How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $ I would like to know how to factor the following polynomial. 
$$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$
What is the method I should use to factor it? If anyone could help.. Thanks in advance.

$ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c =$
$ab^3 - a^3b + a^3c -b^3c -ac^3 + bc^3=$
$ab(b^2-a^2)+c(a^3-b^3)-c^3(a-b)=$
$ab(a+b)(b-a)+c(a-b)(a^2+ab+b^2)-c^3(a-b)=$
$(a-b)[-ab(a+b)+c(a^2+ab+b^2)-c^3]=$
$(a-b)(-a^2b-ab^2+ca^2+cab+cb^2-c^3)=$
$(a-b)(ca^2-a^2b+cab-ab^2+cb^2-c^3)=$
$(a-b)[a^2(c-b)+ab(c-b)+c(b^2-c^2)]= $
$(a-b)(c-b)[a^2+ab-c(b+c)]=$
$(a-b)(c-b)(a^2+ab-bc-c^2)=$
$(a-b)(c-b)(a^2-c^2+ab-bc)=$
$(a-b)(c-b)[(a-c)(a+c)+b(a-c)]=$
$(a-b)(c-b)(a-c)(a+b+c)$
The suggestion of replacing $b$ with $a$ was the following: if in such an expression when you replace $b$ with $a$ you get a $0$ then one of the factors may be $(a-b)$.
In the future if you want an idea of the factors that appear you could write on wolframalpha factor$(ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c)$
