\[ y''(x)=-y'(x) \left (\frac {-\text {a1}-\text {b1}+1}{x-\text {c1}}+\frac {-\text {a2}-\text {b2}+1}{x-\text {c2}}+\frac {-\text {a3}-\text {b3}+1}{x-\text {c3}}\right )-\frac {y(x) \left (\frac {\text {a1} \text {b1} (\text {c1}-\text {c2}) (\text {c1}-\text {c3})}{x-\text {c1}}+\frac {\text {a2} \text {b2} (\text {c2}-\text {c1}) (\text {c2}-\text {c3})}{x-\text {c2}}+\frac {\text {a3} \text {b3} (\text {c3}-\text {c1}) (\text {c3}-\text {c2})}{x-\text {c3}}\right )}{(x-\text {c1}) (x-\text {c2}) (x-\text {c3})} \] ✓ Mathematica : cpu = 14.3933 (sec), leaf count = 316
DSolve[Derivative[2][y][x] == -((((a1*b1*(c1 - c2)*(c1 - c3))/(-c1 + x) + (a2*b2*(-c1 + c2)*(c2 - c3))/(-c2 + x) + (a3*b3*(-c1 + c3)*(-c2 + c3))/(-c3 + x))*y[x])/((-c1 + x)*(-c2 + x)*(-c3 + x))) - ((1 - a1 - b1)/(-c1 + x) + (1 - a2 - b2)/(-c2 + x) + (1 - a3 - b3)/(-c3 + x))*Derivative[1][y][x],y[x],x]
\[\left \{\left \{y(x)\to c_1 (x-\text {c1})^{\text {a1}} (x-\text {c2})^{\text {a2}} (x-\text {c3})^{\text {b3}} \text {HeunG}\left [\frac {\text {c1}-\text {c3}}{\text {c1}-\text {c2}},\frac {\text {a1} (\text {c1} (-\text {a3}-2 \text {b1}-\text {b2}+2)+\text {c2} (\text {a3}+\text {b1}-1)+\text {c3} (\text {b1}+\text {b2}-1))+\text {c2} (-\text {a2} \text {b2}+(\text {a3}-1) \text {b3}+\text {b1} \text {b3})+\text {c3} (\text {a2} \text {b1}+\text {a2} (\text {b2}-1)-\text {a3} \text {b3})+(1-\text {b1}) \text {c1} (\text {a2}+\text {b3})}{\text {c1}-\text {c2}},\text {a1}+\text {a2}+\text {b3},-\text {a3}-\text {b1}-\text {b2}+2,\text {a1}-\text {b1}+1,\text {a2}-\text {b2}+1,\frac {\text {c1}-x}{\text {c1}-\text {c2}}\right ]+c_2 (x-\text {c2})^{\text {a2}} (x-\text {c1})^{\text {b1}} (x-\text {c3})^{\text {b3}} \text {HeunG}\left [\frac {\text {c1}-\text {c3}}{\text {c1}-\text {c2}},\frac {\text {c2} (\text {a1} \text {b3}-\text {a2} \text {b2}+(\text {a3}-1) \text {b3})+\text {c3} (\text {a1} \text {a2}+\text {a2} (\text {b2}-1)-\text {a3} \text {b3})+(1-\text {a1}) \text {c1} (\text {a2}+\text {b3})+\text {b1} (\text {c1} (-2 \text {a1}-\text {a3}-\text {b2}+2)+\text {c2} (\text {a1}+\text {a3}-1)+\text {c3} (\text {a1}+\text {b2}-1))}{\text {c1}-\text {c2}},\text {a2}+\text {b1}+\text {b3},-\text {a1}-\text {a3}-\text {b2}+2,-\text {a1}+\text {b1}+1,\text {a2}-\text {b2}+1,\frac {\text {c1}-x}{\text {c1}-\text {c2}}\right ]\right \}\right \}\] ✓ Maple : cpu = 0.615 (sec), leaf count = 298
dsolve(diff(diff(y(x),x),x) = -((1-a1-b1)/(x-c1)+(1-a2-b2)/(x-c2)+(1-a3-b3)/(x-c3))*diff(y(x),x)-1/(x-c1)/(x-c2)/(x-c3)*(a1*b1*(c1-c3)*(c1-c2)/(x-c1)+a2*b2*(c2-c1)*(c2-c3)/(x-c2)+a3*b3*(c3-c2)*(c3-c1)/(x-c3))*y(x),y(x))
\[y \left (x \right ) = \left (x -\mathit {c2} \right )^{\mathit {a2}} \left (x -\mathit {c3} \right )^{\mathit {b3}} \left (\left (x -\mathit {c1} \right )^{\mathit {a1}} \mathit {HG}\left (\frac {\mathit {c1} -\mathit {c3}}{\mathit {c1} -\mathit {c2}}, \frac {\left (\left (-\mathit {a3} -2 \mathit {b1} -\mathit {b2} +2\right ) \mathit {c1} +\left (\mathit {a3} +\mathit {b1} -1\right ) \mathit {c2} +\mathit {c3} \left (\mathit {b1} +\mathit {b2} -1\right )\right ) \mathit {a1} -\left (\mathit {b1} -1\right ) \left (\mathit {a2} +\mathit {b3} \right ) \mathit {c1} +\left (\mathit {b1} \mathit {b3} -\mathit {a2} \mathit {b2} +\mathit {b3} \left (\mathit {a3} -1\right )\right ) \mathit {c2} +\left (\mathit {b1} \mathit {a2} +\left (\mathit {b2} -1\right ) \mathit {a2} -\mathit {a3} \mathit {b3} \right ) \mathit {c3}}{\mathit {c1} -\mathit {c2}}, \mathit {a1} +\mathit {b3} +\mathit {a2} , 2-\mathit {a3} -\mathit {b1} -\mathit {b2} , \mathit {a1} -\mathit {b1} +1, \mathit {a2} -\mathit {b2} +1, \frac {-x +\mathit {c1}}{\mathit {c1} -\mathit {c2}}\right ) c_{1}+\left (x -\mathit {c1} \right )^{\mathit {b1}} \mathit {HG}\left (\frac {\mathit {c1} -\mathit {c3}}{\mathit {c1} -\mathit {c2}}, \frac {\left (\left (-2 \mathit {a1} -\mathit {a3} -\mathit {b2} +2\right ) \mathit {c1} +\left (\mathit {a1} +\mathit {a3} -1\right ) \mathit {c2} +\mathit {c3} \left (\mathit {a1} +\mathit {b2} -1\right )\right ) \mathit {b1} -\left (\mathit {a2} +\mathit {b3} \right ) \left (\mathit {a1} -1\right ) \mathit {c1} +\left (\mathit {a1} \mathit {b3} -\mathit {a2} \mathit {b2} +\mathit {b3} \left (\mathit {a3} -1\right )\right ) \mathit {c2} +\left (\mathit {a1} \mathit {a2} +\left (\mathit {b2} -1\right ) \mathit {a2} -\mathit {a3} \mathit {b3} \right ) \mathit {c3}}{\mathit {c1} -\mathit {c2}}, \mathit {b1} +\mathit {b3} +\mathit {a2} , 2-\mathit {a1} -\mathit {a3} -\mathit {b2} , -\mathit {a1} +\mathit {b1} +1, \mathit {a2} -\mathit {b2} +1, \frac {-x +\mathit {c1}}{\mathit {c1} -\mathit {c2}}\right ) c_{2}\right )\]