Integrand size = 218, antiderivative size = 26 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4 e^{x^2 \left (2+x-\frac {25}{\left (16+\frac {4}{x}+x\right )^2}\right )^2} x \] Output:
4*x*exp(x^2*(x+2-25/(x+16+4/x)^2)^2)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4 e^{\frac {x^2 \left (32+272 x+631 x^2+328 x^3+34 x^4+x^5\right )^2}{\left (4+16 x+x^2\right )^4}} x \] Input:
Integrate[(E^((1024*x^2 + 17408*x^3 + 114368*x^4 + 364256*x^5 + 578769*x^6 + 432496*x^7 + 151036*x^8 + 23566*x^9 + 1812*x^10 + 68*x^11 + x^12)/(256 + 4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x ^7 + x^8))*(4096 + 81920*x + 693248*x^2 + 3407872*x^3 + 11917824*x^4 + 343 28064*x^5 + 78478560*x^6 + 118319872*x^7 + 97752008*x^8 + 40329056*x^9 + 7 831044*x^10 + 802040*x^11 + 45152*x^12 + 1328*x^13 + 16*x^14))/(1024 + 204 80*x + 165120*x^2 + 675840*x^3 + 1434240*x^4 + 1383936*x^5 + 358560*x^6 + 42240*x^7 + 2580*x^8 + 80*x^9 + x^10),x]
Output:
4*E^((x^2*(32 + 272*x + 631*x^2 + 328*x^3 + 34*x^4 + x^5)^2)/(4 + 16*x + x ^2)^4)*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.
| \(\displaystyle \int \frac {\left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right ) \exp \left (\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}\right )}{x^{10}+80 x^9+2580 x^8+42240 x^7+358560 x^6+1383936 x^5+1434240 x^4+675840 x^3+165120 x^2+20480 x+1024} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {7 e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{663552000 \sqrt {15} \left (-2 x+4 \sqrt {15}-16\right )}-\frac {7 e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{663552000 \sqrt {15} \left (2 x+4 \sqrt {15}+16\right )}-\frac {7 e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{165888000 \left (-2 x+4 \sqrt {15}-16\right )^2}-\frac {7 e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{165888000 \left (2 x+4 \sqrt {15}+16\right )^2}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{460800 \sqrt {15} \left (-2 x+4 \sqrt {15}-16\right )^3}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{460800 \sqrt {15} \left (2 x+4 \sqrt {15}+16\right )^3}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{172800 \left (-2 x+4 \sqrt {15}-16\right )^4}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{172800 \left (2 x+4 \sqrt {15}+16\right )^4}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{7200 \sqrt {15} \left (-2 x+4 \sqrt {15}-16\right )^5}-\frac {e^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^9+151036 x^8+432496 x^7+578769 x^6+364256 x^5+114368 x^4+17408 x^3+1024 x^2}{x^8+64 x^7+1552 x^6+17152 x^5+77920 x^4+68608 x^3+24832 x^2+4096 x+256}} \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^9+97752008 x^8+118319872 x^7+78478560 x^6+34328064 x^5+11917824 x^4+3407872 x^3+693248 x^2+81920 x+4096\right )}{7200 \sqrt {15} \left (2 x+4 \sqrt {15}+16\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (4 x^{14}+332 x^{13}+11288 x^{12}+200510 x^{11}+1957761 x^{10}+10082264 x^9+24438002 x^8+29579968 x^7+19619640 x^6+8582016 x^5+2979456 x^4+851968 x^3+173312 x^2+20480 x+1024\right ) \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right )}{\left (x^2+16 x+4\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {\exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) \left (4 x^{14}+332 x^{13}+11288 x^{12}+200510 x^{11}+1957761 x^{10}+10082264 x^9+24438002 x^8+29579968 x^7+19619640 x^6+8582016 x^5+2979456 x^4+851968 x^3+173312 x^2+20480 x+1024\right )}{\left (x^2+16 x+4\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (4 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^4+12 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^3+8 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^2-50 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x+\exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right )+\frac {50 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (696 x-17849)}{x^2+16 x+4}+\frac {200 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (97936 x-418897)}{\left (x^2+16 x+4\right )^2}+\frac {800 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (1763152 x-441879)}{\left (x^2+16 x+4\right )^3}+\frac {80000 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (169092 x-441139)}{\left (x^2+16 x+4\right )^4}+\frac {1280000 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (468722 x+119071)}{\left (x^2+16 x+4\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) \left (4 x^{14}+332 x^{13}+11288 x^{12}+200510 x^{11}+1957761 x^{10}+10082264 x^9+24438002 x^8+29579968 x^7+19619640 x^6+8582016 x^5+2979456 x^4+851968 x^3+173312 x^2+20480 x+1024\right )}{\left (x^2+16 x+4\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (4 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^4+12 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^3+8 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x^2-50 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) x+\exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right )+\frac {50 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (696 x-17849)}{x^2+16 x+4}+\frac {200 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (97936 x-418897)}{\left (x^2+16 x+4\right )^2}+\frac {800 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (1763152 x-441879)}{\left (x^2+16 x+4\right )^3}+\frac {80000 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (169092 x-441139)}{\left (x^2+16 x+4\right )^4}+\frac {1280000 \exp \left (\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}\right ) (468722 x+119071)}{\left (x^2+16 x+4\right )^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \left (\int e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}dx-\frac {299982080}{27} \left (15-4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^5}dx-\frac {38102720}{9} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^5}dx-7515200 \left (4-\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^4}dx+\frac {37497760}{27} \left (20-3 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^4}dx-\frac {288497600}{27} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^4}dx-\frac {14105216}{9} \left (15-4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^3}dx-\frac {4687220}{27} \left (5-4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^3}dx+\frac {375760}{3} \left (15-8 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^3}dx-\frac {4115587}{9} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^3}dx+\frac {1171805}{162} \left (28-\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^2}dx-\frac {46970}{3} \left (20-\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^2}dx+\frac {1763152}{9} \left (12-\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^2}dx-\frac {1958720}{3} \left (4-\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^2}dx-\frac {927432923}{648} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (-2 x+4 \sqrt {15}-16\right )^2}dx-\frac {77871461}{864} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{-2 x+4 \sqrt {15}-16}dx-50 \int e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}} xdx+8 \int e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}} x^2dx+12 \int e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}} x^3dx+4 \int e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}} x^4dx+\frac {5}{3} \left (20880-23417 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{2 x-4 \sqrt {15}+16}dx+\frac {299982080}{27} \left (15+4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^5}dx-\frac {38102720}{9} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^5}dx+\frac {37497760}{27} \left (20+3 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^4}dx-7515200 \left (4+\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^4}dx-\frac {288497600}{27} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^4}dx-\frac {375760}{3} \left (15+8 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^3}dx+\frac {14105216}{9} \left (15+4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^3}dx+\frac {4687220}{27} \left (5+4 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^3}dx-\frac {4115587}{9} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^3}dx+\frac {1171805}{162} \left (28+\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^2}dx-\frac {46970}{3} \left (20+\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^2}dx+\frac {1763152}{9} \left (12+\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^2}dx-\frac {1958720}{3} \left (4+\sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^2}dx-\frac {927432923}{648} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{\left (2 x+4 \sqrt {15}+16\right )^2}dx+\frac {5}{3} \left (20880+23417 \sqrt {15}\right ) \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{2 x+4 \sqrt {15}+16}dx-\frac {77871461}{864} \sqrt {\frac {5}{3}} \int \frac {e^{\frac {x^2 \left (x^5+34 x^4+328 x^3+631 x^2+272 x+32\right )^2}{\left (x^2+16 x+4\right )^4}}}{2 x+4 \sqrt {15}+16}dx\right )\) |
Input:
Int[(E^((1024*x^2 + 17408*x^3 + 114368*x^4 + 364256*x^5 + 578769*x^6 + 432 496*x^7 + 151036*x^8 + 23566*x^9 + 1812*x^10 + 68*x^11 + x^12)/(256 + 4096 *x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x ^8))*(4096 + 81920*x + 693248*x^2 + 3407872*x^3 + 11917824*x^4 + 34328064* x^5 + 78478560*x^6 + 118319872*x^7 + 97752008*x^8 + 40329056*x^9 + 7831044 *x^10 + 802040*x^11 + 45152*x^12 + 1328*x^13 + 16*x^14))/(1024 + 20480*x + 165120*x^2 + 675840*x^3 + 1434240*x^4 + 1383936*x^5 + 358560*x^6 + 42240* x^7 + 2580*x^8 + 80*x^9 + x^10),x]
Output:
$Aborted
Time = 1.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69
| method | result | size |
| risch | \(4 x \,{\mathrm e}^{\frac {x^{2} \left (x^{5}+34 x^{4}+328 x^{3}+631 x^{2}+272 x +32\right )^{2}}{\left (x^{2}+16 x +4\right )^{4}}}\) | \(44\) |
| gosper | \(4 x \,{\mathrm e}^{\frac {x^{2} \left (x^{10}+68 x^{9}+1812 x^{8}+23566 x^{7}+151036 x^{6}+432496 x^{5}+578769 x^{4}+364256 x^{3}+114368 x^{2}+17408 x +1024\right )}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}\) | \(97\) |
| parallelrisch | \(4 x \,{\mathrm e}^{\frac {x^{2} \left (x^{10}+68 x^{9}+1812 x^{8}+23566 x^{7}+151036 x^{6}+432496 x^{5}+578769 x^{4}+364256 x^{3}+114368 x^{2}+17408 x +1024\right )}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}\) | \(97\) |
| orering | \(\frac {\left (x^{2}+16 x +4\right )^{5} x \left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^{9}+97752008 x^{8}+118319872 x^{7}+78478560 x^{6}+34328064 x^{5}+11917824 x^{4}+3407872 x^{3}+693248 x^{2}+81920 x +4096\right ) {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}}{\left (4 x^{14}+332 x^{13}+11288 x^{12}+200510 x^{11}+1957761 x^{10}+10082264 x^{9}+24438002 x^{8}+29579968 x^{7}+19619640 x^{6}+8582016 x^{5}+2979456 x^{4}+851968 x^{3}+173312 x^{2}+20480 x +1024\right ) \left (x^{10}+80 x^{9}+2580 x^{8}+42240 x^{7}+358560 x^{6}+1383936 x^{5}+1434240 x^{4}+675840 x^{3}+165120 x^{2}+20480 x +1024\right )}\) | \(301\) |
| norman | \(\frac {1024 x \,{\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+16384 x^{2} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+99328 x^{3} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+274432 x^{4} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+311680 x^{5} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+68608 x^{6} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+6208 x^{7} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+256 x^{8} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}+4 x^{9} {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}}{\left (x^{2}+16 x +4\right )^{4}}\) | \(920\) |
Input:
int((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056*x^9+97 752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407872*x^ 3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151036*x^ 8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x^8+64* x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x^10+80 *x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^3+1651 20*x^2+20480*x+1024),x,method=_RETURNVERBOSE)
Output:
4*x*exp(x^2*(x^5+34*x^4+328*x^3+631*x^2+272*x+32)^2/(x^2+16*x+4)^4)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4 \, x e^{\left (\frac {x^{12} + 68 \, x^{11} + 1812 \, x^{10} + 23566 \, x^{9} + 151036 \, x^{8} + 432496 \, x^{7} + 578769 \, x^{6} + 364256 \, x^{5} + 114368 \, x^{4} + 17408 \, x^{3} + 1024 \, x^{2}}{x^{8} + 64 \, x^{7} + 1552 \, x^{6} + 17152 \, x^{5} + 77920 \, x^{4} + 68608 \, x^{3} + 24832 \, x^{2} + 4096 \, x + 256}\right )} \] Input:
integrate((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056* x^9+97752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407 872*x^3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151 036*x^8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x ^8+64*x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x ^10+80*x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^ 3+165120*x^2+20480*x+1024),x, algorithm="fricas")
Output:
4*x*e^((x^12 + 68*x^11 + 1812*x^10 + 23566*x^9 + 151036*x^8 + 432496*x^7 + 578769*x^6 + 364256*x^5 + 114368*x^4 + 17408*x^3 + 1024*x^2)/(x^8 + 64*x^ 7 + 1552*x^6 + 17152*x^5 + 77920*x^4 + 68608*x^3 + 24832*x^2 + 4096*x + 25 6))
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.73 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4 x e^{\frac {x^{12} + 68 x^{11} + 1812 x^{10} + 23566 x^{9} + 151036 x^{8} + 432496 x^{7} + 578769 x^{6} + 364256 x^{5} + 114368 x^{4} + 17408 x^{3} + 1024 x^{2}}{x^{8} + 64 x^{7} + 1552 x^{6} + 17152 x^{5} + 77920 x^{4} + 68608 x^{3} + 24832 x^{2} + 4096 x + 256}} \] Input:
integrate((16*x**14+1328*x**13+45152*x**12+802040*x**11+7831044*x**10+4032 9056*x**9+97752008*x**8+118319872*x**7+78478560*x**6+34328064*x**5+1191782 4*x**4+3407872*x**3+693248*x**2+81920*x+4096)*exp((x**12+68*x**11+1812*x** 10+23566*x**9+151036*x**8+432496*x**7+578769*x**6+364256*x**5+114368*x**4+ 17408*x**3+1024*x**2)/(x**8+64*x**7+1552*x**6+17152*x**5+77920*x**4+68608* x**3+24832*x**2+4096*x+256))/(x**10+80*x**9+2580*x**8+42240*x**7+358560*x* *6+1383936*x**5+1434240*x**4+675840*x**3+165120*x**2+20480*x+1024),x)
Output:
4*x*exp((x**12 + 68*x**11 + 1812*x**10 + 23566*x**9 + 151036*x**8 + 432496 *x**7 + 578769*x**6 + 364256*x**5 + 114368*x**4 + 17408*x**3 + 1024*x**2)/ (x**8 + 64*x**7 + 1552*x**6 + 17152*x**5 + 77920*x**4 + 68608*x**3 + 24832 *x**2 + 4096*x + 256))
Timed out. \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=\text {Timed out} \] Input:
integrate((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056* x^9+97752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407 872*x^3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151 036*x^8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x ^8+64*x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x ^10+80*x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^ 3+165120*x^2+20480*x+1024),x, algorithm="maxima")
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (25) = 50\).
Time = 0.77 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4 \, x e^{\left (\frac {x^{12} + 68 \, x^{11} + 1812 \, x^{10} + 23566 \, x^{9} + 151036 \, x^{8} + 432496 \, x^{7} + 578769 \, x^{6} + 364256 \, x^{5} + 114368 \, x^{4} + 17408 \, x^{3} + 1024 \, x^{2}}{x^{8} + 64 \, x^{7} + 1552 \, x^{6} + 17152 \, x^{5} + 77920 \, x^{4} + 68608 \, x^{3} + 24832 \, x^{2} + 4096 \, x + 256}\right )} \] Input:
integrate((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056* x^9+97752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407 872*x^3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151 036*x^8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x ^8+64*x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x ^10+80*x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^ 3+165120*x^2+20480*x+1024),x, algorithm="giac")
Output:
4*x*e^((x^12 + 68*x^11 + 1812*x^10 + 23566*x^9 + 151036*x^8 + 432496*x^7 + 578769*x^6 + 364256*x^5 + 114368*x^4 + 17408*x^3 + 1024*x^2)/(x^8 + 64*x^ 7 + 1552*x^6 + 17152*x^5 + 77920*x^4 + 68608*x^3 + 24832*x^2 + 4096*x + 25 6))
