y , dx = yx + C_2, \]
where \(C_2\) is another arbitrary constant of integration.
Now we can evaluate the second integral. To simplify \(\sqrt{\frac{y^2 - 8x}{4x}} = \frac{\sqrt{y^2 - 8x}}{2\sqrt{x}}\). Thus,
\[ \int \sqrt{\frac{y^2 - 8x}{4x}} , dx = \int \frac{\sqrt{y^2 - 8x}}{2\sqrt{x}} , dx. \]
The integral \(\int \sqrt{y^2 - 8x} , dx\) can be more complicated and typically involves substitution or integration by parts, depending on the specific context or the function of \(y\).
To summarize, we have:
\[ y + C_1 = yx + \int \frac{\sqrt{y^2 - 8x}}{2\sqrt{x}} , dx + C_2. \]
At this point, we can combine the constant terms \(C_1 - C_2\) and rewrite the equation:
\[ y = yx + \int \frac{\sqrt{y^2 - 8x}}{2\sqrt{x}} , dx + C. \]
Where \(C = C_1 - C_2\).
This gives us an implicit relationship between \(x\) and \(y\). Depending on your specific requirements (whether you seek \(y\) in explicit form, or wish to evaluate for a specific \(x\)), further steps may be needed.