\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =- \left ( {\frac {1-{\it a1}-{\it b1}}{x-{\it c1}}}+{\frac {1-{\it a2}-{\it b2}}{x-{\it c2}}}+{\frac {1-{\it a3}-{\it b3}}{x-{\it c3}}} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{\frac {y \left ( x \right ) }{ \left ( x-{\it c1} \right ) \left ( x-{\it c2} \right ) \left ( x-{\it c3} \right ) } \left ( {\frac {{\it a1}\,{\it b1}\, \left ( {\it c1}-{\it c3} \right ) \left ( {\it c1}-{\it c2} \right ) }{x-{\it c1}}}+{\frac {{\it a2}\,{\it b2}\, \left ( {\it c2}-{\it c1} \right ) \left ( {\it c2}-{\it c3} \right ) }{x-{\it c2}}}+{\frac {{\it a3}\,{\it b3}\, \left ( {\it c3}-{\it c2} \right ) \left ( {\it c3}-{\it c1} \right ) }{x-{\it c3}}} \right ) }=0} \]
Mathematica: cpu = 97.102330 (sec), leaf count = 611 \[ \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 = 0.639 (sec), leaf count = 311 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it HeunG} \left ( {\frac {{ \it c1}-{\it c3}}{{\it c1}-{\it c2}}},{\frac { \left ( \left ( -{\it a3 }-2\,{\it b1}-{\it b2}+2 \right ) {\it c1}+ \left ( {\it a3}+{\it b1}-1 \right ) {\it c2}+{\it c3}\, \left ( {\it b1}+{\it b2}-1 \right ) \right ) {\it a1}- \left ( {\it b1}-1 \right ) \left ( {\it a2}+{\it b3} \right ) {\it c1}+ \left ( {\it b1}\,{\it b3}-{\it a2}\,{\it b2}+{\it b3}\, \left ( {\it a3}-1 \right ) \right ) {\it c2}+ \left ( {\it b1}\,{ \it a2}+ \left ( {\it b2}-1 \right ) {\it a2}-{\it a3}\,{\it b3} \right ) {\it c3}}{{\it c1}-{\it c2}}},{\it a1}+{\it a2}+{\it b3},2-{ \it a3}-{\it b1}-{\it b2},{\it a1}-{\it b1}+1,{\it a2}-{\it b2}+1,{ \frac {-x+{\it c1}}{{\it c1}-{\it c2}}} \right ) \left ( x-{\it c1} \right ) ^{{\it a1}} \left ( x-{\it c2} \right ) ^{{\it a2}} \left ( x-{ \it c3} \right ) ^{{\it b3}}+{\it \_C2}\,{\it HeunG} \left ( {\frac {{ \it c1}-{\it c3}}{{\it c1}-{\it c2}}},{\frac { \left ( \left ( -2\,{ \it a1}-{\it a3}-{\it b2}+2 \right ) {\it c1}+ \left ( {\it a1}+{\it a3} -1 \right ) {\it c2}+{\it c3}\, \left ( {\it a1}+{\it b2}-1 \right ) \right ) {\it b1}- \left ( {\it a2}+{\it b3} \right ) \left ( {\it a1}-1 \right ) {\it c1}+ \left ( {\it a1}\,{\it b3}-{\it a2}\,{\it b2}+{\it b3}\, \left ( {\it a3}-1 \right ) \right ) {\it c2}+{\it c3}\, \left ( { \it a1}\,{\it a2}+ \left ( {\it b2}-1 \right ) {\it a2}-{\it a3}\,{\it b3} \right ) }{{\it c1}-{\it c2}}},{\it b1}+{\it a2}+{\it b3},2-{\it a1 }-{\it a3}-{\it b2},-{\it a1}+{\it b1}+1,{\it a2}-{\it b2}+1,{\frac {- x+{\it c1}}{{\it c1}-{\it c2}}} \right ) \left ( x-{\it c1} \right ) ^{{ \it b1}} \left ( x-{\it c2} \right ) ^{{\it a2}} \left ( x-{\it c3} \right ) ^{{\it b3}} \right \} \]