Integrand size = 316, antiderivative size = 22 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{4+3 \left (81+\left (5+3 e^x\right )^4+x\right )^2} \] Output:
2/(3*(81+x+(5+3*exp(x))^4)^2+4)
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(22)=44\).
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{1495312+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+4236 x+3 x^2+9000 e^x (706+x)+3240 e^{3 x} (4456+x)+486 e^{4 x} (21956+x)+2700 e^{2 x} (4618+3 x)} \] Input:
Integrate[(-8472 - 51030000*E^(5*x) - 18370800*E^(6*x) - 3674160*E^(7*x) - 314928*E^(8*x) + E^(2*x)*(-49890600 - 32400*x) + E^(3*x)*(-86631120 - 194 40*x) + E^x*(-12726000 - 18000*x) + E^(4*x)*(-85365900 - 3888*x) - 12*x)/( 2235957977344 + 1004423490000*E^(13*x) + 129140163000*E^(14*x) + 103312130 40*E^(15*x) + 387420489*E^(16*x) + 12668283264*x + 26915568*x^2 + 25416*x^ 3 + 9*x^4 + E^(12*x)*(5442176919456 + 19131876*x) + E^(11*x)*(217935025891 20 + 382637520*x) + E^(10*x)*(66780663483600 + 3507510600*x) + E^(9*x)*(15 9903511020000 + 19486170000*x) + E^(8*x)*(302780869911648 + 73130533128*x + 354294*x^2) + E^(7*x)*(455606824688640 + 195626975040*x + 4723920*x^2) + E^(6*x)*(543962229350400 + 383361854400*x + 27556200*x^2) + E^(5*x)*(5108 92671168000 + 556162848000*x + 91854000*x^2) + E^(4*x)*(370848773784384 + 594894062016*x + 192071088*x^2 + 2916*x^3) + E^(3*x)*(201627923362560 + 45 9373213440*x + 259873920*x^2 + 19440*x^3) + E^(2*x)*(77662210406400 + 2442 30033600*x + 224434800*x^2 + 48600*x^3) + E^x*(19002424896000 + 8074670400 0*x + 114372000*x^2 + 54000*x^3)),x]
Output:
2/(1495312 + 5103000*E^(5*x) + 1530900*E^(6*x) + 262440*E^(7*x) + 19683*E^ (8*x) + 4236*x + 3*x^2 + 9000*E^x*(706 + x) + 3240*E^(3*x)*(4456 + x) + 48 6*E^(4*x)*(21956 + x) + 2700*E^(2*x)*(4618 + 3*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.
| \(\displaystyle \int \frac {e^{2 x} (-32400 x-49890600)-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{3 x} (-19440 x-86631120)+e^x (-18000 x-12726000)+e^{4 x} (-3888 x-85365900)-12 x-8472}{9 x^4+25416 x^3+26915568 x^2+e^{8 x} \left (354294 x^2+73130533128 x+302780869911648\right )+e^{7 x} \left (4723920 x^2+195626975040 x+455606824688640\right )+e^{6 x} \left (27556200 x^2+383361854400 x+543962229350400\right )+e^{5 x} \left (91854000 x^2+556162848000 x+510892671168000\right )+e^{4 x} \left (2916 x^3+192071088 x^2+594894062016 x+370848773784384\right )+e^{3 x} \left (19440 x^3+259873920 x^2+459373213440 x+201627923362560\right )+e^{2 x} \left (48600 x^3+224434800 x^2+244230033600 x+77662210406400\right )+e^x \left (54000 x^3+114372000 x^2+80746704000 