You are pretty close:
<(?!/?(?:strong|em|i|u|b))[^>]+>
Here is a "formal" explanation:
Remove Other Tags
<(?!/?(?:strong|em|i|u|b))[^>]+>
Match the character “<” literally «<»
Assert that it is impossible to match the regex below starting at this position (negative lookahead) «(?!/?(?:strong|em|i|u|b))»
Match the character “/” literally «/?»
Between zero and one times, as many times as possible, giving back as needed (greedy) «?»
Match the regular expression below «(?:strong|em|i|u|b)»
Match either the regular expression below (attempting the next alternative only if this one fails) «strong»
Match the characters “strong” literally «strong»
Or match regular expression number 2 below (attempting the next alternative only if this one fails) «em»
Match the characters “em” literally «em»
Or match regular expression number 3 below (attempting the next alternative only if this one fails) «i»
Match the character “i” literally «i»
Or match regular expression number 4 below (attempting the next alternative only if this one fails) «u»
Match the character “u” literally «u»
Or match regular expression number 5 below (the entire group fails if this one fails to match) «b»
Match the character “b” literally «b»
Match any character that is NOT a “>” «[^>]+»
Between one and unlimited times, as many times as possible, giving back as needed (greedy) «+»
Match the character “>” literally «>»
Created with RegexBuddy