\[ x y^{(3)}(x) (a x+b)+(\alpha x+\beta ) y''(x)-f(x)+x y'(x)+y(x)=0 \] ✗ Mathematica : cpu = 1.94099 (sec), leaf count = 0 , could not solve
DSolve[-f[x] + y[x] + x*Derivative[1][y][x] + (beta + alpha*x)*Derivative[2][y][x] + x*(b + a*x)*Derivative[3][y][x] == 0, y[x], x]
✓ Maple : cpu = 1.737 (sec), leaf count = 1210
\[ \left \{ y \left ( x \right ) = \left ( {\it HeunC} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) \left ( \int \!-{(\int \!f \left ( x \right ) \,{\rm d}x+{\it \_C1}){\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) \left ( ax+b \right ) ^{{\frac { \left ( -3\,b-\beta \right ) a+b\alpha }{ab}}}{x}^{{\frac {-2\,b+\beta }{b}}} \left ( \left ( \left ( -2\,b+\beta \right ) {\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) -a{\it HeunCPrime} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) x \right ) {\it HeunC} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) +{\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) {\it HeunCPrime} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) xa \right ) ^{-1}}\,{\rm d}xb+{\it \_C2} \right ) {x}^{{\frac {2\,b-\beta }{b}}}-{\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) \left ( \int \!-{(\int \!f \left ( x \right ) \,{\rm d}x+{\it \_C1}){\it HeunC} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) \left ( ax+b \right ) ^{{\frac { \left ( -3\,b-\beta \right ) a+b\alpha }{ab}}} \left ( \left ( \left ( -2\,b+\beta \right ) {\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) -a{\it HeunCPrime} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) x \right ) {\it HeunC} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) +{\it HeunC} \left ( 0,{\frac {-2\,b+\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) {\it HeunCPrime} \left ( 0,{\frac {2\,b-\beta }{b}},{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}},-{\frac {b}{{a}^{2}}},{\frac { \left ( 4\,a-\alpha \right ) {b}^{2}-\alpha \,b\beta +a{\beta }^{2}}{2\,a{b}^{2}}},-{\frac {ax}{b}} \right ) xa \right ) ^{-1}}\,{\rm d}xb-{\it \_C3} \right ) \right ) \left ( ax+b \right ) ^{{\frac { \left ( 2\,b+\beta \right ) a-b\alpha }{ab}}} \right \} \]