4.18.33 \((a+x) y'(x)^2+y'(x) (\text {a1}+\text {b1} x+\text {c1} y(x))+\text {a2}+\text {b2} x+\text {c2} y(x)=0\)

ODE
\[ (a+x) y'(x)^2+y'(x) (\text {a1}+\text {b1} x+\text {c1} y(x))+\text {a2}+\text {b2} x+\text {c2} y(x)=0 \] ODE Classification

[_dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 242.704 (sec), leaf count = 286

\[\text {Solve}\left [\left \{x=(\text {c1} \text {K$\$$1535584}+\text {c2}) (\text {K$\$$1535584} (\text {b1}+\text {c1} \text {K$\$$1535584}+\text {c2}+\text {K$\$$1535584})+\text {b2})^{-\frac {\text {c1}+2}{2 \text {c1}+2}} \exp \left (-\frac {(\text {b1} \text {c1}-(\text {c1}+2) \text {c2}) \tan ^{-1}\left (\frac {\text {b1}+2 (\text {c1}+1) \text {K$\$$1535584}+\text {c2}}{\sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1}+1) \sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right ) \left (c_1-\int \frac {(\text {K$\$$1535584} (\text {b1}+\text {c1} \text {K$\$$1535584}+\text {c2}+\text {K$\$$1535584})+\text {b2})^{-\frac {\text {c1}}{2 \text {c1}+2}} (a \text {K$\$$1535584} (\text {c1} \text {K$\$$1535584}+2 \text {c2})+\text {a1} \text {c2}-\text {a2} \text {c1}) \exp \left (\frac {(\text {b1} \text {c1}-(\text {c1}+2) \text {c2}) \tan ^{-1}\left (\frac {\text {b1}+2 (\text {c1}+1) \text {K$\$$1535584}+\text {c2}}{\sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1}+1) \sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1} \text {K$\$$1535584}+\text {c2})^2} \, d\text {K$\$$1535584}\right ),y(x)=-\frac {a \text {K$\$$1535584}^2+\text {a1} \text {K$\$$1535584}+\text {a2}+\text {b1} \text {K$\$$1535584} x+\text {b2} x+\text {K$\$$1535584}^2 x}{\text {c1} \text {K$\$$1535584}+\text {c2}}\right \},\{y(x),\text {K$\$$1535584}\}\right ]\]

Maple
cpu = 0.131 (sec), leaf count = 2085

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {{\it \_T}\,{\it c1}+{\it c2}}{ \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{ \left ({\it c1}+1 \right ) ^{-1}}} \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{-{\frac {{\it c1}}{2\,{\it c1}+2}}}{{\rm e}^{{\frac {{\it c1}\,{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}}{{\rm e}^{{\frac {{\it c1}\,{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \left ({{\rm e}^{{\frac {{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\rm e}^{{\frac {{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left (\int \!-{\frac { \left ({{\it \_T}}^{2}a{\it c1}+2\,{\it \_T}\,a{\it c2}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1} \right ) \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{ \left ({\it c1}+1 \right ) ^{-1}}}{ \left ({\it \_T}\,{\it c1}+{\it c2} \right ) ^{2} \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) } \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{{\frac {{\it c1}}{2\,{\it c1}+2}}} \left ({{\rm e}^{{{\it b1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\rm e}^{{\frac {{\it c1}\,{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-1} \left ({{\rm e}^{{\frac {{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2} \left ({{\rm e}^{{\frac {{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2} \left ({{\rm e}^{{\frac {{\it c1}\,{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-1}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) \left ({{\rm e}^{{{\it b1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2}},y \left ({\it \_T} \right ) ={\frac {1}{ \left ({\it \_T}\,{\it c1}+{\it c2} \right ) \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{ \left ({\it c1}+1 \right ) ^{-1}}} \left (- \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{-{\frac {{\it c1}}{2\,{\it c1}+2}}} \left ({{\rm e}^{{\frac {{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\rm e}^{{\frac {{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2}{{\rm e}^{{\frac {{\it c1}\,{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}}{{\rm e}^{{\frac {{\it c1}\,{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \left ({\it \_T}\,{\it c1}+{\it c2} \right ) \left ({{\it \_T}}^{2}+{\it \_T}\,{\it b1}+{\it b2} \right ) \int \!-{\frac { \left ({{\it \_T}}^{2}a{\it c1}+2\,{\it \_T}\,a{\it c2}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1} \right ) \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{ \left ({\it c1}+1 \right ) ^{-1}}}{ \left ({\it \_T}\,{\it c1}+{\it c2} \right ) ^{2} \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) } \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{{\frac {{\it c1}}{2\,{\it c1}+2}}} \left ({{\rm e}^{{{\it b1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\rm e}^{{\frac {{\it c1}\,{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-1} \left ({{\rm e}^{{\frac {{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2} \left ({{\rm e}^{{\frac {{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2} \left ({{\rm e}^{{\frac {{\it c1}\,{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-1}}\,{\rm d}{\it \_T}- \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{-{\frac {{\it c1}}{2\,{\it c1}+2}}} \left ({{\rm e}^{{\frac {{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\rm e}^{{\frac {{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2}{{\rm e}^{{\frac {{\it c1}\,{\it c2}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}}{\it \_C1}\, \left ({\it \_T}\,{\it c1}+{\it c2} \right ) \left ({{\it \_T}}^{2}+{\it \_T}\,{\it b1}+{\it b2} \right ) {{\rm e}^{{\frac {{\it c1}\,{\it b1}}{{\it c1}+1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}}- \left (\left ({\it c1}+1 \right ) {{\it \_T}}^{2}+ \left ({\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) ^{ \left ({\it c1}+1 \right ) ^{-1}} \left ({{\rm e}^{{{\it b1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{2} \left ({{\it \_T}}^{2}a+{\it \_T}\,{\it a1}+{\it a2} \right ) \right ) \left ({{\rm e}^{{{\it b1}\arctan \left ({(2\, \left ({\it c1}+1 \right ) {\it \_T}+{\it b1}+{\it c2}){\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}} \right ) {\frac {1}{\sqrt {-{{\it b1}}^{2}-2\,{\it b1}\,{\it c2}+ \left (4\,{\it c1}+4 \right ) {\it b2}-{{\it c2}}^{2}}}}}}} \right ) ^{-2}}] \right \} \] Mathematica raw input

