Integrand size = 155, antiderivative size = 33 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=-4+e^{-x} \left (e^{-\frac {x^2}{1+\frac {1}{5} (5+x)^2}} x+\log (x)\right ) \] Output:
(x/exp(x^2/(1/5*(5+x)^2+1))+ln(x))/exp(x)-4
Time = 5.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=e^{-5-x+\frac {50 (3+x)}{30+10 x+x^2}} x+e^{-x} \log (x) \] Input:
Integrate[(E^(-x - (5*x^2)/(30 + 10*x + x^2))*(900*x - 300*x^2 - 740*x^3 - 190*x^4 - 19*x^5 - x^6 + E^((5*x^2)/(30 + 10*x + x^2))*(900 + 600*x + 160 *x^2 + 20*x^3 + x^4) + E^((5*x^2)/(30 + 10*x + x^2))*(-900*x - 600*x^2 - 1 60*x^3 - 20*x^4 - x^5)*Log[x]))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5) ,x]
Output:
E^(-5 - x + (50*(3 + x))/(30 + 10*x + x^2))*x + Log[x]/E^x
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.
| \(\displaystyle \int \frac {e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{x^5+20 x^4+160 x^3+600 x^2+900 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{x \left (x^4+20 x^3+160 x^2+600 x+900\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {i e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{10 \sqrt {5} \left (-2 x+2 i \sqrt {5}-10\right ) x}-\frac {e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{5 \left (-2 x+2 i \sqrt {5}-10\right )^2 x}+\frac {i e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{10 \sqrt {5} x \left (2 x+2 i \sqrt {5}+10\right )}-\frac {e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (-x^6-19 x^5-190 x^4-740 x^3-300 x^2+e^{\frac {5 x^2}{x^2+10 x+30}} \left (x^4+20 x^3+160 x^2+600 x+900\right )+e^{\frac {5 x^2}{x^2+10 x+30}} \left (-x^5-20 x^4-160 x^3-600 x^2-900 x\right ) \log (x)+900 x\right )}{5 x \left (2 x+2 i \sqrt {5}+10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^{-x} \left (-\frac {e^{-\frac {5 x^2}{x^2+10 x+30}} \left (x^5+19 x^4+190 x^3+740 x^2+300 x-900\right )}{\left (x^2+10 x+30\right )^2}+\frac {1}{x}-\log (x)\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^{-\frac {5 x^2}{x^2+10 x+30}-x} \left (x^5+19 x^4+190 x^3+740 x^2+300 x-900\right )}{\left (x^2+10 x+30\right )^2}+\frac {e^{-x}}{x}-e^{-x} \log (x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}dx-100 \left (5-i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{\left (-2 x+2 i \sqrt {5}-10\right )^2}dx+1200 \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{\left (-2 x+2 i \sqrt {5}-10\right )^2}dx-70 i \sqrt {5} \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{-2 x+2 i \sqrt {5}-10}dx-\int e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}} xdx-10 \left (5+9 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{2 x-2 i \sqrt {5}+10}dx-100 \left (5+i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{\left (2 x+2 i \sqrt {5}+10\right )^2}dx+1200 \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{\left (2 x+2 i \sqrt {5}+10\right )^2}dx-10 \left (5-9 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{2 x+2 i \sqrt {5}+10}dx-70 i \sqrt {5} \int \frac {e^{-\frac {x \left (x^2+15 x+30\right )}{x^2+10 x+30}}}{2 x+2 i \sqrt {5}+10}dx+e^{-x} \log (x)\) |
Input:
Int[(E^(-x - (5*x^2)/(30 + 10*x + x^2))*(900*x - 300*x^2 - 740*x^3 - 190*x ^4 - 19*x^5 - x^6 + E^((5*x^2)/(30 + 10*x + x^2))*(900 + 600*x + 160*x^2 + 20*x^3 + x^4) + E^((5*x^2)/(30 + 10*x + x^2))*(-900*x - 600*x^2 - 160*x^3 - 20*x^4 - x^5)*Log[x]))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5),x]
Output:
$Aborted
Time = 40.40 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \({\mathrm e}^{-x} \ln \left (x \right )+x \,{\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}\) | \(33\) |
| parallelrisch | \(-\frac {\left (-1019160000 x -339720000 x^{2}-33972000 x^{3}-33972000 x^{2} {\mathrm e}^{\frac {5 x^{2}}{x^{2}+10 x +30}} \ln \left (x \right )-339720000 \ln \left (x \right ) {\mathrm e}^{\frac {5 x^{2}}{x^{2}+10 x +30}} x -1019160000 \ln \left (x \right ) {\mathrm e}^{\frac {5 x^{2}}{x^{2}+10 x +30}}\right ) {\mathrm e}^{-x} {\mathrm e}^{-\frac {5 x^{2}}{x^{2}+10 x +30}}}{33972000 \left (x^{2}+10 x +30\right )}\) | \(113\) |
| default | \({\mathrm e}^{\frac {5 x^{2}}{x^{2}+10 x +30}} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}} \ln \left (x \right )+\frac {x^{3} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}+30 x \,{\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}+10 x^{2} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}}{x^{2}+10 x +30}\) | \(133\) |
| orering | \(\text {Expression too large to display}\) | \(4730\) |
Input:
int(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*ln(x)+(x ^4+20*x^3+160*x^2+600*x+900)*exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4-7 40*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/(x^2+10 *x+30))/exp(x),x,method=_RETURNVERBOSE)
Output:
exp(-x)*ln(x)+x*exp(-x*(x^2+15*x+30)/(x^2+10*x+30))
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (27) = 54\).
