\[ y''(x)=-\frac {y'(x) \left (A x^2+B x+C\right )}{(x-a) (x-b) (x-c)}-\frac {(\text {DD} x+e) y(x)}{(x-a) (x-b) (x-c)} \] ✓ Mathematica : cpu = 3.49395 (sec), leaf count = 1175
\[\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 {-(b-c) \left (A^2-3 A-4 \text {DD}+2\right ) a^3+\left (\left (A^2-A-4 \text {DD}+2\right ) b^2+2 B b-(A-1) C+c \left (b \left (A^2-3 A-4 \text {DD}+2\right )-(A+1) B\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 (-(A-2) (b-c) a^2-\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 = 2.894 (sec), leaf count = 1147
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it HeunG} \left ( {\frac {a-c}{a-b}},{\frac {{\it DD}\,a+E}{a-b}},{\frac {A}{2}}-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}},{ \left ( \left ( A \left ( b-c \right ) a-Abc-Bc-C \right ) \sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}- \left ( {A}^{2}-A-4\,{\it DD} \right ) \left ( b-c \right ) a+c \left ( {A}^{2}-A-4\,{\it DD} \right ) b+ \left ( Bc+C \right ) A+4\,{c}^{2}{\it DD}-Bc-C \right ) \left ( 2\, \left ( b-c \right ) \left ( a-c \right ) \sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}+ \left ( -2\,b+2\,c \right ) a+2\,A{c}^{2}+2\,Bc+2\,bc-2\,{c}^{2}+2\,C \right ) ^{-1}},{\frac {A{a}^{2}+aB+C}{ \left ( a-b \right ) \left ( a-c \right ) }},{\frac {-A{b}^{2}-Bb-C}{ \left ( a-b \right ) \left ( b-c \right ) }},{\frac {a-x}{a-b}} \right ) +{\it \_C2}\,{\it HeunG} \left ( {\frac {a-c}{a-b}},{\frac {1}{ \left ( a-b \right ) ^{3} \left ( a-c \right ) ^{2}} \left ( \left ( a \left ( a-2\,b \right ) A-Bb-C+ \left ( -{\it DD}\,a-E \right ) b+{a}^{2}{\it DD}+Ea \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\,Cb-a \left ( a-b \right ) \left ( a-3\,b \right ) \right ) aA-{B}^{2}ab- \left ( a+b \right ) \left ( C- \left ( a-b \right ) a \right ) B-{C}^{2}+ \left ( 3\,{a}^{2}-4\,ab+{b}^{2} \right ) C-2\,a \left ( a-b \right ) ^{2} \left ( {\it DD}\,a+E \right ) \right ) c+{A}^{2}{a}^{4}b+ \left ( \left ( a+b \right ) B-ab+{b}^{2}+2\,C \right ) {a}^{3}A+{B}^{2}{a}^{3}+ \left ( \left ( 3\,{a}^{2}-ab \right ) C-{a}^{3} \left ( a-b \right ) \right ) B+ \left ( 2\,a-b \right ) {C}^{2}-2\,a \left ( a-b \right ) \left ( a-b/2 \right ) C+{a}^{2} \left ( a-b \right ) ^{2} \left ( {\it DD}\,a+E \right ) \right ) },{\frac {1}{ \left ( 2\,a-2\,c \right ) \left ( a-b \right ) } \left ( \left ( a-c \right ) \left ( a-b \right ) \sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}+ \left ( -A+1 \right ) {a}^{2}+ \left ( \left ( -b-c \right ) A-2\,B-b-c \right ) a+Abc+bc-2\,C \right ) },{\frac {1}{ \left ( 2\,a-2\,b \right ) \left ( a-c \right ) } \left ( - \left ( \left ( b-c \right ) \left ( A-2 \right ) {a}^{2}+ \left ( -2\,{c}^{2}-Bc+ \left ( A+2 \right ) {b}^{2}+2\,Bb+C \right ) a+2\,b{c}^{2}+ \left ( \left ( -A-2 \right ) {b}^{2}-Bb-2\,C \right ) c+Cb \right ) \left ( a-c \right ) \sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}- \left ( {A}^{2}-3\,A-4\,{\it DD}+2 \right ) \left ( b-c \right ) {a}^{3}+ \left ( \left ( -3\,{A}^{2}+5\,A+8\,{\it DD}-4 \right ) {c}^{2}+ \left ( \left ( {A}^{2}-3\,A-4\,{\it DD}+2 \right ) b- \left ( 1+A \right ) B \right ) c+ \left ( {A}^{2}-A-4\,{\it DD}+2 \right ) {b}^{2}+2\,Bb-C \left ( A-1 \right ) \right ) {a}^{2}+ \left ( \left ( -2\,A-4\,{\it DD}+2 \right ) {c}^{3}+ \left ( \left ( -2\,A-4\,{\it DD}+2 \right ) b-3\,AB+B \right ) {c}^{2}+ \left ( \left ( -2\,{A}^{2}+2\,A+8\,{\it DD}-4 \right ) {b}^{2}-B \left ( A+3 \right ) b-AC-2\,{B}^{2}-3\,C \right ) c- \left ( \left ( A-1 \right ) b+2\,B \right ) C \right ) a+2\, \left ( A+2\,{\it DD}-1 \right ) b{c}^{3}+ \left ( \left ( {A}^{2}-A-4\,{\it DD}+2 \right ) {b}^{2}+ \left ( 1+A \right ) Bb-2\,C \left ( A-1 \right ) \right ) {c}^{2}+ \left ( \left ( A-1 \right ) b-2\,B \right ) Cc-2\,{C}^{2} \right ) \left ( \left ( b-c \right ) \left ( a-c \right ) \sqrt {{A}^{2}-2\,A-4\,{\it DD}+1}+ \left ( A-1 \right ) {c}^{2}+ \left ( B+a+b \right ) c-ab+C \right ) ^{-1}},{\frac { \left ( -A+2 \right ) {a}^{2}+ \left ( -B-2\,b-2\,c \right ) a+2\,bc-C}{ \left ( a-b \right ) \left ( a-c \right ) }},{\frac {-A{b}^{2}-Bb-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+bc-C}{ \left ( a-b \right ) \left ( a-c \right ) }}} \right \} \]