\[ y''(x)=-\frac {y'(x) \left (A x^2+B x+C\right )}{(x-a) (x-b) (x-c)}-\frac {(\text {DD} x+e) y(x)}{(x-a) (x-b) (x-c)} \] ✓ Mathematica : cpu = 3.25431 (sec), leaf count = 1176
\[\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 = 0.816 (sec), leaf count = 1147
\[y \relax (x ) = c_{1} \mathit {HG}\left (\frac {a -c}{a -b}, \frac {\mathit {DD} a +E}{a -b}, \frac {A}{2}-\frac {1}{2}+\frac {\sqrt {A^{2}-2 A -4 \mathit {DD} +1}}{2}, \frac {\left (A \left (b -c \right ) a -A b c -B c -C \right ) \sqrt {A^{2}-2 A -4 \mathit {DD} +1}-\left (A^{2}-A -4 \mathit {DD} \right ) \left (b -c \right ) a +c \left (A^{2}-A -4 \mathit {DD} \right ) b +\left (B c +C \right ) A +4 c^{2} \mathit {DD} -B c -C}{2 \left (b -c \right ) \left (a -c \right ) \sqrt {A^{2}-2 A -4 \mathit {DD} +1}+\left (-2 b +2 c \right ) a +2 A \,c^{2}+2 B c +2 b c -2 c^{2}+2 C}, \frac {A \,a^{2}+B a +C}{\left (a -b \right ) \left (a -c \right )}, \frac {-A \,b^{2}-B b -C}{\left (a -b \right ) \left (b -c \right )}, \frac {a -x}{a -b}\right )+c_{2} \mathit {HG}\left (\frac {a -c}{a -b}, \frac {\left (a \left (a -2 b \right ) A -B b -C +\left (-\mathit {DD} a -E \right ) b +a^{2} \mathit {DD} +E a \right ) \left (a -b \right ) c^{2}+\left (a^{3} \left (a -2 b \right ) A^{2}+\left (a \left (a -3 b \right ) B -2 C b -a \left (a -b \right ) \left (a -3 b \right )\right ) a A -a \,B^{2} b -\left (C -a \left (a -b \right )\right ) \left (a +b \right ) B -C^{2}+\left (3 a^{2}-4 a b +b^{2}\right ) C -2 a \left (a -b \right )^{2} \left (\mathit {DD} a +E \right )\right ) c +A^{2} a^{4} b +\left (\left (a +b \right ) B -a b +b^{2}+2 C \right ) a^{3} A +B^{2} a^{3}+\left (\left (3 a^{2}-a b \right ) C -a^{3} \left (a -b \right )\right ) B +\left (2 a -b \right ) C^{2}-2 \left (a -\frac {b}{2}\right ) \left (a -b \right ) a C +a^{2} \left (a -b \right )^{2} \left (\mathit {DD} a +E \right )}{\left (a -c \right )^{2} \left (a -b \right )^{3}}, \frac {\left (a -c \right ) \left (a -b \right ) \sqrt {A^{2}-2 A -4 \mathit {DD} +1}+\left (-A +1\right ) a^{2}+\left (\left (-b -c \right ) A -2 B -b -c \right ) a +A b c +b c -2 C}{2 \left (a -c \right ) \left (a -b \right )}, \frac {-\left (\left (b -c \right ) \left (A -2\right ) a^{2}+\left (-2 c^{2}-B c +\left (A +2\right ) b^{2}+2 B b +C \right ) a +2 b \,c^{2}+\left (\left (-A -2\right ) b^{2}-B b -2 C \right ) c +C b \right ) \left (a -c \right ) \sqrt {A^{2}-2 A -4 \mathit {DD} +1}-\left (A^{2}-3 A -4 \mathit {DD} +2\right ) \left (b -c \right ) a^{3}+\left (\left (-3 A^{2}+5 A +8 \mathit {DD} -4\right ) c^{2}+\left (\left (A^{2}-3 A -4 \mathit {DD} +2\right ) b -B \left (1+A \right )\right ) c +\left (A^{2}-A -4 \mathit {DD} +2\right ) b^{2}+2 B b -C \left (A -1\right )\right ) a^{2}+\left (\left (-2 A -4 \mathit {DD} +2\right ) c^{3}+\left (\left (-2 A -4 \mathit {DD} +2\right ) b -3 A B +B \right ) c^{2}+\left (\left (-2 A^{2}+2 A +8 \mathit {DD} -4\right ) b^{2}-B \left (A +3\right ) b -A C -2 B^{2}-3 C \right ) c -\left (\left (A -1\right ) b +2 B \right ) C \right ) a +2 \left (A +2 \mathit {DD} -1\right ) b \,c^{3}+\left (\left (A^{2}-A -4 \mathit {DD} +2\right ) b^{2}+B \left (1+A \right ) b -2 C \left (A -1\right )\right ) c^{2}+\left (\left (A -1\right ) b -2 B \right ) C c -2 C^{2}}{2 \left (\left (b -c \right ) \left (a -c \right ) \sqrt {A^{2}-2 A -4 \mathit {DD} +1}+\left (A -1\right ) c^{2}+\left (B +a +b \right ) c -a b +C \right ) \left (a -c \right ) \left (a -b \right )}, \frac {\left (-A +2\right ) a^{2}+\left (-B -2 b -2 c \right ) a +2 b c -C}{\left (a -c \right ) \left (a -b \right )}, \frac {-A \,b^{2}-B b -C}{\left (a -b \right ) \left (b -c \right )}, \frac {a -x}{a -b}\right ) \left (x -a \right )^{\frac {\left (-A +1\right ) a^{2}+\left (-B -b -c \right ) a +b c -C}{\left (a -c \right ) \left (a -b \right )}}\]