x+19002424896000\right )+12668283264 x+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+e^{12 x} (19131876 x+5442176919456)+e^{11 x} (382637520 x+21793502589120)+e^{10 x} (3507510600 x+66780663483600)+e^{9 x} (19486170000 x+159903511020000)+2235957977344} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 \left (1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}+1\right ) \left (-x-1500 e^x-1350 e^{2 x}-540 e^{3 x}-81 e^{4 x}-706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 12 \int -\frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -12 \int \frac {\left (1+1500 e^x+2700 e^{2 x}+1620 e^{3 x}+324 e^{4 x}\right ) \left (x+1500 e^x+1350 e^{2 x}+540 e^{3 x}+81 e^{4 x}+706\right )}{\left (3 x^2+4236 x+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+9000 e^x (x+706)+3240 e^{3 x} (x+4456)+486 e^{4 x} (x+21956)+2700 e^{2 x} (3 x+4618)+1495312\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -12 \int \left (\frac {4}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )}-\frac {12 x^2+31500 e^x x+24300 e^{2 x} x+8100 e^{3 x} x+972 e^{4 x} x+16941 x+22234500 e^x+37401750 e^{2 x}+36091980 e^{3 x}+21340989 e^{4 x}+7654500 e^{5 x}+1530900 e^{6 x}+131220 e^{7 x}+5979130}{3 \left (3 x^2+9000 e^x x+8100 e^{2 x} x+3240 e^{3 x} x+486 e^{4 x} x+4236 x+6354000 e^x+12468600 e^{2 x}+14437440 e^{3 x}+10670616 e^{4 x}+5103000 e^{5 x}+1530900 e^{6 x}+262440 e^{7 x}+19683 e^{8 x}+1495312\right )^2}\right )dx\) |
Input:
Int[(-8472 - 51030000*E^(5*x) - 18370800*E^(6*x) - 3674160*E^(7*x) - 31492 8*E^(8*x) + E^(2*x)*(-49890600 - 32400*x) + E^(3*x)*(-86631120 - 19440*x) + E^x*(-12726000 - 18000*x) + E^(4*x)*(-85365900 - 3888*x) - 12*x)/(223595 7977344 + 1004423490000*E^(13*x) + 129140163000*E^(14*x) + 10331213040*E^( 15*x) + 387420489*E^(16*x) + 12668283264*x + 26915568*x^2 + 25416*x^3 + 9* x^4 + E^(12*x)*(5442176919456 + 19131876*x) + E^(11*x)*(21793502589120 + 3 82637520*x) + E^(10*x)*(66780663483600 + 3507510600*x) + E^(9*x)*(15990351 1020000 + 19486170000*x) + E^(8*x)*(302780869911648 + 73130533128*x + 3542 94*x^2) + E^(7*x)*(455606824688640 + 195626975040*x + 4723920*x^2) + E^(6* x)*(543962229350400 + 383361854400*x + 27556200*x^2) + E^(5*x)*(5108926711 68000 + 556162848000*x + 91854000*x^2) + E^(4*x)*(370848773784384 + 594894 062016*x + 192071088*x^2 + 2916*x^3) + E^(3*x)*(201627923362560 + 45937321 3440*x + 259873920*x^2 + 19440*x^3) + E^(2*x)*(77662210406400 + 2442300336 00*x + 224434800*x^2 + 48600*x^3) + E^x*(19002424896000 + 80746704000*x + 114372000*x^2 + 54000*x^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(21)=42\).
