\[ y''(x)=-\frac {y'(x) \left (A x^2+B x+C\right )}{(x-a) (x-b) (x-c)}-\frac {(\text {DD} x+e) y(x)}{(x-a) (x-b) (x-c)} \] ✓ Mathematica : cpu = 2.23045 (sec), leaf count = 1176
DSolve[Derivative[2][y][x] == -(((E + DD*x)*y[x])/((-a + x)*(-b + x)*(-c + x))) - ((C + B*x + A*x^2)*Derivative[1][y][x])/((-a + x)*(-b + x)*(-c + x)),y[x],x]
\[\left \{\left \{y(x)\to c_2 \text {HeunG}\left [\frac {a-c}{a-b},\frac {A^2 b a^4+B^2 a^3+A \left (b^2-a b+(a+b) B+2 C\right ) a^3+(a-b)^2 (a \text {DD}+e) a^2-2 (a-b) \left (a-\frac {b}{2}\right ) C a+(2 a-b) C^2+B \left (\left (3 a^2-a b\right ) C-a^3 (a-b)\right )+(a-b) c^2 \left (\text {DD} a^2+A (a-2 b) a+e a-b B-C+b (-a \text {DD}-e)\right )+c \left (A^2 (a-2 b) a^3-b B^2 a+A (-a (a-3 b) (a-b)+a (a-3 b) B-2 b C) a-2 (a-b)^2 (a \text {DD}+e) a-C^2+\left (3 a^2-4 b a+b^2\right ) C-(a+b) B (C-a (a-b))\right )}{(a-b)^3 (a-c)^2},\frac {(1-A) a^2+(-b-2 B+A (-b-c)-c) a+A b c+b c-2 C+(a-b) (a-c) \sqrt {A^2-2 A-4 \text {DD}+1}}{(a-b) (2 a-2 c)},\frac {-\left ((b-c) \left (A^2-3 A-4 \text {DD}+2\right ) a^3\right )+\left (\left (A^2-A-4 \text {DD}+2\right ) b^2+2 B b-(A-1) C+c \left ((-A-1) B+b \left (A^2-3 A-4 \text {DD}+2\right )\right )+c^2 \left (-3 A^2+5 A+8 \text {DD}-4\right )\right ) a^2+\left ((-2 A-4 \text {DD}+2) c^3+(-3 A B+B+b (-2 A-4 \text {DD}+2)) c^2+\left (\left (-2 A^2+2 A+8 \text {DD}-4\right ) b^2-(A+3) B b-2 B^2-A C-3 C\right ) c+(-((A-1) b)-2 B) C\right ) a-2 C^2+((A-1) b-2 B) c C+c^2 \left (\left (A^2-A-4 \text {DD}+2\right ) b^2+(A+1) B b-2 (A-1) C\right )+2 b c^3 (A+2 \text {DD}-1)+(a-c) \left (-\left ((A-2) (b-c) a^2\right )-\left ((A+2) b^2+2 B b-2 c^2-B c+C\right ) a-2 b c^2-c \left ((-A-2) b^2-B b-2 C\right )-b C\right ) \sqrt {A^2-2 A-4 \text {DD}+1}}{(a-b) (2 a-2 c) \left ((A-1) c^2+(a+b+B) c-a b+C+(a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}\right )},\frac {(2-A) a^2+(-2 b-B-2 c) a+2 b c-C}{(a-b) (a-c)},\frac {-A b^2-B b-C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] (x-a)^{\frac {(1-A) a^2+(-b-B-c) a+b c-C}{(a-b) (a-c)}}+c_1 \text {HeunG}\left [\frac {a-c}{a-b},\frac {a \text {DD}+e}{a-b},\frac {A}{2}+\frac {1}{2} \sqrt {A^2-2 A-4 \text {DD}+1}-\frac {1}{2},\frac {4 \text {DD} c^2-B c+b \left (A^2-A-4 \text {DD}\right ) c-C+A (B c+C)-a (b-c) \left (A^2-A-4 \text {DD}\right )+(a A (b-c)-A b c-B c-C) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 A c^2-2 c^2+2 b c+2 B c+a (2 c-2 b)+2 C+2 (a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}},\frac {A a^2+B a+C}{(a-b) (a-c)},\frac {-A b^2-B b-C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ]\right \}\right \}\] ✓ Maple : cpu = 0.911 (sec), leaf count = 1147
dsolve(diff(diff(y(x),x),x) = -(A*x^2+B*x+C)/(x-a)/(x-b)/(x-c)*diff(y(x),x)-(DD*x+E)/(x-a)/(x-b)/(x-c)*y(x),y(x))
\[\text {Expression too large to display}\]