\[ 12 x^3 y''(x)-\left (6 x^2+1\right ) y^{(3)}(x)-\left (9 x^2-7\right ) x^2 y'(x)+2 \left (x^2-3\right ) x^3 y(x)+x y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 2.90353 (sec), leaf count = 270
DSolve[2*x^3*(-3 + x^2)*y[x] - x^2*(-7 + 9*x^2)*Derivative[1][y][x] + 12*x^3*Derivative[2][y][x] - (1 + 6*x^2)*Derivative[3][y][x] + x*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_3 e^{\frac {x^2}{2}} \int _1^x\frac {e^{\frac {K[1]^2}{2}} \left (\int \frac {\exp \left (\frac {1}{4} \sqrt {5} K[1]^2+\frac {1}{2} \left (-\frac {1}{2} K[1]^2-2 \log (K[1])\right )\right ) U\left (-\frac {-9+\sqrt {5}}{4 \sqrt {5}},-\frac {1}{2},-\frac {1}{2} \sqrt {5} K[1]^2\right )}{\sqrt {K[1]} \sqrt [4]{K[1]^2}} \, dK[1]\right ) K[1]}{\sqrt [4]{2}}dK[1]+c_4 e^{\frac {x^2}{2}} \int _1^x\frac {e^{\frac {K[2]^2}{2}} \left (\int \frac {\exp \left (\frac {1}{4} \sqrt {5} K[2]^2+\frac {1}{2} \left (-\frac {1}{2} K[2]^2-2 \log (K[2])\right )\right ) L_{\frac {-9+\sqrt {5}}{4 \sqrt {5}}}^{-\frac {3}{2}}\left (-\frac {1}{2} \sqrt {5} K[2]^2\right )}{\sqrt {K[2]} \sqrt [4]{K[2]^2}} \, dK[2]\right ) K[2]}{\sqrt [4]{2}}dK[2]+c_1 e^{\frac {x^2}{2}}+c_2 e^{x^2}\right \}\right \}\] ✓ Maple : cpu = 2.023 (sec), leaf count = 157
dsolve(x*diff(diff(diff(diff(y(x),x),x),x),x)-(6*x^2+1)*diff(diff(diff(y(x),x),x),x)+12*x^3*diff(diff(y(x),x),x)-(9*x^2-7)*x^2*diff(y(x),x)+2*(x^2-3)*x^3*y(x)=0,y(x))
\[y \left (x \right ) = \left (\int \frac {\WhittakerW \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}} c_{4}-\left (\int \frac {\WhittakerW \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{x^{2}} c_{4}-{\mathrm e}^{x^{2}} \left (\int \frac {\WhittakerM \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) c_{3}+\left (\int \frac {\WhittakerM \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}} c_{3}+c_{1} {\mathrm e}^{x^{2}}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}}\]