\[ y''(x)=-y'(x) \left (\frac {-\text {a1}-\text {b1}+1}{x-\text {c1}}+\frac {-\text {a2}-\text {b2}+1}{x-\text {c2}}+\frac {-\text {a3}-\text {b3}+1}{x-\text {c3}}\right )-\frac {y(x) \left (\frac {\text {a1} \text {b1} (\text {c1}-\text {c2}) (\text {c1}-\text {c3})}{x-\text {c1}}+\frac {\text {a2} \text {b2} (\text {c2}-\text {c1}) (\text {c2}-\text {c3})}{x-\text {c2}}+\frac {\text {a3} \text {b3} (\text {c3}-\text {c1}) (\text {c3}-\text {c2})}{x-\text {c3}}\right )}{(x-\text {c1}) (x-\text {c2}) (x-\text {c3})} \] ✗ Mathematica : cpu = 51.165 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(\text {c1}-\unicode {f817})^2 (\text {c2}-\unicode {f817})^2 \unicode {f818}''(\unicode {f817}) (\text {c3}-\unicode {f817})^2+(\text {c1}-\unicode {f817}) (\text {c2}-\unicode {f817}) \left (\text {a1} \unicode {f817}^2+\text {a2} \unicode {f817}^2+\text {a3} \unicode {f817}^2+\text {b1} \unicode {f817}^2+\text {b2} \unicode {f817}^2+\text {b3} \unicode {f817}^2-3 \unicode {f817}^2-\text {a2} \text {c1} \unicode {f817}-\text {a3} \text {c1} \unicode {f817}-\text {b2} \text {c1} \unicode {f817}-\text {b3} \text {c1} \unicode {f817}+2 \text {c1} \unicode {f817}-\text {a1} \text {c2} \unicode {f817}-\text {a3} \text {c2} \unicode {f817}-\text {b1} \text {c2} \unicode {f817}-\text {b3} \text {c2} \unicode {f817}+2 \text {c2} \unicode {f817}-\text {a1} \text {c3} \unicode {f817}-\text {a2} \text {c3} \unicode {f817}-\text {b1} \text {c3} \unicode {f817}-\text {b2} \text {c3} \unicode {f817}+2 \text {c3} \unicode {f817}+\text {a3} \text {c1} \text {c2}+\text {b3} \text {c1} \text {c2}-\text {c1} \text {c2}+\text {a2} \text {c1} \text {c3}+\text {b2} \text {c1} \text {c3}-\text {c1} \text {c3}+\text {a1} \text {c2} \text {c3}+\text {b1} \text {c2} \text {c3}-\text {c2} \text {c3}\right ) \unicode {f818}'(\unicode {f817}) (\text {c3}-\unicode {f817})+\left (\text {a1} \text {b1} \text {c1}^2 \unicode {f817}^2+\text {a2} \text {b2} \text {c2}^2 \unicode {f817}^2+\text {a3} \text {b3} \text {c3}^2 \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c2} \unicode {f817}^2-\text {a2} \text {b2} \text {c1} \text {c2} \unicode {f817}^2+\text {a3} \text {b3} \text {c1} \text {c2} \unicode {f817}^2-\text {a1} \text {b1} \text {c1} \text {c3} \unicode {f817}^2+\text {a2} \text {b2} \text {c1} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c1} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c2} \text {c3} \unicode {f817}^2-\text {a2} \text {b2} \text {c2} \text {c3} \unicode {f817}^2-\text {a3} \text {b3} \text {c2} \text {c3} \unicode {f817}^2+\text {a1} \text {b1} \text {c1} \text {c2}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c2}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \unicode {f817}+\text {a1} \text {b1} \text {c1} \text {c3}^2 \unicode {f817}-\text {a2} \text {b2} \text {c1} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c1} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c2} \text {c3}^2 \unicode {f817}+\text {a2} \text {b2} \text {c2} \text {c3}^2 \unicode {f817}-\text {a3} \text {b3} \text {c2} \text {c3}^2 \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c2} \unicode {f817}+\text {a2} \text {b2} \text {c1}^2 \text {c2} \unicode {f817}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \unicode {f817}-\text {a1} \text {b1} \text {c1}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c1}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c3} \unicode {f817}-\text {a1} \text {b1} \text {c2}^2 \text {c3} \unicode {f817}-\text {a2} \text {b2} \text {c2}^2 \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c2}^2 \text {c3} \unicode {f817}+2 \text {a1} \text {b1} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a2} \text {b2} \text {c1} \text {c2} \text {c3} \unicode {f817}+2 \text {a3} \text {b3} \text {c1} \text {c2} \text {c3} \unicode {f817}+\text {a3} \text {b3} \text {c1}^2 \text {c2}^2+\text {a2} \text {b2} \text {c1}^2 \text {c3}^2+\text {a1} \text {b1} \text {c2}^2 \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2} \text {c3}^2-\text {a2} \text {b2} \text {c1} \text {c2} \text {c3}^2+\text {a3} \text {b3} \text {c1} \text {c2} \text {c3}^2-\text {a1} \text {b1} \text {c1} \text {c2}^2 \text {c3}+\text {a2} \text {b2} \text {c1} \text {c2}^2 \text {c3}-\text {a3} \text {b3} \text {c1} \text {c2}^2 \text {c3}+\text {a1} \text {b1} \text {c1}^2 \text {c2} \text {c3}-\text {a2} \text {b2} \text {c1}^2 \text {c2} \text {c3}-\text {a3} \text {b3} \text {c1}^2 \text {c2} \text {c3}\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
✓ Maple : cpu = 2.764 (sec), leaf count = 298
\[ \left \{ y \left ( x \right ) = \left ( x-{\it c2} \right ) ^{{\it a2}} \left ( x-{\it c3} \right ) ^{{\it b3}} \left ( \left ( x-{\it c1} \right ) ^{{\it b1}}{\it HeunG} \left ( {\frac {{\it c1}-{\it c3}}{{\it c1}-{\it c2}}},{\frac { \left ( \left ( -2\,{\it a1}-{\it a3}-{\it b2}+2 \right ) {\it c1}+ \left ( {\it a1}+{\it a3}-1 \right ) {\it c2}+{\it c3}\, \left ( {\it a1}+{\it b2}-1 \right ) \right ) {\it b1}- \left ( {\it a2}+{\it b3} \right ) \left ( {\it a1}-1 \right ) {\it c1}+ \left ( {\it a1}\,{\it b3}-{\it a2}\,{\it b2}+{\it b3}\, \left ( {\it a3}-1 \right ) \right ) {\it c2}+ \left ( {\it a1}\,{\it a2}+ \left ( {\it b2}-1 \right ) {\it a2}-{\it a3}\,{\it b3} \right ) {\it c3}}{{\it c1}-{\it c2}}},{\it a2}+{\it b3}+{\it b1},-{\it a3}-{\it b2}+2-{\it a1},-{\it a1}+{\it b1}+1,{\it a2}-{\it b2}+1,{\frac {-x+{\it c1}}{{\it c1}-{\it c2}}} \right ) {\it \_C2}+{\it HeunG} \left ( {\frac {{\it c1}-{\it c3}}{{\it c1}-{\it c2}}},{\frac { \left ( \left ( -{\it a3}-2\,{\it b1}-{\it b2}+2 \right ) {\it c1}+ \left ( {\it a3}+{\it b1}-1 \right ) {\it c2}+{\it c3}\, \left ( {\it b1}+{\it b2}-1 \right ) \right ) {\it a1}- \left ( {\it b1}-1 \right ) \left ( {\it a2}+{\it b3} \right ) {\it c1}+ \left ( {\it b1}\,{\it b3}-{\it a2}\,{\it b2}+{\it b3}\, \left ( {\it a3}-1 \right ) \right ) {\it c2}+ \left ( {\it a2}\,{\it b1}+ \left ( {\it b2}-1 \right ) {\it a2}-{\it a3}\,{\it b3} \right ) {\it c3}}{{\it c1}-{\it c2}}},{\it a1}+{\it a2}+{\it b3},-{\it b1}-{\it a3}-{\it b2}+2,{\it a1}-{\it b1}+1,{\it a2}-{\it b2}+1,{\frac {-x+{\it c1}}{{\it c1}-{\it c2}}} \right ) \left ( x-{\it c1} \right ) ^{{\it a1}}{\it \_C1} \right ) \right \} \]