Integrand size = 103, antiderivative size = 31 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{2+3 x+x^2-\left (-5-x+\frac {x^2}{256 (5+x)^2}\right )^2} \] Output:
exp(3*x+x^2+2-(1/256*x^2/(5+x)^2-5-x)^2)
Time = 0.86 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}} \] Input:
Integrate[(E^((-942080000 - 1040384000*x - 455411200*x^2 - 98920960*x^3 - 10674689*x^4 - 458240*x^5)/(40960000 + 32768000*x + 9830400*x^2 + 1310720* x^3 + 65536*x^4))*(-358400000 - 358240000*x - 143248000*x^2 - 28643205*x^3 - 2864000*x^4 - 114560*x^5))/(51200000 + 51200000*x + 20480000*x^2 + 4096 000*x^3 + 409600*x^4 + 16384*x^5),x]
Output:
E^(-1/65536*(942080000 + 1040384000*x + 455411200*x^2 + 98920960*x^3 + 106 74689*x^4 + 458240*x^5)/(5 + x)^4)
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.
| \(\displaystyle \int \frac {\left (-114560 x^5-2864000 x^4-28643205 x^3-143248000 x^2-358240000 x-358400000\right ) \exp \left (\frac {-458240 x^5-10674689 x^4-98920960 x^3-455411200 x^2-1040384000 x-942080000}{65536 x^4+1310720 x^3+9830400 x^2+32768000 x+40960000}\right )}{16384 x^5+409600 x^4+4096000 x^3+20480000 x^2+51200000 x+51200000} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (-114560 x^5-2864000 x^4-28643205 x^3-143248000 x^2-358240000 x-358400000\right ) \exp \left (\frac {-458240 x^5-10674689 x^4-98920960 x^3-455411200 x^2-1040384000 x-942080000}{65536 x^4+1310720 x^3+9830400 x^2+32768000 x+40960000}\right )}{\left (4\ 2^{4/5} x+20\ 2^{4/5}\right )^5}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {5 \left (-22912 x^5-572800 x^4-5728641 x^3-28649600 x^2-71648000 x-71680000\right ) \exp \left (\frac {-458240 x^5-10674689 x^4-98920960 x^3-455411200 x^2-1040384000 x-942080000}{65536 x^4+1310720 x^3+9830400 x^2+32768000 x+40960000}\right )}{\left (4\ 2^{4/5} x+20\ 2^{4/5}\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right ) \left (22912 x^5+572800 x^4+5728641 x^3+28649600 x^2+71648000 x+71680000\right )}{16384 (x+5)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \int \frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right ) \left (22912 x^5+572800 x^4+5728641 x^3+28649600 x^2+71648000 x+71680000\right )}{(x+5)^5}dx}{16384}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {5 \int \left (22912 \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )+\frac {641 \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^2}-\frac {15 \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^3}+\frac {75 \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^4}-\frac {125 \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^5}\right )dx}{16384}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \left (22912 \int \exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )dx-125 \int \frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^5}dx+75 \int \frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^4}dx-15 \int \frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^3}dx+641 \int \frac {\exp \left (-\frac {458240 x^5+10674689 x^4+98920960 x^3+455411200 x^2+1040384000 x+942080000}{65536 \left (x^4+20 x^3+150 x^2+500 x+625\right )}\right )}{(x+5)^2}dx\right )}{16384}\) |
Input:
Int[(E^((-942080000 - 1040384000*x - 455411200*x^2 - 98920960*x^3 - 106746 89*x^4 - 458240*x^5)/(40960000 + 32768000*x + 9830400*x^2 + 1310720*x^3 + 65536*x^4))*(-358400000 - 358240000*x - 143248000*x^2 - 28643205*x^3 - 286 4000*x^4 - 114560*x^5))/(51200000 + 51200000*x + 20480000*x^2 + 4096000*x^ 3 + 409600*x^4 + 16384*x^5),x]
Output:
$Aborted
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58
\[{\mathrm e}^{-\frac {458240 x^{5}+10674689 x^{4}+98920960 x^{3}+455411200 x^{2}+1040384000 x +942080000}{65536 \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right )}}\]
Input:
int((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400 000)*exp((-458240*x^5-10674689*x^4-98920960*x^3-455411200*x^2-1040384000*x -942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000))/(1638 4*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x)
Output:
exp(-1/65536*(458240*x^5+10674689*x^4+98920960*x^3+455411200*x^2+104038400 0*x+942080000)/(x^4+20*x^3+150*x^2+500*x+625))
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{\left (-\frac {458240 \, x^{5} + 10674689 \, x^{4} + 98920960 \, x^{3} + 455411200 \, x^{2} + 1040384000 \, x + 942080000}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}}\right )} \] Input:
integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x- 358400000)*exp((-458240*x^5-10674689*x^4-98920960*x^3-455411200*x^2-104038 4000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000)) /(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, al gorithm="fricas")
Output:
e^(-1/65536*(458240*x^5 + 10674689*x^4 + 98920960*x^3 + 455411200*x^2 + 10 40384000*x + 942080000)/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625))
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{\frac {- 458240 x^{5} - 10674689 x^{4} - 98920960 x^{3} - 455411200 x^{2} - 1040384000 x - 942080000}{65536 x^{4} + 1310720 x^{3} + 9830400 x^{2} + 32768000 x + 40960000}} \] Input:
integrate((-114560*x**5-2864000*x**4-28643205*x**3-143248000*x**2-35824000 0*x-358400000)*exp((-458240*x**5-10674689*x**4-98920960*x**3-455411200*x** 2-1040384000*x-942080000)/(65536*x**4+1310720*x**3+9830400*x**2+32768000*x +40960000))/(16384*x**5+409600*x**4+4096000*x**3+20480000*x**2+51200000*x+ 51200000),x)
Output:
exp((-458240*x**5 - 10674689*x**4 - 98920960*x**3 - 455411200*x**2 - 10403 84000*x - 942080000)/(65536*x**4 + 1310720*x**3 + 9830400*x**2 + 32768000* x + 40960000))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (28) = 56\).
