\[ y'(x)=\frac {e^{-\frac {3 x^2}{2}} x \left (3 e^{3 x^2} y(x)^3+e^{\frac {9 x^2}{2}} y(x)^3+18 e^{3 x^2} y(x)^2+9 e^{\frac {9 x^2}{2}} y(x)^2+27 e^{3 x^2} y(x)+27 e^{\frac {9 x^2}{2}} y(x)+27 e^{\frac {9 x^2}{2}}+27 y(x)^3\right )}{243 y(x)} \] ✓ Mathematica : cpu = 4.53863 (sec), leaf count = 4323
\[\text {Solve}\left [\frac {x^2}{2}+\frac {27}{2} \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 {2 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right )-3 x^2}{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+\int _1^{y(x)}\left (-27 \left (\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 {2 \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right )-3 x^2}{\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 {9801 e^{\frac {3 x^2}{2}} x^2 K[1]^4+3321 x^2 K[1]^4+123 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^4+531 x^2 \text {$\#$1}^2 K[1]^4-82 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]^4-162 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^4-39690 x^2 \text {$\#$1} 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+58806 e^{\frac {3 x^2}{2}} x^2 K[1]^3+9963 x^2 K[1]^3+1476 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^3+4779 x^2 \text {$\#$1}^2 K[1]^3-984 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]^3-1458 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^3-238140 x^2 \text {$\#$1} 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+88209 e^{\frac {3 x^2}{2}} x^2 K[1]^2+6642 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^2+14337 x^2 \text {$\#$1}^2 K[1]^2-4428 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]^2-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^2-357210 x^2 \text {$\#$1} 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+13284 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]+14337 x^2 \text {$\#$1}^2 K[1]-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]-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]+9963 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2-6642 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 {9801 e^{\frac {3 x^2}{2}} x^2 K[1]^4+3321 x^2 K[1]^4+123 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^4+531 x^2 \text {$\#$1}^2 K[1]^4-82 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]^4-162 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^4-39690 x^2 \text {$\#$1} 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+58806 e^{\frac {3 x^2}{2}} x^2 K[1]^3+9963 x^2 K[1]^3+1476 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^3+4779 x^2 \text {$\#$1}^2 K[1]^3-984 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]^3-1458 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^3-238140 x^2 \text {$\#$1} 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+88209 e^{\frac {3 x^2}{2}} x^2 K[1]^2+6642 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^2+14337 x^2 \text {$\#$1}^2 K[1]^2-4428 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]^2-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^2-357210 x^2 \text {$\#$1} 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+13284 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]+14337 x^2 \text {$\#$1}^2 K[1]-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]-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]+9963 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2-6642 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 {9801 e^{\frac {3 x^2}{2}} x^2 K[1]^4+3321 x^2 K[1]^4+123 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^4+531 x^2 \text {$\#$1}^2 K[1]^4-82 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]^4-162 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^4-39690 x^2 \text {$\#$1} 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+58806 e^{\frac {3 x^2}{2}} x^2 K[1]^3+9963 x^2 K[1]^3+1476 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^3+4779 x^2 \text {$\#$1}^2 K[1]^3-984 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]^3-1458 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^3-238140 x^2 \text {$\#$1} 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+88209 e^{\frac {3 x^2}{2}} x^2 K[1]^2+6642 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]^2+14337 x^2 \text {$\#$1}^2 K[1]^2-4428 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]^2-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]^2-357210 x^2 \text {$\#$1} 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+13284 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2 K[1]+14337 x^2 \text {$\#$1}^2 K[1]-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]-4374 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1} K[1]+2916 e^{\frac {3 x^2}{2}} \log \left (e^{\frac {3 x^2}{2}}-\text {$\#$1}\right ) \text {$\#$1} K[1]+9963 e^{\frac {3 x^2}{2}} x^2 \text {$\#$1}^2-6642 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 = 0.101 (sec), leaf count = 54
\[ \left \{ y \left ( x \right ) =-369\,{\frac {{{\rm e}^{3/2\,{x}^{2}}}}{123+123\,{{\rm e}^{3/2\,{x}^{2}}}-136\,{\it RootOf} \left ( -41\,{x}^{2}-50243409\,\int ^{{\it \_Z}}\! \left ( 9248\,{{\it \_a}}^{3}-1860867\,{\it \_a}+1860867 \right ) ^{-1}{d{\it \_a}}+27\,{\it \_C1} \right ) }} \right \} \]