## Integrating non-constant acceleration to give time

1. The problem statement, all variables and given/known data
Not specifically a homework assignment, but for a personal project - but it's almost entirely parallel with my Physics course at the moment, and is mostly a homework-style question!

I have an object acclerating due to a force, experiencing friction. Both the accelerating and friction forces depend on velocity.

I would expect the result for v to tend to a certain value as time increases, similar to terminal velocity.

I know this should involve integrating acceleration with respect to time - but the combination of questionable integration confidence and a cold mean I just can't fathom the next step.

2. Relevant equations
Accelerating force = $$k/v$$ (decreases as v increases)
Friction force = $$a + bv + cv^{2}$$ (increases as v increases)
Total force = $$k/v - (a + bv + cv^{2})$$
Acceleration = $$\sum F/m$$
Velocity = $$\int a = \int (k/v - (a + bv + cv^2))/m$$

3. The attempt at a solution
My attempts at integration end with a 3rd degree polynomial:
$$(k\;ln(v) - (av + bv^2/2 + cv^3/3))/m$$
Whereas I expect t in the equation, and this does not lead to a limit (for sufficiently large values it gives negative speed).

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