Integrand size = 7, antiderivative size = 163 \[ \int e^x x^{20} \, dx=2432902008176640000 e^x-2432902008176640000 e^x x+1216451004088320000 e^x x^2-405483668029440000 e^x x^3+101370917007360000 e^x x^4-20274183401472000 e^x x^5+3379030566912000 e^x x^6-482718652416000 e^x x^7+60339831552000 e^x x^8-6704425728000 e^x x^9+670442572800 e^x x^{10}-60949324800 e^x x^{11}+5079110400 e^x x^{12}-390700800 e^x x^{13}+27907200 e^x x^{14}-1860480 e^x x^{15}+116280 e^x x^{16}-6840 e^x x^{17}+380 e^x x^{18}-20 e^x x^{19}+e^x x^{20} \]
2432902008176640000*exp(x)-2432902008176640000*exp(x)*x+121645100408832000 0*exp(x)*x^2-405483668029440000*exp(x)*x^3+101370917007360000*exp(x)*x^4-2 0274183401472000*exp(x)*x^5+3379030566912000*exp(x)*x^6-482718652416000*ex p(x)*x^7+60339831552000*exp(x)*x^8-6704425728000*exp(x)*x^9+670442572800*e xp(x)*x^10-60949324800*exp(x)*x^11+5079110400*exp(x)*x^12-390700800*exp(x) *x^13+27907200*exp(x)*x^14-1860480*exp(x)*x^15+116280*exp(x)*x^16-6840*exp (x)*x^17+380*exp(x)*x^18-20*exp(x)*x^19+exp(x)*x^20
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int e^x x^{20} \, dx=e^x \left (2432902008176640000-2432902008176640000 x+1216451004088320000 x^2-405483668029440000 x^3+101370917007360000 x^4-20274183401472000 x^5+3379030566912000 x^6-482718652416000 x^7+60339831552000 x^8-6704425728000 x^9+670442572800 x^{10}-60949324800 x^{11}+5079110400 x^{12}-390700800 x^{13}+27907200 x^{14}-1860480 x^{15}+116280 x^{16}-6840 x^{17}+380 x^{18}-20 x^{19}+x^{20}\right ) \]
E^x*(2432902008176640000 - 2432902008176640000*x + 1216451004088320000*x^2 - 405483668029440000*x^3 + 101370917007360000*x^4 - 20274183401472000*x^5 + 3379030566912000*x^6 - 482718652416000*x^7 + 60339831552000*x^8 - 67044 25728000*x^9 + 670442572800*x^10 - 60949324800*x^11 + 5079110400*x^12 - 39 0700800*x^13 + 27907200*x^14 - 1860480*x^15 + 116280*x^16 - 6840*x^17 + 38 0*x^18 - 20*x^19 + x^20)
Time = 1.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.23, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 3.000, Rules used = {2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2607, 2624}
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.
| \(\displaystyle \int e^x x^{20} \, dx\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \int e^x x^{19}dx\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \int e^x x^{18}dx\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \int e^x x^{17}dx\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \int e^x x^{16}dx\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \int e^x x^{15}dx\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \int e^x x^{14}dx\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \int e^x x^{13}dx\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \int e^x x^{12}dx\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \int e^x x^{11}dx\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \int e^x x^{10}dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \int e^x x^9dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \int e^x x^8dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \int e^x x^7dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \int e^x x^6dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \int e^x x^5dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \int e^x x^4dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \left (e^x x^4-4 \int e^x x^3dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \left (e^x x^4-4 \left (e^x x^3-3 \int e^x x^2dx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \left (e^x x^4-4 \left (e^x x^3-3 \left (e^x x^2-2 \int e^x xdx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \left (e^x x^4-4 \left (e^x x^3-3 \left (e^x x^2-2 \left (e^x x-\int e^xdx\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 2624 |
