ODE No. 1258

$$(ax+b)y^{\prime }(x)+cy(x)+(x-1)x{y}^{″}(x)=0$$ ✓ Mathematica : cpu = 0.108355 (sec), leaf count = 146

DSolve[c*y[x] + (b + a*x)*Derivative[1][y][x] + (-1 + x)*x*Derivative[2][y][x] == 0,y[x],x]
 

$$\left\{\{y(x)\to (-1{)}^{b+1}{c}_2{x}^{b+1}{}_2{F}_1(\frac{a}{2}+b-\frac{1}{2}\sqrt{a^2-2a-4c+1}+\frac{1}{2},\frac{a}{2}+b+\frac{1}{2}\sqrt{a^2-2a-4c+1}+\frac{1}{2};b+2;x)+c_1{}_2{F}_1(\frac{a}{2}-\frac{1}{2}\sqrt{a^2-2a-4c+1}-\frac{1}{2},\frac{a}{2}+\frac{1}{2}\sqrt{a^2-2a-4c+1}-\frac{1}{2};-b;x)\}\right\}$$ ✓ Maple : cpu = 0.039 (sec), leaf count = 110

dsolve(x*(x-1)*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x))
 

$$y\left(x\right)=c_1\mathrm{hypergeom}([-\frac{1}{2}+\frac{\sqrt{a^2-2a-4c+1}}{2}+\frac{a}{2},-\frac{1}{2}-\frac{\sqrt{a^2-2a-4c+1}}{2}+\frac{a}{2}],[-b],x)+c_2{x}^{b+1}\mathrm{hypergeom}([\frac{1}{2}-\frac{\sqrt{a^2-2a-4c+1}}{2}+\frac{a}{2}+b,\frac{1}{2}+\frac{\sqrt{a^2-2a-4c+1}}{2}+\frac{a}{2}+b],[b+2],x)$$