In physics, the radial velocity of a particle around a black hole is given by this equation:
$$ \left(\frac{dr}{d\tau}\right)^2 = \left({E\over m}\right)^2 - \left(1-{2M\over r}\right)\left(1+ \left({l\over r}\right)^2\right) $$
which can be rewritten as
$$ dr = \sqrt{\left({E\over m}\right)^2 - \left(1-{2M\over r}\right)\left(1+ \left({l\over r}\right)^2\right)}\space\space d\tau $$
So
$$ r(\tau) = \int \sqrt{\left({E\over m}\right)^2 - \left(1-{2M\over r}\right)\left(1+ \left({l\over r}\right)^2\right)}\space\space d\tau + K $$
Where
$$ {E\over m} = \sqrt{1-\left({2M\over r}\right)} \space \gamma_0 $$
$$ l = r^2_0 \left( {d\phi\over d\tau}\right) $$
But I have absolutely no idea how to integrate this, as there are terms on the RHS that contain $r$.
I apologise if this is an elementary question, but I haven't done much of this.
PS. I realise this question may seem better suited to physics.SE, but my question is purely from a mathematical perspective.