Time = 1.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95
| method | result | size |
| risch | \(\frac {2}{19683 \,{\mathrm e}^{8 x}+262440 \,{\mathrm e}^{7 x}+1530900 \,{\mathrm e}^{6 x}+5103000 \,{\mathrm e}^{5 x}+486 x \,{\mathrm e}^{4 x}+10670616 \,{\mathrm e}^{4 x}+3240 x \,{\mathrm e}^{3 x}+14437440 \,{\mathrm e}^{3 x}+8100 x \,{\mathrm e}^{2 x}+12468600 \,{\mathrm e}^{2 x}+9000 \,{\mathrm e}^{x} x +3 x^{2}+6354000 \,{\mathrm e}^{x}+4236 x +1495312}\) | \(87\) |
| parallelrisch | \(\frac {2}{19683 \,{\mathrm e}^{8 x}+262440 \,{\mathrm e}^{7 x}+1530900 \,{\mathrm e}^{6 x}+5103000 \,{\mathrm e}^{5 x}+486 x \,{\mathrm e}^{4 x}+10670616 \,{\mathrm e}^{4 x}+3240 x \,{\mathrm e}^{3 x}+14437440 \,{\mathrm e}^{3 x}+8100 x \,{\mathrm e}^{2 x}+12468600 \,{\mathrm e}^{2 x}+9000 \,{\mathrm e}^{x} x +3 x^{2}+6354000 \,{\mathrm e}^{x}+4236 x +1495312}\) | \(87\) |
Input:
int((-314928*exp(x)^8-3674160*exp(x)^7-18370800*exp(x)^6-51030000*exp(x)^5 +(-3888*x-85365900)*exp(x)^4+(-19440*x-86631120)*exp(x)^3+(-32400*x-498906 00)*exp(x)^2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489*exp(x)^16+10 331213040*exp(x)^15+129140163000*exp(x)^14+1004423490000*exp(x)^13+(191318 76*x+5442176919456)*exp(x)^12+(382637520*x+21793502589120)*exp(x)^11+(3507 510600*x+66780663483600)*exp(x)^10+(19486170000*x+159903511020000)*exp(x)^ 9+(354294*x^2+73130533128*x+302780869911648)*exp(x)^8+(4723920*x^2+1956269 75040*x+455606824688640)*exp(x)^7+(27556200*x^2+383361854400*x+54396222935 0400)*exp(x)^6+(91854000*x^2+556162848000*x+510892671168000)*exp(x)^5+(291 6*x^3+192071088*x^2+594894062016*x+370848773784384)*exp(x)^4+(19440*x^3+25 9873920*x^2+459373213440*x+201627923362560)*exp(x)^3+(48600*x^3+224434800* x^2+244230033600*x+77662210406400)*exp(x)^2+(54000*x^3+114372000*x^2+80746 704000*x+19002424896000)*exp(x)+9*x^4+25416*x^3+26915568*x^2+12668283264*x +2235957977344),x,method=_RETURNVERBOSE)
Output:
2/(19683*exp(x)^8+262440*exp(x)^7+1530900*exp(x)^6+5103000*exp(x)^5+486*x* exp(x)^4+10670616*exp(x)^4+3240*x*exp(x)^3+14437440*exp(x)^3+8100*x*exp(x) ^2+12468600*exp(x)^2+9000*exp(x)*x+3*x^2+6354000*exp(x)+4236*x+1495312)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{3 \, x^{2} + 486 \, {\left (x + 21956\right )} e^{\left (4 \, x\right )} + 3240 \, {\left (x + 4456\right )} e^{\left (3 \, x\right )} + 2700 \, {\left (3 \, x + 4618\right )} e^{\left (2 \, x\right )} + 9000 \, {\left (x + 706\right )} e^{x} + 4236 \, x + 19683 \, e^{\left (8 \, x\right )} + 262440 \, e^{\left (7 \, x\right )} + 1530900 \, e^{\left (6 \, x\right )} + 5103000 \, e^{\left (5 \, x\right )} + 1495312} \] Input:
