\[ y'(x)=\frac {e^{-\frac {3 x^2}{2}} x \left (3 e^{3 x^2} y(x)^3+e^{\frac {9 x^2}{2}} y(x)^3+18 e^{3 x^2} y(x)^2+9 e^{\frac {9 x^2}{2}} y(x)^2+27 e^{3 x^2} y(x)+27 e^{\frac {9 x^2}{2}} y(x)+27 e^{\frac {9 x^2}{2}}+27 y(x)^3\right )}{243 y(x)} \] ✓ Mathematica : cpu = 2.81179 (sec), leaf count = 3303
DSolve[Derivative[1][y][x] == (x*(27*E^((9*x^2)/2) + 27*E^(3*x^2)*y[x] + 27*E^((9*x^2)/2)*y[x] + 18*E^(3*x^2)*y[x]^2 + 9*E^((9*x^2)/2)*y[x]^2 + 27*y[x]^3 + 3*E^(3*x^2)*y[x]^3 + E^((9*x^2)/2)*y[x]^3))/(243*E^((3*x^2)/2)*y[x]),y[x],x]
\[\text {Solve}\left [27 \text {RootSum}\left [y(x)^3 \text {$\#$1}^3+9 y(x)^2 \text {$\#$1}^3+27 y(x) \text {$\#$1}^3+27 \text {$\#$1}^3+3 y(x)^3 \text {$\#$1}^2+18 y(x)^2 \text {$\#$1}^2+27 y(x) \text {$\#$1}^2-243 y(x)^3 \text {$\#$1}-729 y(x)^2 \text {$\#$1}+27 y(x)^3\& ,\frac {\log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right )}{y(x)^2 \text {$\#$1}^2+6 y(x) \text {$\#$1}^2+9 \text {$\#$1}^2+2 y(x)^2 \text {$\#$1}+6 y(x) \text {$\#$1}-81 y(x)^2}\& \right ] y(x)^2+\frac {1}{3} \log \left (e^{\frac {3 x^2}{2}}\right )+\int _1^{y(x)}\left (-27 \left (2 \text {RootSum}\left [\text {$\#$1}^3 K[1]^3+3 \text {$\#$1}^2 K[1]^3-243 \text {$\#$1} K[1]^3+27 K[1]^3+9 \text {$\#$1}^3 K[1]^2+18 \text {$\#$1}^2 K[1]^2-729 \text {$\#$1} K[1]^2+27 \text {$\#$1}^3 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1}^3\& ,\frac {\log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right )}{\text {$\#$1}^2 K[1]^2+2 \text {$\#$1} K[1]^2-81 K[1]^2+6 \text {$\#$1}^2 K[1]+6 \text {$\#$1} K[1]+9 \text {$\#$1}^2}\& \right ]+\frac {\text {RootSum}\left [\text {$\#$1}^3 K[1]^3+3 \text {$\#$1}^2 K[1]^3-243 \text {$\#$1} K[1]^3+27 K[1]^3+9 \text {$\#$1}^3 K[1]^2+18 \text {$\#$1}^2 K[1]^2-729 \text {$\#$1} K[1]^2+27 \text {$\#$1}^3 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1}^3\& ,\frac {-82 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-354 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-6534 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4-2214 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4+108 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4+26460 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4-984 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3-3186 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3+531 \text {$\#$1}^2 K[1]^3-39204 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3-6642 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3+972 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3+158760 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3-39690 \text {$\#$1} K[1]^3+3321 K[1]^3-4428 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2+3186 \text {$\#$1}^2 K[1]^2-58806 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^2+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2+238140 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2-119070 \text {$\#$1} K[1]^2-8856 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]+4779 \text {$\#$1}^2 K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]-6642 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-43677 e^{\frac {3 x^2}{2}} K[1]^3+1681 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^3+7257 \text {$\#$1}^2 K[1]^3-2214 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^3-364806 \text {$\#$1} K[1]^3+45387 K[1]^3-131031 e^{\frac {3 x^2}{2}} K[1]^2+15129 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^2+43542 \text {$\#$1}^2 K[1]^2-13284 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^2-1094418 \text {$\#$1} K[1]^2+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]+65313 \text {$\#$1}^2 K[1]-19926 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2}\& \right ]}{6 K[1]^3}\right ) K[1]-\frac {243 e^{\frac {3 x^2}{2}} K[1]}{-243 e^{\frac {3 x^2}{2}} K[1]^3+3 e^{3 x^2} K[1]^3+e^{\frac {9 x^2}{2}} K[1]^3+27 K[1]^3-729 e^{\frac {3 x^2}{2}} K[1]^2+18 e^{3 x^2} K[1]^2+9 e^{\frac {9 x^2}{2}} K[1]^2+27 e^{3 x^2} K[1]+27 e^{\frac {9 x^2}{2}} K[1]+27 e^{\frac {9 x^2}{2}}}-\frac {81 \text {RootSum}\left [\text {$\#$1}^3 K[1]^3+3 \text {$\#$1}^2 K[1]^3-243 \text {$\#$1} K[1]^3+27 K[1]^3+9 \text {$\#$1}^3 K[1]^2+18 \text {$\#$1}^2 K[1]^2-729 \text {$\#$1} K[1]^2+27 \text {$\#$1}^3 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1}^3\& ,\frac {-82 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-354 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-6534 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4-2214 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4+108 