| \(\displaystyle e^x x^{20}-20 \left (e^x x^{19}-19 \left (e^x x^{18}-18 \left (e^x x^{17}-17 \left (e^x x^{16}-16 \left (e^x x^{15}-15 \left (e^x x^{14}-14 \left (e^x x^{13}-13 \left (e^x x^{12}-12 \left (e^x x^{11}-11 \left (e^x x^{10}-10 \left (e^x x^9-9 \left (e^x x^8-8 \left (e^x x^7-7 \left (e^x x^6-6 \left (e^x x^5-5 \left (e^x x^4-4 \left (e^x x^3-3 \left (e^x x^2-2 \left (e^x x-e^x\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\) |
E^x*x^20 - 20*(E^x*x^19 - 19*(E^x*x^18 - 18*(E^x*x^17 - 17*(E^x*x^16 - 16* (E^x*x^15 - 15*(E^x*x^14 - 14*(E^x*x^13 - 13*(E^x*x^12 - 12*(E^x*x^11 - 11 *(E^x*x^10 - 10*(E^x*x^9 - 9*(E^x*x^8 - 8*(E^x*x^7 - 7*(E^x*x^6 - 6*(E^x*x ^5 - 5*(E^x*x^4 - 4*(E^x*x^3 - 3*(E^x*x^2 - 2*(-E^x + E^x*x))))))))))))))) ))))
3.2.59.3.1 Defintions of rubi rules used
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Simp[d*(m/(f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* m] && !TrueQ[$UseGamma]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63
| method | result | size |
| gosper | \(\left (x^{20}-20 x^{19}+380 x^{18}-6840 x^{17}+116280 x^{16}-1860480 x^{15}+27907200 x^{14}-390700800 x^{13}+5079110400 x^{12}-60949324800 x^{11}+670442572800 x^{10}-6704425728000 x^{9}+60339831552000 x^{8}-482718652416000 x^{7}+3379030566912000 x^{6}-20274183401472000 x^{5}+101370917007360000 x^{4}-405483668029440000 x^{3}+1216451004088320000 x^{2}-2432902008176640000 x +2432902008176640000\right ) {\mathrm e}^{x}\) | \(102\) |
| risch | \(\left (x^{20}-20 x^{19}+380 x^{18}-6840 x^{17}+116280 x^{16}-1860480 x^{15}+27907200 x^{14}-390700800 x^{13}+5079110400 x^{12}-60949324800 x^{11}+670442572800 x^{10}-6704425728000 x^{9}+60339831552000 x^{8}-482718652416000 x^{7}+3379030566912000 x^{6}-20274183401472000 x^{5}+101370917007360000 x^{4}-405483668029440000 x^{3}+1216451004088320000 x^{2}-2432902008176640000 x +2432902008176640000\right ) {\mathrm e}^{x}\) | \(102\) |
| meijerg | \(-2432902008176640000+\frac {\left (21 x^{20}-420 x^{19}+7980 x^{18}-143640 x^{17}+2441880 x^{16}-39070080 x^{15}+586051200 x^{14}-8204716800 x^{13}+106661318400 x^{12}-1279935820800 x^{11}+14079294028800 x^{10}-140792940288000 x^{9}+1267136462592000 x^{8}-10137091700736000 x^{7}+70959641905152000 x^{6}-425757851430912000 x^{5}+2128789257154560000 x^{4}-8515157028618240000 x^{3}+25545471085854720000 x^{2}-51090942171709440000 x +51090942171709440000\right ) {\mathrm e}^{x}}{21}\) | \(107\) |
