How would you integrate 1 / (v+v^3) ?
I thought it would be [ 1 / (1+3v^2)] ln |v+v^3| by anti-differentiation but I don't think that is correct?
It appears that you tried to do this:
so that you could use a u substitution.
This is incorrect because you can only pull constants out of the integral, not functions.
Like Mr F said, factorise the denominator then use partial fractions. You should find that your answer has a logarithm term and an arctangent term.