Method: Write px+q = A * d/dx($ax^{2+bx+c}$) + B = A(2ax+b) + B. Solve for A and B: 2aA = p, Ab + B = q. So A = $\frac{p}{2a}$, B = q - $\frac{bp}{2a}$.

Split the integral: integral $\frac{px+q}{sqrt}$($ax^{2+bx+c}$) dx = A * integral $\frac{2ax+b}{sqrt}$($ax^{2+bx+c}$) dx + B * integral dx/sqrt($ax^{2+bx+c}$)

First integral: integral $\frac{2ax+b}{sqrt}$($ax^{2+bx+c}$) dx = 2*sqrt($ax^{2+bx+c}$) + C (since the numerator is the derivative of the expression under the root)

Second integral: Complete the square and use standard forms:

• integral dx/sqrt(X^{2+k}^2) = ln|X+sqrt(X^{2+k}^2)| + C
• integral dx/sqrt(k^{2-X}^2) = arcsin$\frac{X}{k}$ + C
• integral dx/sqrt(X^{2-k}^2) = ln|X+sqrt(X^{2-k}^2)| + C