BUG: `Series.interpolate` does not interpolate `pd.NA`

[attack68]

Pandas version checks

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• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

from pandas import NA, Series
from numpy import nan

s1 = Series([1.0, NA, 2.0])
s2 = Series([1.0, nan, 2.0])
s3 = Series([1.0, None, 2.0])

s1.interpolate()  # Series([1.0, NA, 2.0])
s2.interpolate()  # Series([1.0, 1.5, 2.0])
s3.interpolate()  # Series([1.0, 1.5, 2.0])

Issue Description

This actually works:

s = Series([1, NA, 2], dtype=pd.Float64Dtype())
s.interpolate("ffill")

But there are also error messages about interpolation methods that are not documented in Series.interpolate.
You cannot linearly interpolate over Float64Dtype though.

Expected Behavior

Similar to underlying None and nan versions.

Installed Versions

Replace this line with the output of pd.show_versions()

[lithomas1]

Note:
This is deprecated, I get

<stdin>:1: FutureWarning: Series.interpolate with object dtype is deprecated and will raise in a future version. Call obj.infer_objects(copy=False) before interpolating instead.

running your example.

However, there seems to be another bug where infer_objects is not inferring df/series with pd.NA in it correctly,
s1.infer_objects(copy=False).interpolate() also fails for me.