\[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx \]
Optimal antiderivative \[ \frac {15 \,{\mathrm e}^{-\frac {2 x^{4}}{\left (\left (5+2 x \right )^{2}+x \right )^{2}}}}{4} \]
command
Int[(-750*x^3 - 315*x^4)/(E^((2*x^4)/(625 + 1050*x + 641*x^2 + 168*x^3 + 16*x^4))*(15625 + 39375*x + 40575*x^2 + 21861*x^3 + 6492*x^4 + 1008*x^5 + 64*x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {15}{4} e^{-\frac {2 x^4}{\left (4 x^2+21 x+25\right )^2}} \]