\[ -f(x)+x^2 y^{(3)}(x)+\left (x^2+2\right ) y'(x)+4 x y''(x)+3 x y(x)=0 \] ✓ Mathematica : cpu = 8.32757 (sec), leaf count = 868
\[\left \{\left \{y(x)\to J_0(x) c_1+2 Y_0(x) c_2+J_0(x) \int _1^x -\frac {18 f(K[1]) \left (2 Y_0(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) (Y_0(K[1])-Y_1(K[1]) K[1])\right )}{K[1] \left (2 \left (8 (J_1(K[1]) Y_0(K[1])-J_0(K[1]) Y_1(K[1])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) (2 J_1(K[1]) Y_0(K[1])-J_2(K[1]) K[1] Y_0(K[1])+J_0(K[1]) (Y_2(K[1]) K[1]-2 Y_1(K[1]))) K[1]^2\right )+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) \left (J_2(K[1]) K[1] (Y_1(K[1]) K[1]-Y_0(K[1]))+J_1(K[1]) \left (Y_0(K[1]) \left (K[1]^2+4\right )-Y_2(K[1]) K[1]^2\right )+J_0(K[1]) \left (Y_2(K[1]) K[1]-Y_1(K[1]) \left (K[1]^2+4\right )\right )\right )\right )} \, dK[1]+2 Y_0(x) \int _1^x \frac {9 f(K[2]) \left (2 J_0(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) (J_0(K[2])-J_1(K[2]) K[2])\right )}{K[2] \left (2 \left (8 (J_1(K[2]) Y_0(K[2])-J_0(K[2]) Y_1(K[2])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) (2 J_1(K[2]) Y_0(K[2])-J_2(K[2]) K[2] Y_0(K[2])+J_0(K[2]) (Y_2(K[2]) K[2]-2 Y_1(K[2]))) K[2]^2\right )+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) \left (J_2(K[2]) K[2] (Y_1(K[2]) K[2]-Y_0(K[2]))+J_1(K[2]) \left (Y_0(K[2]) \left (K[2]^2+4\right )-Y_2(K[2]) K[2]^2\right )+J_0(K[2]) \left (Y_2(K[2]) K[2]-Y_1(K[2]) \left (K[2]^2+4\right )\right )\right )\right )} \, dK[2]+\frac {2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \left (c_3+\int _1^x \frac {9 (J_1(K[3]) Y_0(K[3])-J_0(K[3]) Y_1(K[3])) f(K[3]) K[3]}{2 \left (8 (J_1(K[3]) Y_0(K[3])-J_0(K[3]) Y_1(K[3])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) (2 J_1(K[3]) Y_0(K[3])-J_2(K[3]) K[3] Y_0(K[3])+J_0(K[3]) (Y_2(K[3]) K[3]-2 Y_1(K[3]))) K[3]^2\right )+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) \left (J_2(K[3]) K[3] (Y_1(K[3]) K[3]-Y_0(K[3]))+J_1(K[3]) \left (Y_0(K[3]) \left (K[3]^2+4\right )-Y_2(K[3]) K[3]^2\right )+J_0(K[3]) \left (Y_2(K[3]) K[3]-Y_1(K[3]) \left (K[3]^2+4\right )\right )\right )} \, dK[3]\right )}{x}\right \}\right \}\]
✓ Maple : cpu = 0.367 (sec), leaf count = 1033
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( \int \!18\,{\frac { \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) f \left ( x \right ) }{ \left ( \left ( -18\,{x}^{2}{{\sl J}_{0}\left (x\right )}-72\,x{{\sl J}_{1}\left (x\right )}+54\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\,{x}^{2} \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 18\,{x}^{2}{{\sl J}_{0}\left (x\right )}+144\,x{{\sl J}_{1}\left (x\right )}-126\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+16\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+72\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}xG^{3, 1}_{1, 3}\left ({\frac {{x}^{2}}{4}}\, \Big \vert \,^{-{\frac {1}{2}}}_{0, 0, -{\frac {1}{2}}}\right )x+{\it \_C3}\,G^{3, 1}_{1, 3}\left ({\frac {{x}^{2}}{4}}\, \Big \vert \,^{-{\frac {1}{2}}}_{0, 0, -{\frac {1}{2}}}\right )x-\int \!9\,{\frac { \left ( \left ( {x}^{2}{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}-{\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+{\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right ) \right ) f \left ( x \right ) }{ \left ( \left ( -9\,{x}^{2}{{\sl J}_{0}\left (x\right )}-36\,x{{\sl J}_{1}\left (x\right )}+27\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+4\,{x}^{2} \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 9\,{x}^{2}{{\sl J}_{0}\left (x\right )}+72\,x{{\sl J}_{1}\left (x\right )}-63\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+36\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}x{{\sl J}_{0}\left (x\right )}x+{\it \_C1}\,{{\sl J}_{0}\left (x\right )}x-\int \!-9\,{\frac {x \left ( \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+2\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right ){{\sl J}_{0}\left (x\right )} \right ) f \left ( x \right ) }{ \left ( \left ( -18\,{x}^{2}{{\sl J}_{0}\left (x\right )}-72\,x{{\sl J}_{1}\left (x\right )}+54\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\,{x}^{2} \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 18\,{x}^{2}{{\sl J}_{0}\left (x\right )}+144\,x{{\sl J}_{1}\left (x\right )}-126\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+16\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+72\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}x{\mbox {$_1$F$_2$}(1;\,{\frac {1}{2}},{\frac {1}{2}};\,-{\frac {{x}^{2}}{4}})}+{\it \_C2}\,{\mbox {$_1$F$_2$}(1;\,{\frac {1}{2}},{\frac {1}{2}};\,-{\frac {{x}^{2}}{4}})} \right ) } \right \} \]