Integrand size = 172, antiderivative size = 37 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=3-\frac {x^4}{4+x+\frac {x}{-e^{e^4+\frac {3}{2 x}}+5 x^2}} \] Output:
3-x^4/(4+x/(5*x^2-exp(3/2/x+exp(4)))+x)
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-16 x+4 x^2-x^3-\frac {256}{4+x}+\frac {x^5}{(4+x) \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \] Input:
Integrate[(-50*x^6 - 800*x^7 - 150*x^8 + E^((3 + 2*E^4*x)/x)*(-32*x^3 - 6* x^4) + E^((3 + 2*E^4*x)/(2*x))*(-3*x^3 + 6*x^4 + 320*x^5 + 60*x^6))/(2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x^6 + E^((3 + 2*E^4*x)/x)*(32 + 16*x + 2*x^2) + E^((3 + 2*E^4*x)/(2*x))*(-16*x - 324*x^2 - 160*x^3 - 20*x^4)),x]
Output:
-16*x + 4*x^2 - x^3 - 256/(4 + x) + x^5/((4 + x)*(-(E^(E^4 + 3/(2*x))*(4 + x)) + x*(1 + 20*x + 5*x^2)))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-150 x^8-800 x^7-50 x^6+e^{\frac {2 e^4 x+3}{x}} \left (-6 x^4-32 x^3\right )+e^{\frac {2 e^4 x+3}{2 x}} \left (60 x^6+320 x^5+6 x^4-3 x^3\right )}{50 x^6+400 x^5+820 x^4+80 x^3+2 x^2+e^{\frac {2 e^4 x+3}{x}} \left (2 x^2+16 x+32\right )+e^{\frac {2 e^4 x+3}{2 x}} \left (-20 x^4-160 x^3-324 x^2-16 x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-150 x^8-800 x^7-50 x^6+e^{\frac {2 e^4 x+3}{x}} \left (-6 x^4-32 x^3\right )+e^{\frac {2 e^4 x+3}{2 x}} \left (60 x^6+320 x^5+6 x^4-3 x^3\right )}{2 \left (-5 x^3-20 x^2+e^{\frac {3}{2 x}+e^4} x-x+4 e^{\frac {3}{2 x}+e^4}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {150 x^8+800 x^7+50 x^6+2 e^{\frac {2 e^4 x+3}{x}} \left (3 x^4+16 x^3\right )+e^{\frac {2 e^4 x+3}{2 x}} \left (-60 x^6-320 x^5-6 x^4+3 x^3\right )}{\left (-5 x^3-20 x^2+e^{e^4+\frac {3}{2 x}} x-x+4 e^{e^4+\frac {3}{2 x}}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {150 x^8+800 x^7+50 x^6+2 e^{\frac {2 e^4 x+3}{x}} \left (3 x^4+16 x^3\right )+e^{\frac {2 e^4 x+3}{2 x}} \left (-60 x^6-320 x^5-6 x^4+3 x^3\right )}{\left (-5 x^3-20 x^2+e^{e^4+\frac {3}{2 x}} x-x+4 e^{e^4+\frac {3}{2 x}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {\left (20 x^4+175 x^3+440 x^2+251 x+12\right ) x^4}{(x+4)^2 \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}+\frac {2 (3 x+16) x^3}{(x+4)^2}-\frac {\left (6 x^2+43 x+12\right ) x^3}{(x+4)^2 \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2240 \int \frac {1}{\left (-5 x^3-20 x^2+e^{e^4+\frac {3}{2 x}} x-x+4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-432 \int \frac {1}{-5 x^3-20 x^2+e^{e^4+\frac {3}{2 x}} x-x+4 e^{e^4+\frac {3}{2 x}}}dx-432 \int \frac {x}{\left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx+76 \int \frac {x^2}{\left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-11 \int \frac {x^3}{\left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx+8192 \int \frac {1}{(x+4)^2 \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-11008 \int \frac {1}{(x+4) \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-44 \int \frac {x}{5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}}dx-5 \int \frac {x^2}{5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}}dx+6 \int \frac {x^3}{5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}}dx+4096 \int \frac {1}{(x+4)^2 \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )}dx-2752 \int \frac {1}{(x+4) \left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )}dx-20 \int \frac {x^6}{\left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-15 \int \frac {x^5}{\left (5 x^3+20 x^2-e^{e^4+\frac {3}{2 x}} x+x-4 e^{e^4+\frac {3}{2 x}}\right )^2}dx-\frac {2 x^4}{x+4}\right )\) |
Input:
Int[(-50*x^6 - 800*x^7 - 150*x^8 + E^((3 + 2*E^4*x)/x)*(-32*x^3 - 6*x^4) + E^((3 + 2*E^4*x)/(2*x))*(-3*x^3 + 6*x^4 + 320*x^5 + 60*x^6))/(2*x^2 + 80* x^3 + 820*x^4 + 400*x^5 + 50*x^6 + E^((3 + 2*E^4*x)/x)*(32 + 16*x + 2*x^2) + E^((3 + 2*E^4*x)/(2*x))*(-16*x - 324*x^2 - 160*x^3 - 20*x^4)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(33)=66\).
