\[ y''(x)=-\frac {y'(x) \left (A x^2+B x+C\right )}{(x-a) (x-b) (x-c)}-\frac {(\text {DD} x+e) y(x)}{(x-a) (x-b) (x-c)} \] ✗ Mathematica : cpu = 176.457 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{(\unicode {f817} \text {DD}+e) \unicode {f818}(\unicode {f817})+\left (A \unicode {f817}^2+B \unicode {f817}+C\right ) \unicode {f818}'(\unicode {f817})-(a-\unicode {f817}) (b-\unicode {f817}) (c-\unicode {f817}) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
✓ Maple : cpu = 1.131 (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}},{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}+Ba+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-b \right ) \left ( a \left ( a-2\,b \right ) A-Bb-C+ \left ( -{\it DD}\,a-E \right ) b+{a}^{2}{\it DD}+Ea \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-a \left ( a-b \right ) \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+{a}^{3} \left ( \left ( a+b \right ) B-ab+{b}^{2}+2\,C \right ) 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\, \left ( a-b \right ) \left ( a-b/2 \right ) aC+{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 ( A+1 \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 ( A+1 \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 \} \]