\[ y''(x)=-y'(x) \left (\frac {\text {b1} (-\text {al1}-\text {bl1}+1)}{\text {b1} x-\text {a1}}+\frac {\text {b2} (-\text {al2}-\text {bl2}+1)}{\text {b2} x-\text {a2}}+\frac {\text {b3} (-\text {al3}-\text {bl3}+1)}{\text {b3} x-\text {a3}}\right )-\frac {y(x) \left (\frac {\text {al1} \text {bl1} (\text {a1} \text {b2}-\text {a2} \text {b1}) (\text {a3} \text {b1}-\text {a1} \text {b3})}{\text {b1} x-\text {a1}}+\frac {\text {al2} \text {bl2} (\text {a1} \text {b2}-\text {a2} \text {b1}) (\text {a2} \text {b3}-\text {a3} \text {b2})}{\text {b2} x-\text {a2}}+\frac {\text {al3} \text {bl3} (\text {a3} \text {b1}-\text {a1} \text {b3}) (\text {a2} \text {b3}-\text {a3} \text {b2})}{\text {b3} x-\text {a3}}\right )}{(\text {b1} x-\text {a1}) (\text {b2} x-\text {a2}) (\text {b3} x-\text {a3})} \] ✓ Mathematica : cpu = 88.7641 (sec), leaf count = 2002
\[\left \{\left \{y(x)\to \left (x-\frac {\text {a2}}{\text {b2}}\right )^{\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )} \left (x-\frac {\text {a3}}{\text {b3}}\right )^{\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )} c_1 \text {HeunG}\left [\frac {\frac {\text {a1}}{\text {b1}}-\frac {\text {a3}}{\text {b3}}}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}},\frac {\frac {\text {a1} \left (\frac {1}{2} \left (-\text {al1}-\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+1\right ) \left (\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )\right )}{\text {b1}}+\frac {\text {a3} \left (\frac {1}{4} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-1\right ) \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-\frac {1}{4} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ) \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )\right )}{\text {b3}}+\frac {\text {a2} \left (\frac {1}{4} \left (-\text {al2}-\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right ) \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{4} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ) \left (\frac {1}{2} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )-1\right )\right )}{\text {b2}}+\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\frac {\text {a3} \left (\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-1\right )}{\text {b3}}+\frac {\text {a2} \left (\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )-1\right )}{\text {b2}}+\frac {\text {a1} \left (-\text {al1}-\text {bl1}+\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (-\text {al3}-\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}+2\right )}{\text {b1}}\right )}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}},\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ),\frac {1}{2} \left (-\text {al1}-\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (-\text {al3}-\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )+2,\frac {1}{2} \left (-\text {al1}-\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+1,\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+1,\frac {\frac {\text {a1}}{\text {b1}}-x}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}}\right ] \left (x-\frac {\text {a1}}{\text {b1}}\right )^{\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )}+\left (x-\frac {\text {a2}}{\text {b2}}\right )^{\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )} \left (x-\frac {\text {a3}}{\text {b3}}\right )^{\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )} c_2 \text {HeunG}\left [\frac {\frac {\text {a1}}{\text {b1}}-\frac {\text {a3}}{\text {b3}}}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}},\frac {\frac {\text {a1} \left (\frac {1}{2} \left (-\text {al1}-\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+1\right ) \left (\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )\right )}{\text {b1}}+\frac {\text {a3} \left (\frac {1}{4} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-1\right ) \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-\frac {1}{4} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ) \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )\right )}{\text {b3}}+\frac {\text {a2} \left (\frac {1}{4} \left (-\text {al2}-\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right ) \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{4} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ) \left (\frac {1}{2} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )-1\right )\right )}{\text {b2}}+\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right ) \left (\frac {\text {a3} \left (\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}+\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )-1\right )}{\text {b3}}+\frac {\text {a2} \left (\frac {1}{2} \left (\text {al1}+\text {bl1}-\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}-\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )-1\right )}{\text {b2}}+\frac {\text {a1} \left (-\text {al1}-\text {bl1}+\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (-\text {al3}-\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}+2\right )}{\text {b1}}\right )}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}},\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al3}+\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right ),\frac {1}{2} \left (-\text {al1}-\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (-\text {al3}-\text {bl3}+\sqrt {\text {al3}^2+6 \text {bl3} \text {al3}+\text {bl3}^2}\right )+2,\frac {1}{2} \left (-\text {al1}-\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )+1,\frac {1}{2} \left (-\text {al2}-\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+\frac {1}{2} \left (\text {al2}+\text {bl2}-\sqrt {\text {al2}^2+6 \text {bl2} \text {al2}+\text {bl2}^2}\right )+1,\frac {\frac {\text {a1}}{\text {b1}}-x}{\frac {\text {a1}}{\text {b1}}-\frac {\text {a2}}{\text {b2}}}\right ] \left (x-\frac {\text {a1}}{\text {b1}}\right )^{\frac {1}{2} \left (\text {al1}+\text {bl1}+\sqrt {\text {al1}^2+6 \text {bl1} \text {al1}+\text {bl1}^2}\right )}\right \}\right \}\] ✓ Maple : cpu = 1.664 (sec), leaf count = 2607
\[\text {Expression too large to display}\]