Time = 1.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89
| method | result | size |
| norman | \(\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}} x^{4}-5 x^{6}}{5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}}\) | \(70\) |
| parallelrisch | \(-\frac {10 x^{6}-2 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}} x^{4}}{2 \left (5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}\right )}\) | \(72\) |
| risch | \(-x^{3}+4 x^{2}-16 x -\frac {256}{4+x}+\frac {x^{5}}{\left (4+x \right ) \left (5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}\right )}\) | \(76\) |
Input:
int(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x ^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp (1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x*exp(4 )+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x,method=_RETURNVERBOSE)
Output:
(exp(1/2*(2*x*exp(4)+3)/x)*x^4-5*x^6)/(5*x^3+20*x^2-x*exp(1/2*(2*x*exp(4)+ 3)/x)+x-4*exp(1/2*(2*x*exp(4)+3)/x))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (33) = 66\).
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.05 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} + 64 \, x + 256\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x + 4\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x} \] Input:
integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x ^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+3 2)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x *exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="fricas")
Output:
-(5*x^6 + 320*x^3 + 1280*x^2 - (x^4 + 64*x + 256)*e^(1/2*(2*x*e^4 + 3)/x) + 64*x)/(5*x^3 + 20*x^2 - (x + 4)*e^(1/2*(2*x*e^4 + 3)/x) + x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.57 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=- \frac {x^{5}}{- 5 x^{4} - 40 x^{3} - 81 x^{2} - 4 x + \left (x^{2} + 8 x + 16\right ) e^{\frac {x e^{4} + \frac {3}{2}}{x}}} - x^{3} + 4 x^{2} - 16 x - \frac {256}{x + 4} \] Input:
integrate(((-6*x**4-32*x**3)*exp(1/2*(2*x*exp(4)+3)/x)**2+(60*x**6+320*x** 5+6*x**4-3*x**3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x**8-800*x**7-50*x**6)/((2* x**2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)**2+(-20*x**4-160*x**3-324*x**2-16* x)*exp(1/2*(2*x*exp(4)+3)/x)+50*x**6+400*x**5+820*x**4+80*x**3+2*x**2),x)
Output:
-x**5/(-5*x**4 - 40*x**3 - 81*x**2 - 4*x + (x**2 + 8*x + 16)*exp((x*exp(4) + 3/2)/x)) - x**3 + 4*x**2 - 16*x - 256/(x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (33) = 66\).
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.19 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} e^{\left (e^{4}\right )} + 64 \, x e^{\left (e^{4}\right )} + 256 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x e^{\left (e^{4}\right )} + 4 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + x} \] Input:
integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x ^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+3 2)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x *exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="maxima")
Output:
-(5*x^6 + 320*x^3 + 1280*x^2 - (x^4*e^(e^4) + 64*x*e^(e^4) + 256*e^(e^4))* e^(3/2/x) + 64*x)/(5*x^3 + 20*x^2 - (x*e^(e^4) + 4*e^(e^4))*e^(3/2/x) + x)
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (33) = 66\).
Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.11 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} - x^{4} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 320 \, x^{3} + 1280 \, x^{2} - 64 \, x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x - 256 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}}{5 \, x^{3} + 20 \, x^{2} - x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x - 4 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}} \] Input:
integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x ^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+3 2)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x *exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="giac")
Output:
-(5*x^6 - x^4*e^(1/2*(2*x*e^4 + 3)/x) + 320*x^3 + 1280*x^2 - 64*x*e^(1/2*( 2*x*e^4 + 3)/x) + 64*x - 256*e^(1/2*(2*x*e^4 + 3)/x))/(5*x^3 + 20*x^2 - x* e^(1/2*(2*x*e^4 + 3)/x) + x - 4*e^(1/2*(2*x*e^4 + 3)/x))
Timed out. \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=\int -\frac {{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (6\,x^4+32\,x^3\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (60\,x^6+320\,x^5+6\,x^4-3\,x^3\right )+50\,x^6+800\,x^7+150\,x^8}{{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (2\,x^2+16\,x+32\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (20\,x^4+160\,x^3+324\,x^2+16\,x\right )+2\,x^2+80\,x^3+820\,x^4+400\,x^5+50\,x^6} \,d x \] Input:
int(-(exp((2*(x*exp(4) + 3/2))/x)*(32*x^3 + 6*x^4) - exp((x*exp(4) + 3/2)/ x)*(6*x^4 - 3*x^3 + 320*x^5 + 60*x^6) + 50*x^6 + 800*x^7 + 150*x^8)/(exp(( 2*(x*exp(4) + 3/2))/x)*(16*x + 2*x^2 + 32) - exp((x*exp(4) + 3/2)/x)*(16*x + 324*x^2 + 160*x^3 + 20*x^4) + 2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x ^6),x)
Output:
int(-(exp((2*(x*exp(4) + 3/2))/x)*(32*x^3 + 6*x^4) - exp((x*exp(4) + 3/2)/ x)*(6*x^4 - 3*x^3 + 320*x^5 + 60*x^6) + 50*x^6 + 800*x^7 + 150*x^8)/(exp(( 2*(x*exp(4) + 3/2))/x)*(16*x + 2*x^2 + 32) - exp((x*exp(4) + 3/2)/x)*(16*x + 324*x^2 + 160*x^3 + 20*x^4) + 2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x ^6), x)
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.08 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=\frac {x^{4} \left (-e^{\frac {2 e^{4} x +3}{2 x}}+5 x^{2}\right )}{e^{\frac {2 e^{4} x +3}{2 x}} x +4 e^{\frac {2 e^{4} x +3}{2 x}}-5 x^{3}-20 x^{2}-x} \] Input:
int(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x ^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp (1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x*exp(4 )+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x)
Output:
(x**4*( - e**((2*e**4*x + 3)/(2*x)) + 5*x**2))/(e**((2*e**4*x + 3)/(2*x))* x + 4*e**((2*e**4*x + 3)/(2*x)) - 5*x**3 - 20*x**2 - x)