\[ y''(x)=-\frac {y(x) (a x+b)}{4 (x-1)^2 x}-\frac {(3 x-1) y'(x)}{2 (x-1) x} \] ✓ Mathematica : cpu = 0.22539 (sec), leaf count = 893
\[\left \{\left \{y(x)\to e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} \sqrt [4]{x} c_1 \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right ),\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4};\frac {1}{2};x\right ) (x-1)^{\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2}\right )}+i e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} x^{3/4} c_2 \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {1}{2},\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2};\frac {3}{2};x\right ) (x-1)^{\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+1\right )+\frac {\left (-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1\right )^{3/2}+16 a \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+8 b \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}-2 \sqrt {-8 a-4 b-4 \sqrt {4 a^2+4 b a-a-b}+1}+4 b+1}{16 b+4}+\frac {1}{2}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.056 (sec), leaf count = 57
\[y \relax (x ) = c_{1} \LegendreP \left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )+c_{2} \LegendreQ \left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )\]