An integrating factor of the equation $\left(1+y+x^{2}y\right)dx+\left(x+x^3\right)dy=0$ is

$\frac{dy}{dx}=-\left[\frac{1+y+x^{2}y}{x+x^3}\right]$

$\frac{dy}{dx}=-\left[\frac{1}{x\left(1+x^2\right)}+\frac{\left(1+x^2\right)y}{x\left(1+x^2\right)}\right]$

$\frac{dy}{dx}+\frac{y}{x}=-\frac{1}{x(1+x^2)}$

here $P=\frac{1}{x},Q=-\frac{1}{x(1+x^2)}$

Integrating factor

$=e^{\int pdx}$

$=e^{\int \frac{1}{x}dx}=e^{\log x}$

=x