There are general algorithms to compute the integral basis of any number field. For example, see here.
That said, it's an exercise in most standard number theory courses to find integral bases for any biquadratic extension. There is an exercise (with hints) in Marcus's Number Fields which discusses this (it is exercise 42 on page 51)--you should do it.
It tells you, in his notation, that if we let $m=69$, $k=23$, and $n=3$, then your field has integral basis
Also, SAGE is your friend. It has the ability to compute the integral basis of any number field. For example, it spat out the following:
K.<a,b> = NumberField([x^2-23,x^22-3]);
[1, 5/2*a - 11/2*b, -1/2*b*a + 13/2, 4*a - 9*b]