Integrand size = 127, antiderivative size = 20 \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=e^{\left (2+\left (x+\frac {45 (5+x)}{1+x}\right )^2\right )^2} \] Output:
exp(((45*(5+x)/(1+x)+x)^2+2)^2)
\[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=\int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx \] Input:
Integrate[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x^6 + 184*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x^3 + 2569600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10* x^2 + 10*x^3 + 5*x^4 + x^5),x]
Output:
Integrate[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 513920*x^5 + 13600*x^6 + 184*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x^3 + 2569600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10* x^2 + 10*x^3 + 5*x^4 + x^5), 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.
| \(\displaystyle \int \frac {\left (4 x^8+560 x^7+28488 x^6+595520 x^5+2569600 x^4-73629456 x^3-1030398440 x^2-4911736672 x-8156009700\right ) \exp \left (\frac {x^8+184 x^7+13600 x^6+513920 x^5+10505414 x^4+115651112 x^3+688675888 x^2+2096362816 x+2563093129}{x^4+4 x^3+6 x^2+4 x+1}\right )}{x^5+5 x^4+10 x^3+10 x^2+5 x+1} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (4 x^8+560 x^7+28488 x^6+595520 x^5+2569600 x^4-73629456 x^3-1030398440 x^2-4911736672 x-8156009700\right ) \exp \left (\frac {x^8+184 x^7+13600 x^6+513920 x^5+10505414 x^4+115651112 x^3+688675888 x^2+2096362816 x+2563093129}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^5}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 \left (x^8+140 x^7+7122 x^6+148880 x^5+642400 x^4-18407364 x^3-257599610 x^2-1227934168 x-2039002425\right ) \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) \left (-x^8-140 x^7-7122 x^6-148880 x^5-642400 x^4+18407364 x^3+257599610 x^2+1227934168 x+2039002425\right )}{(x+1)^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) \left (-x^8-140 x^7-7122 x^6-148880 x^5-642400 x^4+18407364 x^3+257599610 x^2+1227934168 x+2039002425\right )}{(x+1)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) x^3-135 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) x^2-6437 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) x-115335 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )+\frac {19625760 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^2}+\frac {199908000 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^3}+\frac {769824000 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^4}+\frac {1049760000 \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-115335 \int \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )dx-6437 \int \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) xdx-135 \int \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) x^2dx-\int \exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right ) x^3dx+1049760000 \int \frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^5}dx+769824000 \int \frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^4}dx+199908000 \int \frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^3}dx+19625760 \int \frac {\exp \left (\frac {\left (x^4+92 x^3+2568 x^2+20704 x+50627\right )^2}{x^4+4 x^3+6 x^2+4 x+1}\right )}{(x+1)^2}dx\right )\) |
Input:
Int[(E^((2563093129 + 2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505 414*x^4 + 513920*x^5 + 13600*x^6 + 184*x^7 + x^8)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4))*(-8156009700 - 4911736672*x - 1030398440*x^2 - 73629456*x^3 + 256 9600*x^4 + 595520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8))/(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(19)=38\).
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05
\[{\mathrm e}^{\frac {x^{8}+184 x^{7}+13600 x^{6}+513920 x^{5}+10505414 x^{4}+115651112 x^{3}+688675888 x^{2}+2096362816 x +2563093129}{x^{4}+4 x^{3}+6 x^{2}+4 x +1}}\]
Input:
int((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103039844 0*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505 414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4*x^3+6* x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x)
Output:
exp((x^8+184*x^7+13600*x^6+513920*x^5+10505414*x^4+115651112*x^3+688675888 *x^2+2096362816*x+2563093129)/(x^4+4*x^3+6*x^2+4*x+1))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.00 \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=e^{\left (\frac {x^{8} + 184 \, x^{7} + 13600 \, x^{6} + 513920 \, x^{5} + 10505414 \, x^{4} + 115651112 \, x^{3} + 688675888 \, x^{2} + 2096362816 \, x + 2563093129}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right )} \] Input:
integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103 0398440*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5 +10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4* x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="fricas")
Output:
e^((x^8 + 184*x^7 + 13600*x^6 + 513920*x^5 + 10505414*x^4 + 115651112*x^3 + 688675888*x^2 + 2096362816*x + 2563093129)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (15) = 30\).
