\[ y''(x) \left (-6 x (\text {a1}+\text {a2}+\text {a3})+3 \text {a1} \text {a2}+3 \text {a1} \text {a3}+3 \text {a2} \text {a3}+9 x^2\right )+2 (x-\text {a1}) (x-\text {a2}) (x-\text {a3}) y^{(3)}(x)-2 \left (b+\left (n^2+n-3\right ) x\right ) y'(x)-n (n+1) y(x)=0 \] ✗ Mathematica : cpu = 72.442 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{-n (n+1) \unicode {f818}(\unicode {f817})-2 \left (\unicode {f817} n^2+\unicode {f817} n-3 \unicode {f817}+b\right ) \unicode {f818}'(\unicode {f817})+3 \left (3 \unicode {f817}^2-2 \text {a1} \unicode {f817}-2 \text {a2} \unicode {f817}-2 \text {a3} \unicode {f817}+\text {a1} \text {a2}+\text {a1} \text {a3}+\text {a2} \text {a3}\right ) \unicode {f818}''(\unicode {f817})-2 (\text {a1}-\unicode {f817}) (\text {a2}-\unicode {f817}) (\text {a3}-\unicode {f817}) \unicode {f818}^{(3)}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2,\unicode {f818}''(0)=c_3\right \},\langle \langle \rangle \rangle \right )(x)\right \}\right \}\]
✓ Maple : cpu = 0.764 (sec), leaf count = 288
\[ \left \{ y \left ( x \right ) =-{\it \_C2}\, \left ( x-{\it a1} \right ) \left ( {\it HeunG} \left ( {\frac {-{\it a3}+{\it a1}}{-{\it a2}+{\it a1}}},{\frac { \left ( -{n}^{2}-n+3 \right ) {\it a1}-b}{-4\,{\it a2}+4\,{\it a1}}},{\frac {n}{2}}+1,-{\frac {n}{2}}+{\frac {1}{2}},{\frac {3}{2}},{\frac {1}{2}},{\frac {-x+{\it a1}}{-{\it a2}+{\it a1}}} \right ) \right ) ^{2}+{\it \_C3}\,{\it HeunG} \left ( {\frac {-{\it a3}+{\it a1}}{-{\it a2}+{\it a1}}},{\frac { \left ( -{n}^{2}-n+1 \right ) {\it a1}-b+{\it a2}+{\it a3}}{-4\,{\it a2}+4\,{\it a1}}},-{\frac {n}{2}},{\frac {n}{2}}+{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {-x+{\it a1}}{-{\it a2}+{\it a1}}} \right ) {\it HeunG} \left ( {\frac {-{\it a3}+{\it a1}}{-{\it a2}+{\it a1}}},{\frac { \left ( -{n}^{2}-n+3 \right ) {\it a1}-b}{-4\,{\it a2}+4\,{\it a1}}},{\frac {n}{2}}+1,-{\frac {n}{2}}+{\frac {1}{2}},{\frac {3}{2}},{\frac {1}{2}},{\frac {-x+{\it a1}}{-{\it a2}+{\it a1}}} \right ) \sqrt {-x+{\it a1}}+{\it \_C1}\, \left ( {\it HeunG} \left ( {\frac {-{\it a3}+{\it a1}}{-{\it a2}+{\it a1}}},{\frac { \left ( -{n}^{2}-n+1 \right ) {\it a1}-b+{\it a2}+{\it a3}}{-4\,{\it a2}+4\,{\it a1}}},-{\frac {n}{2}},{\frac {n}{2}}+{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}},{\frac {-x+{\it a1}}{-{\it a2}+{\it a1}}} \right ) \right ) ^{2} \right \} \]