\[ y''(x)-\frac {h\left (\frac {y(x)}{\sqrt {x}}\right )}{x^{3/2}}=0 \] ✓ Mathematica : cpu = 945.186 (sec), leaf count = 734
\[\left \{\text {Solve}\left [\int _1^{y(x)} \frac {2}{\sqrt {x} \sqrt {\frac {8 x \int _1^{\frac {K[3]}{\sqrt {x}}} h(K[2]) \, dK[2]+K[3]^2+4 c_1 x}{x}}} \, dK[3]-\int _1^x \left (\int _1^{y(x)} \left (-\frac {\frac {-\frac {4 K[3] h\left (\frac {K[3]}{\sqrt {K[4]}}\right )}{\sqrt {K[4]}}+8 \int _1^{\frac {K[3]}{\sqrt {K[4]}}} h(K[2]) \, dK[2]+4 c_1}{K[4]}-\frac {4 c_1 K[4]+8 K[4] \left (\int _1^{\frac {K[3]}{\sqrt {K[4]}}} h(K[2]) \, dK[2]\right )+K[3]^2}{K[4]^2}}{\sqrt {K[4]} \left (\frac {4 c_1 K[4]+8 K[4] \left (\int _1^{\frac {K[3]}{\sqrt {K[4]}}} h(K[2]) \, dK[2]\right )+K[3]^2}{K[4]}\right ){}^{3/2}}-\frac {1}{K[4]^{3/2} \sqrt {\frac {4 c_1 K[4]+8 K[4] \left (\int _1^{\frac {K[3]}{\sqrt {K[4]}}} h(K[2]) \, dK[2]\right )+K[3]^2}{K[4]}}}\right ) \, dK[3]+\frac {2 \left (\frac {y(x)}{2 \sqrt {K[4]}}-\frac {\sqrt {4 \int _1^{\frac {y(x)}{\sqrt {K[4]}}} h(K[2]) \, dK[2]+\frac {y(x)^2}{2 K[4]}+2 c_1}}{\sqrt {2}}\right )}{K[4] \sqrt {\frac {4 c_1 K[4]+8 K[4] \left (\int _1^{\frac {y(x)}{\sqrt {K[4]}}} h(K[2]) \, dK[2]\right )+y(x)^2}{K[4]}}}\right ) \, dK[4]=c_2,y(x)\right ],\text {Solve}\left [\int _1^{y(x)} -\frac {2}{\sqrt {x} \sqrt {\frac {8 x \int _1^{\frac {K[5]}{\sqrt {x}}} h(K[2]) \, dK[2]+K[5]^2+4 c_1 x}{x}}} \, dK[5]-\int _1^x \left (\int _1^{y(x)} \left (\frac {\frac {-\frac {4 K[5] h\left (\frac {K[5]}{\sqrt {K[6]}}\right )}{\sqrt {K[6]}}+8 \int _1^{\frac {K[5]}{\sqrt {K[6]}}} h(K[2]) \, dK[2]+4 c_1}{K[6]}-\frac {4 c_1 K[6]+8 K[6] \left (\int _1^{\frac {K[5]}{\sqrt {K[6]}}} h(K[2]) \, dK[2]\right )+K[5]^2}{K[6]^2}}{\sqrt {K[6]} \left (\frac {4 c_1 K[6]+8 K[6] \left (\int _1^{\frac {K[5]}{\sqrt {K[6]}}} h(K[2]) \, dK[2]\right )+K[5]^2}{K[6]}\right ){}^{3/2}}+\frac {1}{K[6]^{3/2} \sqrt {\frac {4 c_1 K[6]+8 K[6] \left (\int _1^{\frac {K[5]}{\sqrt {K[6]}}} h(K[2]) \, dK[2]\right )+K[5]^2}{K[6]}}}\right ) \, dK[5]-\frac {2 \left (\frac {\sqrt {4 \int _1^{\frac {y(x)}{\sqrt {K[6]}}} h(K[2]) \, dK[2]+\frac {y(x)^2}{2 K[6]}+2 c_1}}{\sqrt {2}}+\frac {y(x)}{2 \sqrt {K[6]}}\right )}{K[6] \sqrt {\frac {4 c_1 K[6]+8 K[6] \left (\int _1^{\frac {y(x)}{\sqrt {K[6]}}} h(K[2]) \, dK[2]\right )+y(x)^2}{K[6]}}}\right ) \, dK[6]=c_2,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.289 (sec), leaf count = 92
\[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) -2\,\int ^{{\it \_Z}}\!{\frac {1}{\sqrt {{\it \_C1}+8\,\int \!h \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{{\it \_g}}^{2}}}}{d{\it \_g}}+2\,{\it \_C2} \right ) \sqrt {x},y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +2\,\int ^{{\it \_Z}}\!{\frac {1}{\sqrt {{\it \_C1}+8\,\int \!h \left ( {\it \_g} \right ) \,{\rm d}{\it \_g}+{{\it \_g}}^{2}}}}{d{\it \_g}}+2\,{\it \_C2} \right ) \sqrt {x},y \left ( x \right ) ={\it RootOf} \left ( {\it \_Z}\,{x}^{{\frac {3}{2}}}+4\,h \left ( {\frac {{\it \_Z}}{\sqrt {x}}} \right ) {x}^{2} \right ) \right \} \]