| default | \(2432902008176640000 \,{\mathrm e}^{x}-2432902008176640000 \,{\mathrm e}^{x} x +1216451004088320000 \,{\mathrm e}^{x} x^{2}-405483668029440000 \,{\mathrm e}^{x} x^{3}+101370917007360000 \,{\mathrm e}^{x} x^{4}-20274183401472000 \,{\mathrm e}^{x} x^{5}+3379030566912000 \,{\mathrm e}^{x} x^{6}-482718652416000 \,{\mathrm e}^{x} x^{7}+60339831552000 \,{\mathrm e}^{x} x^{8}-6704425728000 \,{\mathrm e}^{x} x^{9}+670442572800 \,{\mathrm e}^{x} x^{10}-60949324800 \,{\mathrm e}^{x} x^{11}+5079110400 \,{\mathrm e}^{x} x^{12}-390700800 \,{\mathrm e}^{x} x^{13}+27907200 \,{\mathrm e}^{x} x^{14}-1860480 \,{\mathrm e}^{x} x^{15}+116280 \,{\mathrm e}^{x} x^{16}-6840 \,{\mathrm e}^{x} x^{17}+380 \,{\mathrm e}^{x} x^{18}-20 \,{\mathrm e}^{x} x^{19}+{\mathrm e}^{x} x^{20}\) | \(143\) |
| parallelrisch | \(2432902008176640000 \,{\mathrm e}^{x}-2432902008176640000 \,{\mathrm e}^{x} x +1216451004088320000 \,{\mathrm e}^{x} x^{2}-405483668029440000 \,{\mathrm e}^{x} x^{3}+101370917007360000 \,{\mathrm e}^{x} x^{4}-20274183401472000 \,{\mathrm e}^{x} x^{5}+3379030566912000 \,{\mathrm e}^{x} x^{6}-482718652416000 \,{\mathrm e}^{x} x^{7}+60339831552000 \,{\mathrm e}^{x} x^{8}-6704425728000 \,{\mathrm e}^{x} x^{9}+670442572800 \,{\mathrm e}^{x} x^{10}-60949324800 \,{\mathrm e}^{x} x^{11}+5079110400 \,{\mathrm e}^{x} x^{12}-390700800 \,{\mathrm e}^{x} x^{13}+27907200 \,{\mathrm e}^{x} x^{14}-1860480 \,{\mathrm e}^{x} x^{15}+116280 \,{\mathrm e}^{x} x^{16}-6840 \,{\mathrm e}^{x} x^{17}+380 \,{\mathrm e}^{x} x^{18}-20 \,{\mathrm e}^{x} x^{19}+{\mathrm e}^{x} x^{20}\) | \(143\) |
| parts | \(2432902008176640000 \,{\mathrm e}^{x}-2432902008176640000 \,{\mathrm e}^{x} x +1216451004088320000 \,{\mathrm e}^{x} x^{2}-405483668029440000 \,{\mathrm e}^{x} x^{3}+101370917007360000 \,{\mathrm e}^{x} x^{4}-20274183401472000 \,{\mathrm e}^{x} x^{5}+3379030566912000 \,{\mathrm e}^{x} x^{6}-482718652416000 \,{\mathrm e}^{x} x^{7}+60339831552000 \,{\mathrm e}^{x} x^{8}-6704425728000 \,{\mathrm e}^{x} x^{9}+670442572800 \,{\mathrm e}^{x} x^{10}-60949324800 \,{\mathrm e}^{x} x^{11}+5079110400 \,{\mathrm e}^{x} x^{12}-390700800 \,{\mathrm e}^{x} x^{13}+27907200 \,{\mathrm e}^{x} x^{14}-1860480 \,{\mathrm e}^{x} x^{15}+116280 \,{\mathrm e}^{x} x^{16}-6840 \,{\mathrm e}^{x} x^{17}+380 \,{\mathrm e}^{x} x^{18}-20 \,{\mathrm e}^{x} x^{19}+{\mathrm e}^{x} x^{20}\) | \(143\) |
(x^20-20*x^19+380*x^18-6840*x^17+116280*x^16-1860480*x^15+27907200*x^14-39 0700800*x^13+5079110400*x^12-60949324800*x^11+670442572800*x^10-6704425728 000*x^9+60339831552000*x^8-482718652416000*x^7+3379030566912000*x^6-202741 83401472000*x^5+101370917007360000*x^4-405483668029440000*x^3+121645100408 8320000*x^2-2432902008176640000*x+2432902008176640000)*exp(x)
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.62 \[ \int e^x x^{20} \, dx={\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]
(x^20 - 20*x^19 + 380*x^18 - 6840*x^17 + 116280*x^16 - 1860480*x^15 + 2790 7200*x^14 - 390700800*x^13 + 5079110400*x^12 - 60949324800*x^11 + 67044257 2800*x^10 - 6704425728000*x^9 + 60339831552000*x^8 - 482718652416000*x^7 + 3379030566912000*x^6 - 20274183401472000*x^5 + 101370917007360000*x^4 - 4 05483668029440000*x^3 + 1216451004088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x
Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int e^x x^{20} \, dx=\left (x^{20} - 20 x^{19} + 380 x^{18} - 6840 x^{17} + 116280 x^{16} - 1860480 x^{15} + 27907200 x^{14} - 390700800 x^{13} + 5079110400 x^{12} - 60949324800 x^{11} + 670442572800 x^{10} - 6704425728000 x^{9} + 60339831552000 x^{8} - 482718652416000 x^{7} + 3379030566912000 x^{6} - 20274183401472000 x^{5} + 101370917007360000 x^{4} - 405483668029440000 x^{3} + 1216451004088320000 x^{2} - 2432902008176640000 x + 2432902008176640000\right ) e^{x} \]
