Integrand size = 80, antiderivative size = 26 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=e^{4 \left (4 e^{-(-5+x)^2} x^2+\frac {2+x}{5}\right )} \] Output:
exp(16*x^2/exp((-5+x)^2)+8/5+4/5*x)
Time = 1.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=e^{\frac {8}{5}+\frac {4 x}{5}+16 e^{-(-5+x)^2} x^2} \] Input:
Integrate[(E^(-25 + 10*x - x^2 + (E^(-25 + 10*x - x^2)*(80*x^2 + E^(25 - 1 0*x + x^2)*(8 + 4*x)))/5)*(4*E^(25 - 10*x + x^2) + 160*x + 800*x^2 - 160*x ^3))/5,x]
Output:
E^(8/5 + (4*x)/5 + (16*x^2)/E^(-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.
\(\displaystyle \int \frac {1}{5} \left (-160 x^3+800 x^2+4 e^{x^2-10 x+25}+160 x\right ) \exp \left (-x^2+\frac {1}{5} e^{-x^2+10 x-25} \left (80 x^2+e^{x^2-10 x+25} (4 x+8)\right )+10 x-25\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int 4 \exp \left (-x^2+10 x+\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )-25\right ) \left (-40 x^3+200 x^2+40 x+e^{x^2-10 x+25}\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} \int \exp \left (-x^2+10 x+\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )-25\right ) \left (-40 x^3+200 x^2+40 x+e^{x^2-10 x+25}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {4}{5} \int \left (-40 \exp \left (-x^2+10 x+\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )-25\right ) x^3+200 \exp \left (-x^2+10 x+\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )-25\right ) x^2+40 \exp \left (-x^2+10 x+\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )-25\right ) x+\exp \left (\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{5} \left (\int \exp \left (\frac {4}{5} e^{-x^2+10 x-25} \left (20 x^2+e^{x^2-10 x+25} (x+2)\right )\right )dx+40 \int \exp \left (\left (-1+16 e^{-(x-5)^2}\right ) x^2+\frac {54 x}{5}-\frac {117}{5}\right ) xdx+200 \int \exp \left (\left (-1+16 e^{-(x-5)^2}\right ) x^2+\frac {54 x}{5}-\frac {117}{5}\right ) x^2dx-40 \int \exp \left (\left (-1+16 e^{-(x-5)^2}\right ) x^2+\frac {54 x}{5}-\frac {117}{5}\right ) x^3dx\right )\) |
Input:
Int[(E^(-25 + 10*x - x^2 + (E^(-25 + 10*x - x^2)*(80*x^2 + E^(25 - 10*x + x^2)*(8 + 4*x)))/5)*(4*E^(25 - 10*x + x^2) + 160*x + 800*x^2 - 160*x^3))/5 ,x]
Output:
$Aborted
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31
method | result | size |
risch | \({\mathrm e}^{\frac {4 \left ({\mathrm e}^{\left (-5+x \right )^{2}} x +20 x^{2}+2 \,{\mathrm e}^{\left (-5+x \right )^{2}}\right ) {\mathrm e}^{-\left (-5+x \right )^{2}}}{5}}\) | \(34\) |
default | \({\mathrm e}^{\frac {\left (\left (4 x +8\right ) {\mathrm e}^{x^{2}-10 x +25}+80 x^{2}\right ) {\mathrm e}^{-x^{2}+10 x -25}}{5}}\) | \(36\) |
norman | \({\mathrm e}^{\frac {\left (\left (4 x +8\right ) {\mathrm e}^{x^{2}-10 x +25}+80 x^{2}\right ) {\mathrm e}^{-x^{2}+10 x -25}}{5}}\) | \(36\) |
parallelrisch | \({\mathrm e}^{\frac {\left (\left (4 x +8\right ) {\mathrm e}^{x^{2}-10 x +25}+80 x^{2}\right ) {\mathrm e}^{-x^{2}+10 x -25}}{5}}\) | \(36\) |
Input:
int(1/5*(4*exp(x^2-10*x+25)-160*x^3+800*x^2+160*x)*exp(1/5*((4*x+8)*exp(x^ 2-10*x+25)+80*x^2)/exp(x^2-10*x+25))/exp(x^2-10*x+25),x,method=_RETURNVERB OSE)
Output:
