\[ y'(x)-\sqrt {\frac {y(x)^3+1}{x^3+1}}=0 \] ✓ Mathematica : cpu = 51.7435 (sec), leaf count = 312
DSolve[-Sqrt[(1 + y[x]^3)/(1 + x^3)] + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {i (\text {$\#$1}+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (\text {$\#$1}+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (\text {$\#$1}+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {\text {$\#$1}+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {\text {$\#$1}^2-\text {$\#$1}+1}}\& \right ]\left [\frac {i (x+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x^2-x+1}}+c_1\right ]\right \}\right \}\] ✓ Maple : cpu = 0.062 (sec), leaf count = 47
dsolve(diff(y(x),x)-((y(x)^3+1)/(x^3+1))^(1/2) = 0,y(x))
\[\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {\frac {y \left (x \right )^{3}+1}{\textit {\_a}^{3}+1}}}{\sqrt {y \left (x \right )^{3}+1}}d \textit {\_a} +c_{1} = 0\]