DSolve[a2 + b2*x + c2*y[x] + (a1 + b1*x + c1*y[x])*y'[x] + (a + x)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{x == ((c2 + c1*K$1535584)*(C[1] - Integrate[(E^(((b1*c1 - (2 + c1)*c2)*Ar
cTan[(b1 + c2 + 2*(1 + c1)*K$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^
2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(-(a2*c1) + a1*c2 
+ a*K$1535584*(2*c2 + c1*K$1535584)))/((c2 + c1*K$1535584)^2*(b2 + K$1535584*(b1
 + c2 + K$1535584 + c1*K$1535584))^(c1/(2 + 2*c1))), K$1535584]))/(E^(((b1*c1 - 
(2 + c1)*c2)*ArcTan[(b1 + c2 + 2*(1 + c1)*K$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1) 
- 2*b1*c2 - c2^2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(b2
 + K$1535584*(b1 + c2 + K$1535584 + c1*K$1535584))^((2 + c1)/(2 + 2*c1))), y[x] 
== -((a2 + a1*K$1535584 + a*K$1535584^2 + b2*x + b1*K$1535584*x + K$1535584^2*x)
/(c2 + c1*K$1535584))}, {y[x], K$1535584}]

Maple raw input

dsolve((a+x)*diff(y(x),x)^2+(a1+b1*x+c1*y(x))*diff(y(x),x)+a2+b2*x+c2*y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = ((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4
)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/
2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*
_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*exp(1/(-b1^2-2*b
1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan
((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*(_T*c1
+c2)*(Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1)*_T^2+(b1+c2)
*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)
*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/((c1+1)*_T^2+(b
1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1
+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^
2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c
1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2),_T)+_C1)/((
(c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1
/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2, y(
_T) = (-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)
*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T
+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*(_T*c1+
c2)*(_T^2+_T*b1+b2)*Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1
)*_T^2+(b1+c2)*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1
/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*ar
ctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/(
(c1+1)*_T^2+(b1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-
b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c
2),_T)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)
)/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+
b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c2
+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c
2^2)^(1/2))/(c1+1)*c1*c2)*_C1*(_T*c1+c2)*(_T^2+_T*b1+b2)*exp(1/(-b1^2-2*b1*c2+(4
*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2
)^(1/2))/(c1+1)*c1*b1)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*
c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2
-c2^2)^(1/2))*b1)^2*(_T^2*a+_T*a1+a2))/(_T*c1+c2)/(((c1+1)*_T^2+(b1+c2)*_T+b2)^(
1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c
2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2]