This is a separable diff.eq. You may rewrite it as:
[tex]\frac{rdr}{\sqrt{a+br+cr^{2}}}=\pm{dt}[/tex]
This can be readily integrated, yielding an implicit equation for the function r(t).
As a hint, you ought to rewrite the left-hand side as:
[tex]\frac{rdr}{\sqrt{cr^{2}+br+a}}=\frac{dr}{2c}(\frac{2cr+b}{\sqrt{cr^{2}+br+a}}-\frac{b}{\sqrt{cr^{2}+br+a}})[/tex]
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