Time = 2.81 (sec) , antiderivative size = 508, normalized size of antiderivative = 19.54 \[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=4\,x\,{\mathrm {e}}^{\frac {x^{12}}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {68\,x^{11}}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {1024\,x^2}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {1812\,x^{10}}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {17408\,x^3}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {23566\,x^9}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {114368\,x^4}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {151036\,x^8}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {364256\,x^5}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {432496\,x^7}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}}\,{\mathrm {e}}^{\frac {578769\,x^6}{x^8+64\,x^7+1552\,x^6+17152\,x^5+77920\,x^4+68608\,x^3+24832\,x^2+4096\,x+256}} \] Input:
int((exp((1024*x^2 + 17408*x^3 + 114368*x^4 + 364256*x^5 + 578769*x^6 + 43 2496*x^7 + 151036*x^8 + 23566*x^9 + 1812*x^10 + 68*x^11 + x^12)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*(81920*x + 693248*x^2 + 3407872*x^3 + 11917824*x^4 + 34328064*x^5 + 78478560*x^6 + 118319872*x^7 + 97752008*x^8 + 40329056*x^9 + 7831044*x^10 + 802040*x^11 + 45152*x^12 + 1328*x^13 + 16*x^14 + 4096))/(20480*x + 16512 0*x^2 + 675840*x^3 + 1434240*x^4 + 1383936*x^5 + 358560*x^6 + 42240*x^7 + 2580*x^8 + 80*x^9 + x^10 + 1024),x)
Output:
4*x*exp(x^12/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 155 2*x^6 + 64*x^7 + x^8 + 256))*exp((68*x^11)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((1024*x^2)/ (4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^ 7 + x^8 + 256))*exp((1812*x^10)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^ 4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((17408*x^3)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((23566*x^9)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152* x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((114368*x^4)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp ((151036*x^8)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 15 52*x^6 + 64*x^7 + x^8 + 256))*exp((364256*x^5)/(4096*x + 24832*x^2 + 68608 *x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((432496 *x^7)/(4096*x + 24832*x^2 + 68608*x^3 + 77920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))*exp((578769*x^6)/(4096*x + 24832*x^2 + 68608*x^3 + 7 7920*x^4 + 17152*x^5 + 1552*x^6 + 64*x^7 + x^8 + 256))
\[ \int \frac {e^{\frac {1024 x^2+17408 x^3+114368 x^4+364256 x^5+578769 x^6+432496 x^7+151036 x^8+23566 x^9+1812 x^{10}+68 x^{11}+x^{12}}{256+4096 x+24832 x^2+68608 x^3+77920 x^4+17152 x^5+1552 x^6+64 x^7+x^8}} \left (4096+81920 x+693248 x^2+3407872 x^3+11917824 x^4+34328064 x^5+78478560 x^6+118319872 x^7+97752008 x^8+40329056 x^9+7831044 x^{10}+802040 x^{11}+45152 x^{12}+1328 x^{13}+16 x^{14}\right )}{1024+20480 x+165120 x^2+675840 x^3+1434240 x^4+1383936 x^5+358560 x^6+42240 x^7+2580 x^8+80 x^9+x^{10}} \, dx=\int \frac {\left (16 x^{14}+1328 x^{13}+45152 x^{12}+802040 x^{11}+7831044 x^{10}+40329056 x^{9}+97752008 x^{8}+118319872 x^{7}+78478560 x^{6}+34328064 x^{5}+11917824 x^{4}+3407872 x^{3}+693248 x^{2}+81920 x +4096\right ) {\mathrm e}^{\frac {x^{12}+68 x^{11}+1812 x^{10}+23566 x^{9}+151036 x^{8}+432496 x^{7}+578769 x^{6}+364256 x^{5}+114368 x^{4}+17408 x^{3}+1024 x^{2}}{x^{8}+64 x^{7}+1552 x^{6}+17152 x^{5}+77920 x^{4}+68608 x^{3}+24832 x^{2}+4096 x +256}}}{x^{10}+80 x^{9}+2580 x^{8}+42240 x^{7}+358560 x^{6}+1383936 x^{5}+1434240 x^{4}+675840 x^{3}+165120 x^{2}+20480 x +1024}d x \] Input:
int((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056*x^9+97 752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407872*x^ 3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151036*x^ 8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x^8+64* x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x^10+80 *x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^3+1651 20*x^2+20480*x+1024),x)
Output:
int((16*x^14+1328*x^13+45152*x^12+802040*x^11+7831044*x^10+40329056*x^9+97 752008*x^8+118319872*x^7+78478560*x^6+34328064*x^5+11917824*x^4+3407872*x^ 3+693248*x^2+81920*x+4096)*exp((x^12+68*x^11+1812*x^10+23566*x^9+151036*x^ 8+432496*x^7+578769*x^6+364256*x^5+114368*x^4+17408*x^3+1024*x^2)/(x^8+64* x^7+1552*x^6+17152*x^5+77920*x^4+68608*x^3+24832*x^2+4096*x+256))/(x^10+80 *x^9+2580*x^8+42240*x^7+358560*x^6+1383936*x^5+1434240*x^4+675840*x^3+1651 20*x^2+20480*x+1024),x)