(x**20 - 20*x**19 + 380*x**18 - 6840*x**17 + 116280*x**16 - 1860480*x**15 + 27907200*x**14 - 390700800*x**13 + 5079110400*x**12 - 60949324800*x**11 + 670442572800*x**10 - 6704425728000*x**9 + 60339831552000*x**8 - 48271865 2416000*x**7 + 3379030566912000*x**6 - 20274183401472000*x**5 + 1013709170 07360000*x**4 - 405483668029440000*x**3 + 1216451004088320000*x**2 - 24329 02008176640000*x + 2432902008176640000)*exp(x)
Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.62 \[ \int e^x x^{20} \, dx={\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]
(x^20 - 20*x^19 + 380*x^18 - 6840*x^17 + 116280*x^16 - 1860480*x^15 + 2790 7200*x^14 - 390700800*x^13 + 5079110400*x^12 - 60949324800*x^11 + 67044257 2800*x^10 - 6704425728000*x^9 + 60339831552000*x^8 - 482718652416000*x^7 + 3379030566912000*x^6 - 20274183401472000*x^5 + 101370917007360000*x^4 - 4 05483668029440000*x^3 + 1216451004088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.62 \[ \int e^x x^{20} \, dx={\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]
(x^20 - 20*x^19 + 380*x^18 - 6840*x^17 + 116280*x^16 - 1860480*x^15 + 2790 7200*x^14 - 390700800*x^13 + 5079110400*x^12 - 60949324800*x^11 + 67044257 2800*x^10 - 6704425728000*x^9 + 60339831552000*x^8 - 482718652416000*x^7 + 3379030566912000*x^6 - 20274183401472000*x^5 + 101370917007360000*x^4 - 4 05483668029440000*x^3 + 1216451004088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.62 \[ \int e^x x^{20} \, dx={\mathrm {e}}^x\,\left (x^{20}-20\,x^{19}+380\,x^{18}-6840\,x^{17}+116280\,x^{16}-1860480\,x^{15}+27907200\,x^{14}-390700800\,x^{13}+5079110400\,x^{12}-60949324800\,x^{11}+670442572800\,x^{10}-6704425728000\,x^9+60339831552000\,x^8-482718652416000\,x^7+3379030566912000\,x^6-20274183401472000\,x^5+101370917007360000\,x^4-405483668029440000\,x^3+1216451004088320000\,x^2-2432902008176640000\,x+2432902008176640000\right ) \]
exp(x)*(1216451004088320000*x^2 - 2432902008176640000*x - 4054836680294400 00*x^3 + 101370917007360000*x^4 - 20274183401472000*x^5 + 3379030566912000 *x^6 - 482718652416000*x^7 + 60339831552000*x^8 - 6704425728000*x^9 + 6704 42572800*x^10 - 60949324800*x^11 + 5079110400*x^12 - 390700800*x^13 + 2790 7200*x^14 - 1860480*x^15 + 116280*x^16 - 6840*x^17 + 380*x^18 - 20*x^19 + x^20 + 2432902008176640000)
Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int e^x x^{20} \, dx=e^{x} \left (x^{20}-20 x^{19}+380 x^{18}-6840 x^{17}+116280 x^{16}-1860480 x^{15}+27907200 x^{14}-390700800 x^{13}+5079110400 x^{12}-60949324800 x^{11}+670442572800 x^{10}-6704425728000 x^{9}+60339831552000 x^{8}-482718652416000 x^{7}+3379030566912000 x^{6}-20274183401472000 x^{5}+101370917007360000 x^{4}-405483668029440000 x^{3}+1216451004088320000 x^{2}-2432902008176640000 x +2432902008176640000\right ) \]
e**x*(x**20 - 20*x**19 + 380*x**18 - 6840*x**17 + 116280*x**16 - 1860480*x **15 + 27907200*x**14 - 390700800*x**13 + 5079110400*x**12 - 60949324800*x **11 + 670442572800*x**10 - 6704425728000*x**9 + 60339831552000*x**8 - 482 718652416000*x**7 + 3379030566912000*x**6 - 20274183401472000*x**5 + 10137 0917007360000*x**4 - 405483668029440000*x**3 + 1216451004088320000*x**2 - 2432902008176640000*x + 2432902008176640000)