integrate((-314928*exp(x)^8-3674160*exp(x)^7-18370800*exp(x)^6-51030000*ex p(x)^5+(-3888*x-85365900)*exp(x)^4+(-19440*x-86631120)*exp(x)^3+(-32400*x- 49890600)*exp(x)^2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489*exp(x) ^16+10331213040*exp(x)^15+129140163000*exp(x)^14+1004423490000*exp(x)^13+( 19131876*x+5442176919456)*exp(x)^12+(382637520*x+21793502589120)*exp(x)^11 +(3507510600*x+66780663483600)*exp(x)^10+(19486170000*x+159903511020000)*e xp(x)^9+(354294*x^2+73130533128*x+302780869911648)*exp(x)^8+(4723920*x^2+1 95626975040*x+455606824688640)*exp(x)^7+(27556200*x^2+383361854400*x+54396 2229350400)*exp(x)^6+(91854000*x^2+556162848000*x+510892671168000)*exp(x)^ 5+(2916*x^3+192071088*x^2+594894062016*x+370848773784384)*exp(x)^4+(19440* x^3+259873920*x^2+459373213440*x+201627923362560)*exp(x)^3+(48600*x^3+2244 34800*x^2+244230033600*x+77662210406400)*exp(x)^2+(54000*x^3+114372000*x^2 +80746704000*x+19002424896000)*exp(x)+9*x^4+25416*x^3+26915568*x^2+1266828 3264*x+2235957977344),x, algorithm="fricas")
Output:
2/(3*x^2 + 486*(x + 21956)*e^(4*x) + 3240*(x + 4456)*e^(3*x) + 2700*(3*x + 4618)*e^(2*x) + 9000*(x + 706)*e^x + 4236*x + 19683*e^(8*x) + 262440*e^(7 *x) + 1530900*e^(6*x) + 5103000*e^(5*x) + 1495312)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.45 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{3 x^{2} + 4236 x + \left (486 x + 10670616\right ) e^{4 x} + \left (3240 x + 14437440\right ) e^{3 x} + \left (8100 x + 12468600\right ) e^{2 x} + \left (9000 x + 6354000\right ) e^{x} + 19683 e^{8 x} + 262440 e^{7 x} + 1530900 e^{6 x} + 5103000 e^{5 x} + 1495312} \] Input:
integrate((-314928*exp(x)**8-3674160*exp(x)**7-18370800*exp(x)**6-51030000 *exp(x)**5+(-3888*x-85365900)*exp(x)**4+(-19440*x-86631120)*exp(x)**3+(-32 400*x-49890600)*exp(x)**2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489 *exp(x)**16+10331213040*exp(x)**15+129140163000*exp(x)**14+1004423490000*e xp(x)**13+(19131876*x+5442176919456)*exp(x)**12+(382637520*x+2179350258912 0)*exp(x)**11+(3507510600*x+66780663483600)*exp(x)**10+(19486170000*x+1599 03511020000)*exp(x)**9+(354294*x**2+73130533128*x+302780869911648)*exp(x)* *8+(4723920*x**2+195626975040*x+455606824688640)*exp(x)**7+(27556200*x**2+ 383361854400*x+543962229350400)*exp(x)**6+(91854000*x**2+556162848000*x+51 0892671168000)*exp(x)**5+(2916*x**3+192071088*x**2+594894062016*x+37084877 3784384)*exp(x)**4+(19440*x**3+259873920*x**2+459373213440*x+2016279233625 60)*exp(x)**3+(48600*x**3+224434800*x**2+244230033600*x+77662210406400)*ex p(x)**2+(54000*x**3+114372000*x**2+80746704000*x+19002424896000)*exp(x)+9* x**4+25416*x**3+26915568*x**2+12668283264*x+2235957977344),x)
Output:
2/(3*x**2 + 4236*x + (486*x + 10670616)*exp(4*x) + (3240*x + 14437440)*exp (3*x) + (8100*x + 12468600)*exp(2*x) + (9000*x + 6354000)*exp(x) + 19683*e xp(8*x) + 262440*exp(7*x) + 1530900*exp(6*x) + 5103000*exp(5*x) + 1495312)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (21) = 42\).
