As well

I think it all depends on whether it is used as part of an adverbial phrase when making comparisons, in which case the infinitive or simple form of the verb is the norm, or whether it is used as a conjunction introducing clauses of comparison and similar in meaning to in addition to, in which case the '-ing' pattern is required. Let us compare the two usages:

as + adj + as + clause/phrase

as + adv + as + clause/phrase

For example:

• 'I saw as many as three thousand people at the concert.'
• 'He was badly injured, but I did as much as I could to make him comfortable.'
• 'I waited for as long as I dared, but when it got dark, I went home.'
• 'Please come as quickly as you can. My father is very ill.'
• 'The Irish played as well as the Scots but didn't convert as many attempts on goal.'
• 'Richardson was as good an actor as Gieldgud (was).'

Note that if an adjective is placed between as and the noun,

a / an must be placed after the adjective.

What is interesting in your example, Van Anh, is that if you say: 'I can swim as well as cook well', you are stating that these are two things that you can do, whereas if you say: 'I can swim as well as I can cook', you are stating that you can do both these things to an equal degree of proficiency.

When we use as well as - similar in meaning and usage to in addition to - as a subordinating conjunction, the '-ing' form in the verb which follows is required:

• 'As well as playing tennis with Steve three times during the week, I (also) play badminton with my wife at the weekend.'
• 'In addition to working on his Ph.D. dissertation, he (also) translates articles for The Weekly Review.' 

 Note here that the adverb as well is similar in meaning to as well as and is often used as an alternative to too. Note that both as well and too must be used at the end of the sentence:

• 'We enjoyed a rare night out last night. We ate at Luigi's, and went to the cinema as well.' ( = As well as eating out, we (also) went to the cinema.)
• 'We eat well here in Sardinia. The wine is excellent too.'
• 'My wife is a chemist and both her parents were chemists as well.'

 Note the special use of as long as which is similar in meaning and use to provided that. Both are more emphatic forms of if and are used to introduce conditions:

• 'As long as you promise to help me, I don't mind cooking for twelve people on Easter Sunday.'
• 'I'll join you on this skiing holiday, provided I can have my own room at the hotel.'