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4+26460 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4-984 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3-3186 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3+531 \text {$\#$1}^2 K[1]^3-39204 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3-6642 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3+972 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3+158760 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3-39690 \text {$\#$1} K[1]^3+3321 K[1]^3-4428 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2+3186 \text {$\#$1}^2 K[1]^2-58806 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^2+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2+238140 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2-119070 \text {$\#$1} K[1]^2-8856 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]+4779 \text {$\#$1}^2 K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]-6642 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-43677 e^{\frac {3 x^2}{2}} K[1]^3+1681 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^3+7257 \text {$\#$1}^2 K[1]^3-2214 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^3-364806 \text {$\#$1} K[1]^3+45387 K[1]^3-131031 e^{\frac {3 x^2}{2}} K[1]^2+15129 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^2+43542 \text {$\#$1}^2 K[1]^2-13284 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^2-1094418 \text {$\#$1} K[1]^2+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]+65313 \text {$\#$1}^2 K[1]-19926 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2}\& \right ]}{2 (K[1]+3) K[1]^3}+\frac {27 \text {RootSum}\left [\text {$\#$1}^3 K[1]^3+3 \text {$\#$1}^2 K[1]^3-243 \text {$\#$1} K[1]^3+27 K[1]^3+9 \text {$\#$1}^3 K[1]^2+18 \text {$\#$1}^2 K[1]^2-729 \text {$\#$1} K[1]^2+27 \text {$\#$1}^3 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1}^3\& ,\frac {-82 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-354 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^4-6534 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4-2214 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^4+108 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4+26460 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^4-984 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3-3186 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^3+531 \text {$\#$1}^2 K[1]^3-39204 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3-6642 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^3+972 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3+158760 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^3-39690 \text {$\#$1} K[1]^3+3321 K[1]^3-4428 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]^2+3186 \text {$\#$1}^2 K[1]^2-58806 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) K[1]^2+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2+238140 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]^2-119070 \text {$\#$1} K[1]^2-8856 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]-9558 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2 K[1]+4779 \text {$\#$1}^2 K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]-6642 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-43677 e^{\frac {3 x^2}{2}} K[1]^3+1681 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^3+7257 \text {$\#$1}^2 K[1]^3-2214 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^3-364806 \text {$\#$1} K[1]^3+45387 K[1]^3-131031 e^{\frac {3 x^2}{2}} K[1]^2+15129 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]^2+43542 \text {$\#$1}^2 K[1]^2-13284 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]^2-1094418 \text {$\#$1} K[1]^2+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2 K[1]+65313 \text {$\#$1}^2 K[1]-19926 e^{\frac {3 x^2}{2}} \text {$\#$1} K[1]+45387 e^{\frac {3 x^2}{2}} \text {$\#$1}^2}\& \right ]}{2 K[1]^3}\right )dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.104 (sec), leaf count = 54
dsolve(diff(y(x),x) = 1/243*(27*y(x)^3+27*exp(3*x^2)*y(x)+18*exp(3*x^2)*y(x)^2+3*y(x)^3*exp(3*x^2)+27*exp(9/2*x^2)+27*exp(9/2*x^2)*y(x)+9*exp(9/2*x^2)*y(x)^2+exp(9/2*x^2)*y(x)^3)*exp(3*x^2)*x/y(x)/exp(9/2*x^2),y(x))
\[y \left (x \right ) = -\frac {369 \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{123+123 \,{\mathrm e}^{\frac {3 x^{2}}{2}}-136 \operatorname {RootOf}\left (-41 x^{2}-50243409 \left (\int _{}^{\textit {\_Z}}\frac {1}{9248 \textit {\_a}^{3}-1860867 \textit {\_a} +1860867}d \textit {\_a} \right )+27 c_{1}\right )}\]