\[ y''(x)=-y'(x) \left (\frac {\text {b1} (-\text {al1}-\text {bl1}+1)}{\text {b1} x-\text {a1}}+\frac {\text {b2} (-\text {al2}-\text {bl2}+1)}{\text {b2} x-\text {a2}}+\frac {\text {b3} (-\text {al3}-\text {bl3}+1)}{\text {b3} x-\text {a3}}\right )-\frac {y(x) \left (\frac {\text {al1} \text {bl1} (\text {a1} \text {b2}-\text {a2} \text {b1}) (\text {a3} \text {b1}-\text {a1} \text {b3})}{\text {b1} x-\text {a1}}+\frac {\text {al2} \text {bl2} (\text {a1} \text {b2}-\text {a2} \text {b1}) (\text {a2} \text {b3}-\text {a3} \text {b2})}{\text {b2} x-\text {a2}}+\frac {\text {al3} \text {bl3} (\text {a3} \text {b1}-\text {a1} \text {b3}) (\text {a2} \text {b3}-\text {a3} \text {b2})}{\text {b3} x-\text {a3}}\right )}{(\text {b1} x-\text {a1}) (\text {b2} x-\text {a2}) (\text {b3} x-\text {a3})} \] ✓ Mathematica : cpu = 84.1822 (sec), leaf count = 2002
DSolve[Derivative[2][y][x] == -((((al1*(-(a2*b1) + a1*b2)*(a3*b1 - a1*b3)*bl1)/(-a1 + b1*x) + (al2*(-(a2*b1) + a1*b2)*(-(a3*b2) + a2*b3)*bl2)/(-a2 + b2*x) + (al3*(a3*b1 - a1*b3)*(-(a3*b2) + a2*b3)*bl3)/(-a3 + b3*x))*y[x])/((-a1 + b1*x)*(-a2 + b2*x)*(-a3 + b3*x))) - ((b1*(1 - al1 - bl1))/(-a1 + b1*x) + (b2*(1 - al2 - bl2))/(-a2 + b2*x) + (b3*(1 - al3 - bl3))/(-a3 + b3*x))*Derivative[1][y][x],y[x],x]
\[\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 = 1.764 (sec), leaf count = 2607
dsolve(diff(diff(y(x),x),x) = -((1-al1-bl1)*b1/(b1*x-a1)+(1-al2-bl2)*b2/(b2*x-a2)+(1-al3-bl3)*b3/(b3*x-a3))*diff(y(x),x)-1/(b1*x-a1)/(b2*x-a2)/(b3*x-a3)*(al1*bl1*(a1*b2-a2*b1)*(-a1*b3+a3*b1)/(b1*x-a1)+al2*bl2*(a2*b3-a3*b2)*(a1*b2-a2*b1)/(b2*x-a2)+al3*bl3*(-a1*b3+a3*b1)*(a2*b3-a3*b2)/(b3*x-a3))*y(x),y(x))
\[\text {Expression too large to display}\]