exp(4/5*(exp((-5+x)^2)*x+20*x^2+2*exp((-5+x)^2))*exp(-(-5+x)^2))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=e^{\left (x^{2} + \frac {1}{5} \, {\left (80 \, x^{2} - {\left (5 \, x^{2} - 54 \, x + 117\right )} e^{\left (x^{2} - 10 \, x + 25\right )}\right )} e^{\left (-x^{2} + 10 \, x - 25\right )} - 10 \, x + 25\right )} \] Input:
integrate(1/5*(4*exp(x^2-10*x+25)-160*x^3+800*x^2+160*x)*exp(1/5*((4*x+8)* exp(x^2-10*x+25)+80*x^2)/exp(x^2-10*x+25))/exp(x^2-10*x+25),x, algorithm=" fricas")
Output:
e^(x^2 + 1/5*(80*x^2 - (5*x^2 - 54*x + 117)*e^(x^2 - 10*x + 25))*e^(-x^2 + 10*x - 25) - 10*x + 25)
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=e^{\left (16 x^{2} + \frac {\left (4 x + 8\right ) e^{x^{2} - 10 x + 25}}{5}\right ) e^{- x^{2} + 10 x - 25}} \] Input:
integrate(1/5*(4*exp(x**2-10*x+25)-160*x**3+800*x**2+160*x)*exp(1/5*((4*x+ 8)*exp(x**2-10*x+25)+80*x**2)/exp(x**2-10*x+25))/exp(x**2-10*x+25),x)
Output:
exp((16*x**2 + (4*x + 8)*exp(x**2 - 10*x + 25)/5)*exp(-x**2 + 10*x - 25))
\[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=\int { -\frac {4}{5} \, {\left (40 \, x^{3} - 200 \, x^{2} - 40 \, x - e^{\left (x^{2} - 10 \, x + 25\right )}\right )} e^{\left (-x^{2} + \frac {4}{5} \, {\left (20 \, x^{2} + {\left (x + 2\right )} e^{\left (x^{2} - 10 \, x + 25\right )}\right )} e^{\left (-x^{2} + 10 \, x - 25\right )} + 10 \, x - 25\right )} \,d x } \] Input:
integrate(1/5*(4*exp(x^2-10*x+25)-160*x^3+800*x^2+160*x)*exp(1/5*((4*x+8)* exp(x^2-10*x+25)+80*x^2)/exp(x^2-10*x+25))/exp(x^2-10*x+25),x, algorithm=" maxima")
Output:
-4/5*integrate((40*x^3 - 200*x^2 - 40*x - e^(x^2 - 10*x + 25))*e^(-x^2 + 4 /5*(20*x^2 + (x + 2)*e^(x^2 - 10*x + 25))*e^(-x^2 + 10*x - 25) + 10*x - 25 ), x)
\[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=\int { -\frac {4}{5} \, {\left (40 \, x^{3} - 200 \, x^{2} - 40 \, x - e^{\left (x^{2} - 10 \, x + 25\right )}\right )} e^{\left (-x^{2} + \frac {4}{5} \, {\left (20 \, x^{2} + {\left (x + 2\right )} e^{\left (x^{2} - 10 \, x + 25\right )}\right )} e^{\left (-x^{2} + 10 \, x - 25\right )} + 10 \, x - 25\right )} \,d x } \] Input:
integrate(1/5*(4*exp(x^2-10*x+25)-160*x^3+800*x^2+160*x)*exp(1/5*((4*x+8)* exp(x^2-10*x+25)+80*x^2)/exp(x^2-10*x+25))/exp(x^2-10*x+25),x, algorithm=" giac")
Output:
integrate(-4/5*(40*x^3 - 200*x^2 - 40*x - e^(x^2 - 10*x + 25))*e^(-x^2 + 4 /5*(20*x^2 + (x + 2)*e^(x^2 - 10*x + 25))*e^(-x^2 + 10*x - 25) + 10*x - 25 ), x)
Time = 3.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx={\mathrm {e}}^{\frac {4\,x}{5}}\,{\mathrm {e}}^{8/5}\,{\mathrm {e}}^{16\,x^2\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{-25}\,{\mathrm {e}}^{-x^2}} \] Input:
int((exp(exp(10*x - x^2 - 25)*((exp(x^2 - 10*x + 25)*(4*x + 8))/5 + 16*x^2 ))*exp(10*x - x^2 - 25)*(160*x + 4*exp(x^2 - 10*x + 25) + 800*x^2 - 160*x^ 3))/5,x)
Output:
exp((4*x)/5)*exp(8/5)*exp(16*x^2*exp(10*x)*exp(-25)*exp(-x^2))
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {1}{5} e^{-25+10 x-x^2+\frac {1}{5} e^{-25+10 x-x^2} \left (80 x^2+e^{25-10 x+x^2} (8+4 x)\right )} \left (4 e^{25-10 x+x^2}+160 x+800 x^2-160 x^3\right ) \, dx=e^{\frac {4 e^{x^{2}} e^{25} x +3 e^{x^{2}} e^{25}+80 e^{10 x} x^{2}}{5 e^{x^{2}} e^{25}}} e \] Input:
int(1/5*(4*exp(x^2-10*x+25)-160*x^3+800*x^2+160*x)*exp(1/5*((4*x+8)*exp(x^ 2-10*x+25)+80*x^2)/exp(x^2-10*x+25))/exp(x^2-10*x+25),x)
Output:
e**((4*e**(x**2)*e**25*x + 3*e**(x**2)*e**25 + 80*e**(10*x)*x**2)/(5*e**(x **2)*e**25))*e