Time = 2.61 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{\left (-\frac {895}{128} \, x - \frac {625}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} + \frac {125}{16384 \, {\left (x^{3} + 15 \, x^{2} + 75 \, x + 125\right )}} - \frac {75}{32768 \, {\left (x^{2} + 10 \, x + 25\right )}} + \frac {3205}{16384 \, {\left (x + 5\right )}} - \frac {1509889}{65536}\right )} \] Input:
integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x- 358400000)*exp((-458240*x^5-10674689*x^4-98920960*x^3-455411200*x^2-104038 4000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000)) /(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, al gorithm="maxima")
Output:
e^(-895/128*x - 625/65536/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) + 125/163 84/(x^3 + 15*x^2 + 75*x + 125) - 75/32768/(x^2 + 10*x + 25) + 3205/16384/( x + 5) - 1509889/65536)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (28) = 56\).
Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.74 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=e^{\left (-\frac {895 \, x^{5}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {10674689 \, x^{4}}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {193205 \, x^{3}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {889475 \, x^{2}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {15875 \, x}{x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625} - \frac {14375}{x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625}\right )} \] Input:
integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x- 358400000)*exp((-458240*x^5-10674689*x^4-98920960*x^3-455411200*x^2-104038 4000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000)) /(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, al gorithm="giac")
Output:
e^(-895/128*x^5/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 10674689/65536*x^ 4/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 193205/128*x^3/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 889475/128*x^2/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 15875*x/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 14375/(x^4 + 20*x^ 3 + 150*x^2 + 500*x + 625))
Time = 7.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 5.16 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx={\mathrm {e}}^{-\frac {15875\,x}{x^4+20\,x^3+150\,x^2+500\,x+625}}\,{\mathrm {e}}^{-\frac {14375}{x^4+20\,x^3+150\,x^2+500\,x+625}}\,{\mathrm {e}}^{-\frac {895\,x^5}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {193205\,x^3}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {889475\,x^2}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {10674689\,x^4}{65536\,x^4+1310720\,x^3+9830400\,x^2+32768000\,x+40960000}} \] Input:
int(-(exp(-(1040384000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 4 58240*x^5 + 942080000)/(32768000*x + 9830400*x^2 + 1310720*x^3 + 65536*x^4 + 40960000))*(358240000*x + 143248000*x^2 + 28643205*x^3 + 2864000*x^4 + 114560*x^5 + 358400000))/(51200000*x + 20480000*x^2 + 4096000*x^3 + 409600 *x^4 + 16384*x^5 + 51200000),x)
Output:
exp(-(15875*x)/(500*x + 150*x^2 + 20*x^3 + x^4 + 625))*exp(-14375/(500*x + 150*x^2 + 20*x^3 + x^4 + 625))*exp(-(895*x^5)/(64000*x + 19200*x^2 + 2560 *x^3 + 128*x^4 + 80000))*exp(-(193205*x^3)/(64000*x + 19200*x^2 + 2560*x^3 + 128*x^4 + 80000))*exp(-(889475*x^2)/(64000*x + 19200*x^2 + 2560*x^3 + 1 28*x^4 + 80000))*exp(-(10674689*x^4)/(32768000*x + 9830400*x^2 + 1310720*x ^3 + 65536*x^4 + 40960000))
Time = 11.86 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx=\frac {e^{\frac {29515 x^{3}+300925 x^{2}+1136000 x +1520000}{128 x^{4}+2560 x^{3}+19200 x^{2}+64000 x +80000}}}{e^{\frac {458240 x^{5}+7922177 x^{4}+58982400 x^{3}+196608000 x^{2}+245760000 x}{65536 x^{4}+1310720 x^{3}+9830400 x^{2}+32768000 x +40960000}} e^{42}} \] Input:
int((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400 000)*exp((-458240*x^5-10674689*x^4-98920960*x^3-455411200*x^2-1040384000*x -942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000))/(1638 4*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x)
Output:
e**((29515*x**3 + 300925*x**2 + 1136000*x + 1520000)/(128*x**4 + 2560*x**3 + 19200*x**2 + 64000*x + 80000))/(e**((458240*x**5 + 7922177*x**4 + 58982 400*x**3 + 196608000*x**2 + 245760000*x)/(65536*x**4 + 1310720*x**3 + 9830 400*x**2 + 32768000*x + 40960000))*e**42)