Time = 0.37 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.90 \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=e^{\frac {x^{8} + 184 x^{7} + 13600 x^{6} + 513920 x^{5} + 10505414 x^{4} + 115651112 x^{3} + 688675888 x^{2} + 2096362816 x + 2563093129}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}} \] Input:
integrate((4*x**8+560*x**7+28488*x**6+595520*x**5+2569600*x**4-73629456*x* *3-1030398440*x**2-4911736672*x-8156009700)*exp((x**8+184*x**7+13600*x**6+ 513920*x**5+10505414*x**4+115651112*x**3+688675888*x**2+2096362816*x+25630 93129)/(x**4+4*x**3+6*x**2+4*x+1))/(x**5+5*x**4+10*x**3+10*x**2+5*x+1),x)
Output:
exp((x**8 + 184*x**7 + 13600*x**6 + 513920*x**5 + 10505414*x**4 + 11565111 2*x**3 + 688675888*x**2 + 2096362816*x + 2563093129)/(x**4 + 4*x**3 + 6*x* *2 + 4*x + 1))
Timed out. \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=\text {Timed out} \] Input:
integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103 0398440*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5 +10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4* x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="maxima")
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 11.05 \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=e^{\left (\frac {x^{8}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {184 \, x^{7}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {13600 \, x^{6}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {513920 \, x^{5}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {10505414 \, x^{4}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {115651112 \, x^{3}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {688675888 \, x^{2}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {2096362816 \, x}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} + \frac {2563093129}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right )} \] Input:
integrate((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103 0398440*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5 +10505414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4* x^3+6*x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x, algorithm="giac")
Output:
e^(x^8/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 184*x^7/(x^4 + 4*x^3 + 6*x^2 + 4* x + 1) + 13600*x^6/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 513920*x^5/(x^4 + 4*x ^3 + 6*x^2 + 4*x + 1) + 10505414*x^4/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 115 651112*x^3/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 688675888*x^2/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 2096362816*x/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 25630931 29/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1))
Time = 2.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 11.45 \[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx={\mathrm {e}}^{\frac {2563093129}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {x^8}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {184\,x^7}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {13600\,x^6}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {513920\,x^5}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {10505414\,x^4}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {115651112\,x^3}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {2096362816\,x}{x^4+4\,x^3+6\,x^2+4\,x+1}}\,{\mathrm {e}}^{\frac {688675888\,x^2}{x^4+4\,x^3+6\,x^2+4\,x+1}} \] Input:
int((exp((2096362816*x + 688675888*x^2 + 115651112*x^3 + 10505414*x^4 + 51 3920*x^5 + 13600*x^6 + 184*x^7 + x^8 + 2563093129)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*(2569600*x^4 - 1030398440*x^2 - 73629456*x^3 - 4911736672*x + 59 5520*x^5 + 28488*x^6 + 560*x^7 + 4*x^8 - 8156009700))/(5*x + 10*x^2 + 10*x ^3 + 5*x^4 + x^5 + 1),x)
Output:
exp(2563093129/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp(x^8/(4*x + 6*x^2 + 4*x ^3 + x^4 + 1))*exp((184*x^7)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((13600*x ^6)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((513920*x^5)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((10505414*x^4)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((1156 51112*x^3)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))*exp((2096362816*x)/(4*x + 6*x^ 2 + 4*x^3 + x^4 + 1))*exp((688675888*x^2)/(4*x + 6*x^2 + 4*x^3 + x^4 + 1))
\[ \int \frac {e^{\frac {2563093129+2096362816 x+688675888 x^2+115651112 x^3+10505414 x^4+513920 x^5+13600 x^6+184 x^7+x^8}{1+4 x+6 x^2+4 x^3+x^4}} \left (-8156009700-4911736672 x-1030398440 x^2-73629456 x^3+2569600 x^4+595520 x^5+28488 x^6+560 x^7+4 x^8\right )}{1+5 x+10 x^2+10 x^3+5 x^4+x^5} \, dx=\int \frac {\left (4 x^{8}+560 x^{7}+28488 x^{6}+595520 x^{5}+2569600 x^{4}-73629456 x^{3}-1030398440 x^{2}-4911736672 x -8156009700\right ) {\mathrm e}^{\frac {x^{8}+184 x^{7}+13600 x^{6}+513920 x^{5}+10505414 x^{4}+115651112 x^{3}+688675888 x^{2}+2096362816 x +2563093129}{x^{4}+4 x^{3}+6 x^{2}+4 x +1}}}{x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+5 x +1}d x \] Input:
int((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103039844 0*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505 414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4*x^3+6* x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x)
Output:
int((4*x^8+560*x^7+28488*x^6+595520*x^5+2569600*x^4-73629456*x^3-103039844 0*x^2-4911736672*x-8156009700)*exp((x^8+184*x^7+13600*x^6+513920*x^5+10505 414*x^4+115651112*x^3+688675888*x^2+2096362816*x+2563093129)/(x^4+4*x^3+6* x^2+4*x+1))/(x^5+5*x^4+10*x^3+10*x^2+5*x+1),x)