Time = 0.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{3 \, x^{2} + 486 \, {\left (x + 21956\right )} e^{\left (4 \, x\right )} + 3240 \, {\left (x + 4456\right )} e^{\left (3 \, x\right )} + 2700 \, {\left (3 \, x + 4618\right )} e^{\left (2 \, x\right )} + 9000 \, {\left (x + 706\right )} e^{x} + 4236 \, x + 19683 \, e^{\left (8 \, x\right )} + 262440 \, e^{\left (7 \, x\right )} + 1530900 \, e^{\left (6 \, x\right )} + 5103000 \, e^{\left (5 \, x\right )} + 1495312} \] Input:
integrate((-314928*exp(x)^8-3674160*exp(x)^7-18370800*exp(x)^6-51030000*ex p(x)^5+(-3888*x-85365900)*exp(x)^4+(-19440*x-86631120)*exp(x)^3+(-32400*x- 49890600)*exp(x)^2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489*exp(x) ^16+10331213040*exp(x)^15+129140163000*exp(x)^14+1004423490000*exp(x)^13+( 19131876*x+5442176919456)*exp(x)^12+(382637520*x+21793502589120)*exp(x)^11 +(3507510600*x+66780663483600)*exp(x)^10+(19486170000*x+159903511020000)*e xp(x)^9+(354294*x^2+73130533128*x+302780869911648)*exp(x)^8+(4723920*x^2+1 95626975040*x+455606824688640)*exp(x)^7+(27556200*x^2+383361854400*x+54396 2229350400)*exp(x)^6+(91854000*x^2+556162848000*x+510892671168000)*exp(x)^ 5+(2916*x^3+192071088*x^2+594894062016*x+370848773784384)*exp(x)^4+(19440* x^3+259873920*x^2+459373213440*x+201627923362560)*exp(x)^3+(48600*x^3+2244 34800*x^2+244230033600*x+77662210406400)*exp(x)^2+(54000*x^3+114372000*x^2 +80746704000*x+19002424896000)*exp(x)+9*x^4+25416*x^3+26915568*x^2+1266828 3264*x+2235957977344),x, algorithm="maxima")
Output:
2/(3*x^2 + 486*(x + 21956)*e^(4*x) + 3240*(x + 4456)*e^(3*x) + 2700*(3*x + 4618)*e^(2*x) + 9000*(x + 706)*e^x + 4236*x + 19683*e^(8*x) + 262440*e^(7 *x) + 1530900*e^(6*x) + 5103000*e^(5*x) + 1495312)
Timed out. \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((-314928*exp(x)^8-3674160*exp(x)^7-18370800*exp(x)^6-51030000*ex p(x)^5+(-3888*x-85365900)*exp(x)^4+(-19440*x-86631120)*exp(x)^3+(-32400*x- 49890600)*exp(x)^2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489*exp(x) ^16+10331213040*exp(x)^15+129140163000*exp(x)^14+1004423490000*exp(x)^13+( 19131876*x+5442176919456)*exp(x)^12+(382637520*x+21793502589120)*exp(x)^11 +(3507510600*x+66780663483600)*exp(x)^10+(19486170000*x+159903511020000)*e xp(x)^9+(354294*x^2+73130533128*x+302780869911648)*exp(x)^8+(4723920*x^2+1 95626975040*x+455606824688640)*exp(x)^7+(27556200*x^2+383361854400*x+54396 2229350400)*exp(x)^6+(91854000*x^2+556162848000*x+510892671168000)*exp(x)^ 5+(2916*x^3+192071088*x^2+594894062016*x+370848773784384)*exp(x)^4+(19440* x^3+259873920*x^2+459373213440*x+201627923362560)*exp(x)^3+(48600*x^3+2244 34800*x^2+244230033600*x+77662210406400)*exp(x)^2+(54000*x^3+114372000*x^2 +80746704000*x+19002424896000)*exp(x)+9*x^4+25416*x^3+26915568*x^2+1266828 3264*x+2235957977344),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\int -\frac {12\,x+51030000\,{\mathrm {e}}^{5\,x}+18370800\,{\mathrm {e}}^{6\,x}+3674160\,{\mathrm {e}}^{7\,x}+314928\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^x\,\left (18000\,x+12726000\right )+{\mathrm {e}}^{2\,x}\,\left (32400\,x+49890600\right )+{\mathrm {e}}^{4\,x}\,\left (3888\,x+85365900\right )+{\mathrm {e}}^{3\,x}\,\left (19440\,x+86631120\right )+8472}{12668283264\,x+1004423490000\,{\mathrm {e}}^{13\,x}+129140163000\,{\mathrm {e}}^{14\,x}+10331213040\,{\mathrm {e}}^{15\,x}+387420489\,{\mathrm {e}}^{16\,x}+{\mathrm {e}}^{5\,x}\,\left (91854000\,x^2+556162848000\,x+510892671168000\right )+{\mathrm {e}}^x\,\left (54000\,x^3+114372000\,x^2+80746704000\,x+19002424896000\right )+{\mathrm {e}}^{11\,x}\,\left (382637520\,x+21793502589120\right )+{\mathrm {e}}^{10\,x}\,\left (3507510600\,x+66780663483600\right )+{\mathrm {e}}^{2\,x}\,\left (48600\,x^3+224434800\,x^2+244230033600\,x+77662210406400\right )+{\mathrm {e}}^{6\,x}\,\left (27556200\,x^2+383361854400\,x+543962229350400\right )+{\mathrm {e}}^{3\,x}\,\left (19440\,x^3+259873920\,x^2+459373213440\,x+201627923362560\right )+{\mathrm {e}}^{8\,x}\,\left (354294\,x^2+73130533128\,x+302780869911648\right )+{\mathrm {e}}^{4\,x}\,\left (2916\,x^3+192071088\,x^2+594894062016\,x+370848773784384\right )+{\mathrm {e}}^{9\,x}\,\left (19486170000\,x+159903511020000\right )+{\mathrm {e}}^{7\,x}\,\left (4723920\,x^2+195626975040\,x+455606824688640\right )+26915568\,x^2+25416\,x^3+9\,x^4+{\mathrm {e}}^{12\,x}\,\left (19131876\,x+5442176919456\right )+2235957977344} \,d x \] Input:
int(-(12*x + 51030000*exp(5*x) + 18370800*exp(6*x) + 3674160*exp(7*x) + 31 4928*exp(8*x) + exp(x)*(18000*x + 12726000) + exp(2*x)*(32400*x + 49890600 ) + exp(4*x)*(3888*x + 85365900) + exp(3*x)*(19440*x + 86631120) + 8472)/( 12668283264*x + 1004423490000*exp(13*x) + 129140163000*exp(14*x) + 1033121 3040*exp(15*x) + 387420489*exp(16*x) + exp(5*x)*(556162848000*x + 91854000 *x^2 + 510892671168000) + exp(x)*(80746704000*x + 114372000*x^2 + 54000*x^ 3 + 19002424896000) + exp(11*x)*(382637520*x + 21793502589120) + exp(10*x) *(3507510600*x + 66780663483600) + exp(2*x)*(244230033600*x + 224434800*x^ 2 + 48600*x^3 + 77662210406400) + exp(6*x)*(383361854400*x + 27556200*x^2 + 543962229350400) + exp(3*x)*(459373213440*x + 259873920*x^2 + 19440*x^3 + 201627923362560) + exp(8*x)*(73130533128*x + 354294*x^2 + 30278086991164 8) + exp(4*x)*(594894062016*x + 192071088*x^2 + 2916*x^3 + 370848773784384 ) + exp(9*x)*(19486170000*x + 159903511020000) + exp(7*x)*(195626975040*x + 4723920*x^2 + 455606824688640) + 26915568*x^2 + 25416*x^3 + 9*x^4 + exp( 12*x)*(19131876*x + 5442176919456) + 2235957977344),x)
Output:
int(-(12*x + 51030000*exp(5*x) + 18370800*exp(6*x) + 3674160*exp(7*x) + 31 4928*exp(8*x) + exp(x)*(18000*x + 12726000) + exp(2*x)*(32400*x + 49890600 ) + exp(4*x)*(3888*x + 85365900) + exp(3*x)*(19440*x + 86631120) + 8472)/( 12668283264*x + 1004423490000*exp(13*x) + 129140163000*exp(14*x) + 1033121 3040*exp(15*x) + 387420489*exp(16*x) + exp(5*x)*(556162848000*x + 91854000 *x^2 + 510892671168000) + exp(x)*(80746704000*x + 114372000*x^2 + 54000*x^ 3 + 19002424896000) + exp(11*x)*(382637520*x + 21793502589120) + exp(10*x) *(3507510600*x + 66780663483600) + exp(2*x)*(244230033600*x + 224434800*x^ 2 + 48600*x^3 + 