\[ f\left (x-\frac {3}{2} y'(x)^2\right )+y'(x)^3-y(x)=0 \] ✓ Mathematica : cpu = 0.0339602 (sec), leaf count = 102
DSolve[f[x - (3*Derivative[1][y][x]^2)/2] - y[x] + Derivative[1][y][x]^3 == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{9} \left (9 f(c_1)+2 \sqrt {6} x \sqrt {x-c_1}-2 \sqrt {6} c_1 \sqrt {x-c_1}\right ),y(x)\to \frac {1}{9} \left (9 f(c_1)-2 \sqrt {6} x \sqrt {x-c_1}+2 \sqrt {6} c_1 \sqrt {x-c_1}\right )\right \}\right \}\] ✓ Maple : cpu = 0.141 (sec), leaf count = 41
dsolve(f(x-3/2*diff(y(x),x)^2)+diff(y(x),x)^3-y(x)=0,y(x))
\[y \left (x \right ) = f \left (c_{1}\right )-\frac {2 \sqrt {6}\, \sqrt {\left (-c_{1}+x \right )^{3}}}{9}\]