\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( a{x}^{2\,c}+b{x}^{c-1} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.126016 (sec), leaf count = 312 \[ \left \{\left \{y(x)\to 2^{\frac {c}{2 (c+1)}} c_1 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} U\left (\frac {\frac {\sqrt {a} c b}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (c a+a)},\frac {c}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )+2^{\frac {c}{2 (c+1)}} c_2 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} L_{-\frac {\frac {\sqrt {a} b c}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (a c+a)}}^{\frac {c}{c+1}-1}\left (\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )\right \}\right \} \]
Maple: cpu = 0.141 (sec), leaf count = 95 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {c}{2}}}{{\sl M} _{{\frac {-ib}{2\,c+2}{\frac {1}{\sqrt {a}}}},\, \left ( 2\,c+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{c+1}}{c+1}\sqrt {a}}\right )}+{ \it \_C2}\,{x}^{-{\frac {c}{2}}}{{\sl W}_{{\frac {-ib}{2\,c+2}{\frac { 1}{\sqrt {a}}}},\, \left ( 2\,c+2 \right ) ^{-1}}\left ({\frac {2\,i{x}^{ c+1}}{c+1}\sqrt {a}}\right )} \right \} \]