ODE
\[ 2 a (a+1) y(x)+2 (1-x) x y''(x)-(3 x+1) y'(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.231939 (sec), leaf count = 114
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (1-\sqrt {16 a^2+16 a+1}\right ),\frac {1}{4} \left (\sqrt {16 a^2+16 a+1}+1\right );-\frac {1}{2};x\right )-i c_2 x^{3/2} \, _2F_1\left (\frac {1}{4} \left (7-\sqrt {16 a^2+16 a+1}\right ),\frac {1}{4} \left (\sqrt {16 a^2+16 a+1}+7\right );\frac {5}{2};x\right )\right \}\right \}\]
Maple ✓
cpu = 0.462 (sec), leaf count = 86
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [\frac {1}{4}-\frac {\sqrt {16 a^{2}+16 a +1}}{4}, \frac {1}{4}+\frac {\sqrt {16 a^{2}+16 a +1}}{4}\right ], \left [-\frac {1}{2}\right ], x\right )+\textit {\_C2} \,x^{\frac {3}{2}} \hypergeom \left (\left [\frac {7}{4}+\frac {\sqrt {16 a^{2}+16 a +1}}{4}, \frac {7}{4}-\frac {\sqrt {16 a^{2}+16 a +1}}{4}\right ], \left [\frac {5}{2}\right ], x\right )\right ]\] Mathematica raw input
DSolve[2*a*(1 + a)*y[x] - (1 + 3*x)*y'[x] + 2*(1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(1 - Sqrt[1 + 16*a + 16*a^2])/4, (1 + Sqrt[1 + 16*a + 16*a^2])/4, -1/2, x] - I*x^(3/2)*C[2]*Hypergeometric2F1[(7 - Sqrt[1 + 16*a + 16*a^2])/4, (7 + Sqrt[1 + 16*a + 16*a^2])/4, 5/2, x]}}
Maple raw input
dsolve(2*x*(1-x)*diff(diff(y(x),x),x)-(1+3*x)*diff(y(x),x)+2*a*(1+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([1/4-1/4*(16*a^2+16*a+1)^(1/2), 1/4+1/4*(16*a^2+16*a+1)^(1/2)],[-1/2],x)+_C2*x^(3/2)*hypergeom([7/4+1/4*(16*a^2+16*a+1)^(1/2), 7/4-1/4*(16*a^2+16*a+1)^(1/2)],[5/2],x)]