2.741   ODE No. 741

\[ y'(x)=\frac {x \left (a y(x)^2+b x^2\right )^3}{a^{5/2} y(x) \left (a y(x)^2+a+b x^2\right )} \]

✓ Mathematica : cpu = 0.949029 (sec), leaf count = 175

DSolve[Derivative[1][y][x] == (x*(b*x^2 + a*y[x]^2)^3)/(a^(5/2)*y[x]*(a + b*x^2 + a*y[x]^2)),y[x],x]
 

\[\text {Solve}\left [\frac {1}{2} \left (x^2-a^{3/2} \text {RootSum}\left [\text {$\#$1}^3 b^3+3 \text {$\#$1}^2 a b^2 y(x)^2+\text {$\#$1} a^{3/2} b^2+3 \text {$\#$1} a^2 b y(x)^4+a^{5/2} b y(x)^2+a^{5/2} b+a^3 y(x)^6\& ,\frac {a y(x)^2 \log \left (x^2-\text {$\#$1}\right )+a \log \left (x^2-\text {$\#$1}\right )+\text {$\#$1} b \log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 b^2+6 \text {$\#$1} a b y(x)^2+a^{3/2} b+3 a^2 y(x)^4}\& \right ]\right )=c_1,y(x)\right ]\]

✓ Maple : cpu = 0.806 (sec), leaf count = 246

dsolve(diff(y(x),x) = (a*y(x)^2+b*x^2)^3/a^(5/2)*x/(a*y(x)^2+b*x^2+a)/y(x),y(x))
 

\[\int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a}^{2} b +a y \left (x \right )^{2}\right )^{3} \textit {\_a}}{\left (b \left (y \left (x \right )^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} \textit {\_a}^{2}+\left (\textit {\_a}^{2} b +a y \left (x \right )^{2}\right )^{3}\right ) a^{3}}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {\left (\left (-\textit {\_f}^{2}-1\right ) b \,a^{{5}/{2}}-a^{{3}/{2}} b^{2} x^{2}-\left (\textit {\_f}^{2} a +b \,x^{2}\right )^{3}\right ) \left (\int _{\textit {\_b}}^{x}\frac {4 \textit {\_f} \textit {\_a} \left (\textit {\_f}^{2} a +\textit {\_a}^{2} b +\frac {3}{2} a \right ) \left (\textit {\_a}^{2} b +\textit {\_f}^{2} a \right )^{2} b}{\sqrt {a}\, {\left (b \left (\textit {\_f}^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} \textit {\_a}^{2}+\left (\textit {\_a}^{2} b +\textit {\_f}^{2} a \right )^{3}\right )}^{2}}d \textit {\_a} \right )-\frac {\left (\textit {\_f}^{2} a +b \,x^{2}+a \right ) \textit {\_f}}{\sqrt {a}}}{b \left (\textit {\_f}^{2}+1\right ) a^{{5}/{2}}+a^{{3}/{2}} b^{2} x^{2}+\left (\textit {\_f}^{2} a +b \,x^{2}\right )^{3}}d \textit {\_f} +c_{1} = 0\]