\[ y(x) \left (a x^{2 c}+b x^{c-1}\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.10514 (sec), leaf count = 312
DSolve[(b*x^(-1 + c) + a*x^(2*c))*y[x] + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to 2^{\frac {c}{2 (c+1)}} c_1 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} U\left (\frac {\frac {\sqrt {a} c b}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (c a+a)},\frac {c}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )+2^{\frac {c}{2 (c+1)}} c_2 \left (x^{c+1}\right )^{\frac {c}{2 (c+1)}} x^{-c/2} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}} L_{-\frac {\frac {\sqrt {a} c b}{\sqrt {-(c+1)^2}}+\frac {\sqrt {a} b}{\sqrt {-(c+1)^2}}+a c}{2 (c a+a)}}^{\frac {c}{c+1}-1}\left (\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-c^2-2 c-1}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.218 (sec), leaf count = 91
dsolve(diff(diff(y(x),x),x)+(a*x^(2*c)+b*x^(c-1))*y(x)=0,y(x))
\[y \left (x \right ) = x^{-\frac {c}{2}} \left (\WhittakerW \left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) c_{2}+\WhittakerM \left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) c_{1}\right )\]