\[ \boxed { x \left ( 1+x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( ax+b \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +cy \left ( x \right ) =0} \]
Mathematica: cpu = 0.161521 (sec), leaf count = 151 \[ \left \{\left \{y(x)\to c_2 x^{1-b} \, _2F_1\left (\frac {a}{2}-b-\frac {1}{2} \sqrt {a^2-2 a-4 c+1}+\frac {1}{2},\frac {a}{2}-b+\frac {1}{2} \sqrt {a^2-2 a-4 c+1}+\frac {1}{2};2-b;-x\right )+c_1 \, _2F_1\left (\frac {a}{2}-\frac {1}{2} \sqrt {a^2-2 a-4 c+1}-\frac {1}{2},\frac {a}{2}+\frac {1}{2} \sqrt {a^2-2 a-4 c+1}-\frac {1}{2};b;-x\right )\right \}\right \} \]
Maple: cpu = 0.032 (sec), leaf count = 124 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_2$F$_1$}(-{\frac {1}{2}}-{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\,c+1}}+{\frac {a}{2}},-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\,c+1}}+{\frac {a}{2}};\,a-b;\,1+x)} +{\it \_C2}\, \left ( 1+x \right ) ^{-a+b+1} {\mbox {$_2$F$_1$}({\frac {1}{2}}+{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\,c+1}}-{\frac {a}{2}}+b,{\frac {1}{2}}-{\frac {1}{2}\sqrt {{a}^{2}-2\,a-4\,c+1}}-{\frac {a}{2}}+b;\,2-a+b;\,1+x)} \right \} \]