Volume enclosed by Implicit Surface

Question

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta \cdot \cos\varphi$, $y=r\cdot\sin\vartheta\cdot\sin\varphi$, $z=r\cdot\cos\vartheta$ and could solve for $r$: $$r=\sin ^2\vartheta\cdot\cos\vartheta\cdot\sin\varphi\cdot\cos\varphi$$ With this $r$ i could parametrize my $z$ giving me $$z(\vartheta,\varphi)=\sin^2\vartheta\cos^2\vartheta\cos\varphi\sin\varphi$$ The volume of a solid can be calculated with a Surface Integral so $\iint_S z \ dx \ dy $ should give me what i want. The Functional Determninant of going from cartesian to spherical is: $$\frac{D(x,y)}{D(\vartheta,\varphi)}=\begin{vmatrix}x_\vartheta & x_\varphi\\y_\vartheta & y_\varphi\end{vmatrix}=r^2\cdot\cos\vartheta\cdot\sin\vartheta$$ So my Integral would look like $$V=\int_{\varphi=o}^{2\pi}\int_{\vartheta=0}^\pi \sin^7\vartheta\cos^5\vartheta \ \cos^3\varphi\sin^3\varphi \ d\vartheta \ d\varphi$$ Where did i go wrong?