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=x e^{\left (-\frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} + e^{\left (\frac {5 \, x^{2}}{x^{2} + 10 \, x + 30} - \frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} \log \left (x\right ) \] Input:
integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*lo g(x)+(x^4+20*x^3+160*x^2+600*x+900)*exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-19 0*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/ (x^2+10*x+30))/exp(x),x, algorithm="fricas")
Output:
x*e^(-(x^3 + 15*x^2 + 30*x)/(x^2 + 10*x + 30)) + e^(5*x^2/(x^2 + 10*x + 30 ) - (x^3 + 15*x^2 + 30*x)/(x^2 + 10*x + 30))*log(x)
Timed out. \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=\text {Timed out} \] Input:
integrate(((-x**5-20*x**4-160*x**3-600*x**2-900*x)*exp(5*x**2/(x**2+10*x+3 0))*ln(x)+(x**4+20*x**3+160*x**2+600*x+900)*exp(5*x**2/(x**2+10*x+30))-x** 6-19*x**5-190*x**4-740*x**3-300*x**2+900*x)/(x**5+20*x**4+160*x**3+600*x** 2+900*x)/exp(5*x**2/(x**2+10*x+30))/exp(x),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx={\left (x e^{\left (\frac {50 \, x}{x^{2} + 10 \, x + 30} + \frac {150}{x^{2} + 10 \, x + 30}\right )} + e^{5} \log \left (x\right )\right )} e^{\left (-x - 5\right )} \] Input:
integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*lo g(x)+(x^4+20*x^3+160*x^2+600*x+900)*exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-19 0*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/ (x^2+10*x+30))/exp(x),x, algorithm="maxima")
Output:
(x*e^(50*x/(x^2 + 10*x + 30) + 150/(x^2 + 10*x + 30)) + e^5*log(x))*e^(-x - 5)
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx={\left (x e^{\left (x - \frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} + \log \left (x\right )\right )} e^{\left (-x\right )} \] Input:
integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*lo g(x)+(x^4+20*x^3+160*x^2+600*x+900)*exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-19 0*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/ (x^2+10*x+30))/exp(x),x, algorithm="giac")
Output:
(x*e^(x - (x^3 + 15*x^2 + 30*x)/(x^2 + 10*x + 30)) + log(x))*e^(-x)
Timed out. \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {5\,x^2}{x^2+10\,x+30}}\,{\mathrm {e}}^{-x}\,\left (300\,x^2-900\,x+740\,x^3+190\,x^4+19\,x^5+x^6-{\mathrm {e}}^{\frac {5\,x^2}{x^2+10\,x+30}}\,\left (x^4+20\,x^3+160\,x^2+600\,x+900\right )+{\mathrm {e}}^{\frac {5\,x^2}{x^2+10\,x+30}}\,\ln \left (x\right )\,\left (x^5+20\,x^4+160\,x^3+600\,x^2+900\,x\right )\right )}{x^5+20\,x^4+160\,x^3+600\,x^2+900\,x} \,d x \] Input:
int(-(exp(-(5*x^2)/(10*x + x^2 + 30))*exp(-x)*(300*x^2 - 900*x + 740*x^3 + 190*x^4 + 19*x^5 + x^6 - exp((5*x^2)/(10*x + x^2 + 30))*(600*x + 160*x^2 + 20*x^3 + x^4 + 900) + exp((5*x^2)/(10*x + x^2 + 30))*log(x)*(900*x + 600 *x^2 + 160*x^3 + 20*x^4 + x^5)))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5 ),x)
Output:
int(-(exp(-(5*x^2)/(10*x + x^2 + 30))*exp(-x)*(300*x^2 - 900*x + 740*x^3 + 190*x^4 + 19*x^5 + x^6 - exp((5*x^2)/(10*x + x^2 + 30))*(600*x + 160*x^2 + 20*x^3 + x^4 + 900) + exp((5*x^2)/(10*x + x^2 + 30))*log(x)*(900*x + 600 *x^2 + 160*x^3 + 20*x^4 + x^5)))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5 ), x)
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx=\frac {e^{\frac {50 x +150}{x^{2}+10 x +30}} x +\mathrm {log}\left (x \right ) e^{5}}{e^{x} e^{5}} \] Input:
int(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*log(x)+( x^4+20*x^3+160*x^2+600*x+900)*exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4- 740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/(x^2+1 0*x+30))/exp(x),x)
Output:
(e**((50*x + 150)/(x**2 + 10*x + 30))*x + log(x)*e**5)/(e**x*e**5)