2.1.63.3 ✓ Mathematica. Time used: 0.429 (sec). Leaf size: 163

ode=D[y[x],x]==(a+b*x+y[x])^(4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{gather*} \text {Solve}\left [\frac {2 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} (a+b x+y(x))}{\sqrt [4]{b}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} (a+b x+y(x))}{\sqrt [4]{b}}+1\right )+\sqrt {2} \log \left ((a+b x+y(x))^2-\sqrt {2} \sqrt [4]{b} (a+b x+y(x))+\sqrt {b}\right )-\sqrt {2} \log \left ((a+b x+y(x))^2+\sqrt {2} \sqrt [4]{b} (a+b x+y(x))+\sqrt {b}\right )+8 b^{3/4} x}{8 b^{3/4}}=c_1,y(x)\right ] \end{gather*}