I want to convert $r=3\tan(\theta)$ to rectangular coordinates. According to this site the answer is $x^3(x(x^2 - 3y^2)\sqrt{x^2 + y^2} - y(3x^2 - y^2)) = 0$
However, when I draw implicit graph here, it misses the part of the graph, drawn in polar coordinates.
Where am I wrong?
You need to be precise with your definition of a polar plot.
When you write $r = \tan 3\theta$, you implicitly allow $r$ to be negative. Thus, the point $(r,\theta)$ may appear as $(-r,\theta+\pi)$.
When you convert to Cartesian coordinates, you take $r=\sqrt{x^2+y^2}$, thus prohibiting $r$ to become negative. No wonder, you lose some branches.
If the polar plot is what you want, you need to write $\pm\sqrt{x^2+y^2}$ or effectively get rid of $\pm$ sign by squaring again: $$ x^2(x^2 - 3y^2)^2(x^2 + y^2) = y^2(3x^2 - y^2)^2 $$