\[ y(x) \left (a x^{2 c}+b x^{c-1}\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.108631 (sec), leaf count = 312
\[\left \{\left \{y(x)\to 2^{\frac {c}{2 (c+1)}} c_1 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} U\left (\frac {\frac {\sqrt {a} c b}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (c a+a)},\frac {c}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )+2^{\frac {c}{2 (c+1)}} c_2 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} L_{-\frac {\frac {\sqrt {a} c b}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (c a+a)}}^{\frac {c}{c+1}-1}\left (\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )\right \}\right \}\] ✓ Maple : cpu = 2.377 (sec), leaf count = 91
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {c}{2}}} \left ( {{\sl W}_{{\frac {-ib}{2\,c+2}{\frac {1}{\sqrt {a}}}},\, \left ( 2\,c+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{c+1}}{c+1}\sqrt {a}}\right )}{\it \_C2}+{{\sl M}_{{\frac {-ib}{2\,c+2}{\frac {1}{\sqrt {a}}}},\, \left ( 2\,c+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{c+1}}{c+1}\sqrt {a}}\right )}{\it \_C1} \right ) \right \} \]