77662210406400) + exp(6*x)*(383361854400*x + 27556200*x^2 + 543962229350400) + exp(3*x)*(459373213440*x + 259873920*x^2 + 19440*x^3 + 201627923362560) + exp(8*x)*(73130533128*x + 354294*x^2 + 30278086991164 8) + exp(4*x)*(594894062016*x + 192071088*x^2 + 2916*x^3 + 370848773784384 ) + exp(9*x)*(19486170000*x + 159903511020000) + exp(7*x)*(195626975040*x + 4723920*x^2 + 455606824688640) + 26915568*x^2 + 25416*x^3 + 9*x^4 + exp( 12*x)*(19131876*x + 5442176919456) + 2235957977344), x)
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.45 \[ \int \frac {-8472-51030000 e^{5 x}-18370800 e^{6 x}-3674160 e^{7 x}-314928 e^{8 x}+e^{2 x} (-49890600-32400 x)+e^{3 x} (-86631120-19440 x)+e^x (-12726000-18000 x)+e^{4 x} (-85365900-3888 x)-12 x}{2235957977344+1004423490000 e^{13 x}+129140163000 e^{14 x}+10331213040 e^{15 x}+387420489 e^{16 x}+12668283264 x+26915568 x^2+25416 x^3+9 x^4+e^{12 x} (5442176919456+19131876 x)+e^{11 x} (21793502589120+382637520 x)+e^{10 x} (66780663483600+3507510600 x)+e^{9 x} (159903511020000+19486170000 x)+e^{8 x} \left (302780869911648+73130533128 x+354294 x^2\right )+e^{7 x} \left (455606824688640+195626975040 x+4723920 x^2\right )+e^{6 x} \left (543962229350400+383361854400 x+27556200 x^2\right )+e^{5 x} \left (510892671168000+556162848000 x+91854000 x^2\right )+e^{4 x} \left (370848773784384+594894062016 x+192071088 x^2+2916 x^3\right )+e^{3 x} \left (201627923362560+459373213440 x+259873920 x^2+19440 x^3\right )+e^{2 x} \left (77662210406400+244230033600 x+224434800 x^2+48600 x^3\right )+e^x \left (19002424896000+80746704000 x+114372000 x^2+54000 x^3\right )} \, dx=\frac {2}{19683 e^{8 x}+262440 e^{7 x}+1530900 e^{6 x}+5103000 e^{5 x}+486 e^{4 x} x +10670616 e^{4 x}+3240 e^{3 x} x +14437440 e^{3 x}+8100 e^{2 x} x +12468600 e^{2 x}+9000 e^{x} x +6354000 e^{x}+3 x^{2}+4236 x +1495312} \] Input:
int((-314928*exp(x)^8-3674160*exp(x)^7-18370800*exp(x)^6-51030000*exp(x)^5 +(-3888*x-85365900)*exp(x)^4+(-19440*x-86631120)*exp(x)^3+(-32400*x-498906 00)*exp(x)^2+(-18000*x-12726000)*exp(x)-12*x-8472)/(387420489*exp(x)^16+10 331213040*exp(x)^15+129140163000*exp(x)^14+1004423490000*exp(x)^13+(191318 76*x+5442176919456)*exp(x)^12+(382637520*x+21793502589120)*exp(x)^11+(3507 510600*x+66780663483600)*exp(x)^10+(19486170000*x+159903511020000)*exp(x)^ 9+(354294*x^2+73130533128*x+302780869911648)*exp(x)^8+(4723920*x^2+1956269 75040*x+455606824688640)*exp(x)^7+(27556200*x^2+383361854400*x+54396222935 0400)*exp(x)^6+(91854000*x^2+556162848000*x+510892671168000)*exp(x)^5+(291 6*x^3+192071088*x^2+594894062016*x+370848773784384)*exp(x)^4+(19440*x^3+25 9873920*x^2+459373213440*x+201627923362560)*exp(x)^3+(48600*x^3+224434800* x^2+244230033600*x+77662210406400)*exp(x)^2+(54000*x^3+114372000*x^2+80746 704000*x+19002424896000)*exp(x)+9*x^4+25416*x^3+26915568*x^2+12668283264*x +2235957977344),x)
Output:
2/(19683*e**(8*x) + 262440*e**(7*x) + 1530900*e**(6*x) + 5103000*e**(5*x) + 486*e**(4*x)*x + 10670616*e**(4*x) + 3240*e**(3*x)*x + 14437440*e**(3*x) + 8100*e**(2*x)*x + 12468600*e**(2*x) + 9000*e**x*x + 6354000*e**x + 3*x* *2 + 4236*x + 1495312)