Utilitarianism

Utilitarianism was developed in the 18th century by Hutchenson, who used the phrase "the greatest good for the greatest number" to describe his theory. The promotion of happiness, however, can be found much earlier in history, in the works of Aristotle, in the Bible, in the teachings of Buddha, in the works of Confucius and elsewhere. Hutchenson's idea of Utilitarianism, however, seeks to find arational means of assessing how best to put this promotion of happiness into practice, and is split into two types; Act Utilitarianism is the earliest form, in which what is deemed right is based on the assessment of results of a particular action, and Rule Utilitarianism, which allows to be taken into account the general benefit to society that occurs when people follow general rules.
A good,clear introduction, giving precise definititions of key terms (see the Glossary on the site!).  There is also the preference utilitarianism of Peter Singer.  However, if you look back at the question, the first four lines aren't really relevant, as the question only asks you to discuss Bentham and Mill!
Bentham and Mill each argued respectively for these types of Utilitarianism, and thus theirbeliefs differ.Bentham was born in London at a time of great scientific and social change. He argued for Act Utilitarianism, and maintained that human beings were motivated by pleasure and by pain. He believed that everyone had an equal right to happiness, irrespective of their situation or status in life and argued that everyone counted equally in the assessment of the benefits of an action. Hebelieved that overall, this would also benefit the individual who did so and this would lead to that person's greatest happiness as well. His theory is democratic as pleasure cannot be for one person and one person alone.
You bring out two key points rather well: that utilitarianism is rationalistic (para 1) and that it isdemocratic (each to count as one, and no-one as more than one), thus providingthe basis for radical social reform.
Bentham wrote down his beliefs in his major work, ‘The principles of morals and legislation', written in1789, which is divided into three sections; The motivation of human beings and the concept of good and bad - "Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we shall do, aswell as to determine what we shall do." ; The Principle of Utility -The greatest good for the greatest number (The most useful course of action if trying to maximise pleasure and minimize pain and in a given situation, one must examine the consequential pain/pleasure resultant for all concerned.) ; and lastly, his Hedonic Calculus - this weighs up the pain and pleasure generated by the availablemoral actions to find the best option. It asks us to consider(s) several factors:
a. Intensity
b. Duration
c. Certainty or uncertainty
d. Nearness or remoteness
e. Consequences
f. Purity
g. Extent
I would prefer to see the hedonic calculus applied with an example which brings out some of the problems with this sort of idea.

If the probable pain of an action outweighs its pleasure thenBentham says that it is morally wrong. He believed that this was the way of calculating happiness as a result of the course of an action and by this he was making the basis of deciding whether an action should be considered right or wrong.
Also implied in Bentham's views, Mill argued that not all forms of happiness or pleasure were of equal value. This isn't expressed very clearly: you mean, Benthamimplied any pleasure is of equal value eg pushpin is equal to poetry, whereas Mill, who had a nervous breakdown at the age of 23, was saved by reading Wordsworth eg Intimations of Immortality and so, from his superior vantage point, saw pleasures as higher and lower. He recognised that it was easy to settle for the more immediate and sensual pleasures rather than the nobler and perhaps more...