Solution to the Integral of the Multiplication of Two Polynomials Under the Square Root

Question

Anybody has an idea of the generic solution of this integral? $$ \int_0^pdx \sqrt{A+(p-x)^2}\sqrt{B+x^2} $$ where $A,B < p$. If $A$ and $B$ is zero, we can find a definite solution. However, when they are not zero, we have the square root of a 4th power polynomial. I need a new approach for this 4th power polynomial that does not have elliptic function after the integration. [Wolfram Mathematica throws a solution with an Elliptic function, which has imaginary parts. $A$ and $B$ represent the masses of the particles that cannot be imaginary and $p$ represents the momentum of the particle.]