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As you would be playing on Immortal, which I believe is harder than King, we know that your opponents will have bonuses to progress faster.
That being said, you can do things to ensure your science output remains high, if not on top. To do this, you would have to play as a certain civilization. Babylon is your best bet for a science victory, because they get a free Great Scientist upon researching Writing, as well as a few other Unique Buildings, I believe.
Also, in Civ5, population is a great factor in science output. You can either create a bunch of cities (not recommended because of unhappiness), or build tall (small number of cities, large population). India is the best civ for this, as they have Unique Abilities for population growth and happiness, so that you can have a large, happy empire that produces a bunch of science.
Also, Korea gets +2 science for every specialist worked, so that may be an alternative, albeit not as great. The maya (Pacal) are a great civ, as you get +2 science on pyramids, which you can actually use to your advantage by building wide (many cities). The maya also get a free great person after every 394 years (Long Count calendar) after researching Theology.
One last civ that is great all around, surprisingly, is Austria. You will almost definitely start slow with this civ, but Austria has a Unique Ability which allows you to buy allied city-states (and make them your own city) using gold. If you wait until a later era (Industrial, perhaps) you can literally get a successful, strong city with a BUNCH of science output for 500 gold.