my $xmlish = <<'END';
<whatever><book-part type='closed'>
Chapter 1
</book-part>
<book-part type='unclosed'>
Chapter 2
<book-part type='final'>
Chapter 3
</whatever>
END

my $prev = '/';
$xmlish =~ s{(?=<\s*(/?)\s*book-part\b)}{
    (   ! $prev  &&  ! $1  ?  '</book-part>'  :  '',
        $prev= $1,
    )[0]
}ge;

print $xmlish;
__END__
<whatever><book-part type='closed'>
Chapter 1
</book-part>
<book-part type='unclosed'>
Chapter 2
</book-part><book-part type='final'>
Chapter 3
</whatever>