Expression x^m(bx^n+a)^p (1) is called binomial differential. We can convert your expression into following: x^2(bx^2-1)^1/2; in your particular case m=2, n=2, p=1/2. As it was proved by Chebyshev binomial differential has antiderivative expressed by elementary functions only in the following 3 cases:
In other cases, the integral can not be calculated in terms of elementary functions. Your case is #3. Use next substitution: a+b(x^n)=(x^n)(u^2). Making the change of variable, solve and return to the old variable.