3.159 \(\int e^x x^{20} \, dx\)

Optimal. Leaf size=163 \[ e^x x^{20}-20 e^x x^{19}+380 e^x x^{18}-6840 e^x x^{17}+116280 e^x x^{16}-1860480 e^x x^{15}[phone] e^x x^{14}-390700800 e^x x^{13}[phone] e^x x^{12}-60949324800 e^x x^{11}[phone] e^x x^{10}-6704425728000 e^x x^9+60339831552000 e^x x^8-482718652416000 e^x x^7+3379030566912000 e^x x^6-20274183401472000 e^x x^5+101370917007360000 e^x x^4-405483668029440000 e^x x^3+1216451004088320000 e^x x^2-2432902008176640000 e^x x+2432902008176640000 e^x \]

[Out]

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 + 5
079110400*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

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Rubi [A]  time = 0.378312, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ e^x x^{20}-20 e^x x^{19}+380 e^x x^{18}-6840 e^x x^{17}+116280 e^x x^{16}-1860480 e^x x^{15}[phone] e^x x^{14}-390700800 e^x x^{13}[phone] e^x x^{12}-60949324800 e^x x^{11}[phone] e^x x^{10}-6704425728000 e^x x^9+60339831552000 e^x x^8-482718652416000 e^x x^7+3379030566912000 e^x x^6-20274183401472000 e^x x^5+101370917007360000 e^x x^4-405483668029440000 e^x x^3+1216451004088320000 e^x x^2-2432902008176640000 e^x x+2432902008176640000 e^x \]

Antiderivative was successfully verified.

[In]  Int[E^x*x^20,x]

[Out]

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 + 5
079110400*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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x^{20} e^{x} - 20 x^{19} e^{x} + 380 x^{18} e^{x} - 6840 x^{17} e^{x} + 116280 x^{16} e^{x} - 1860480 x^{15} e^{x} + 27907200 x^{14} e^{x} - 390700800 x^{13} e^{x} + 5079110400 x^{12} e^{x} - 60949324800 x^{11} e^{x} + 670442572800 x^{10} e^{x} - 6704425728000 \int x^{9} e^{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)*x**20,x)

[Out]

x**20*exp(x) - 20*x**19*exp(x) + 380*x**18*exp(x) - 6840*x**17*exp(x) + 116280*x
**16*exp(x) - 1860480*x**15*exp(x) + 27907200*x**14*exp(x) - 390700800*x**13*exp
(x) + 5079110400*x**12*exp(x) - 60949324800*x**11*exp(x) + 670442572800*x**10*ex
p(x) - 6704425728000*Integral(x**9*exp(x), x)

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Mathematica [A]  time = 0.0112535, size = 102, normalized size = 0.63 \[ e^x \left (x^{20}-20 x^{19}+380 x^{18}-6840 x^{17}+116280 x^{16}-1860480 x^{15}[phone] x^{14}-390700800 x^{13}[phone] x^{12}-60949324800 x^{11}[phone] x^{10}-6704425728000 x^9+60339831552000 x^8-482718652416000 x^7+3379030566912000 x^6-20274183401472000 x^5+101370917007360000 x^4-405483668029440000 x^3+1216451004088320000 x^2-2432902008176640000 x+2432902008176640000\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x*x^20,x]

[Out]

E^x*(2432902008176640000 - 2432902008176640000*x + 1216451004088320000*x^2 - 405
483668029440000*x^3 + 101370917007360000*x^4 - 20274183401472000*x^5 + 337903056
6912000*x^6 - 482718652416000*x^7 + 60339831552000*x^8 - 6704425728000*x^9 + 670
442572800*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)

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Maple [A]  time = 0.002, size = 102, normalized size = 0.6 \[ \left ({x}^{20}-20\,{x}^{19}+380\,{x}^{18}-6840\,{x}^{17}+116280\,{x}^{16}-1860480\,{x}^{15}[phone]\,{x}^{14}-390700800\,{x}^{13}[phone]\,{x}^{12}-60949324800\,{x}^{11}[phone]\,{x}^{10}-6704425728000\,{x}^{9}[phone]\,{x}^{8}-482718652416000\,{x}^{7}+3379030566912000\,{x}^{6}-20274183401472000\,{x}^{5}+101370917007360000\,{x}^{4}-405483668029440000\,{x}^{3}+1216451004088320000\,{x}^{2}-2432902008176640000\,x+2432902008176640000 \right ){{\rm e}^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*x^20,x)

[Out]

(x^20-20*x^19+380*x^18-6840*x^17+116280*x^16-1860480*x^15+27907200*x^14-39070080
0*x^13+5079110400*x^12-60949324800*x^11+670442572800*x^10-6704425728000*x^9+6033
9831552000*x^8-482718652416000*x^7+3379030566912000*x^6-20274183401472000*x^5+10
1370917007360000*x^4-405483668029440000*x^3+1216451004088320000*x^2-243290200817
6640000*x+2432902008176640000)*exp(x)

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Maxima [A]  time = 1.42532, size = 136, normalized size = 0.83 \[{\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^20*e^x,x, algorithm="maxima")

[Out]

(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)*e^x

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Fricas [A]  time = 0.214026, size = 136, normalized size = 0.83 \[{\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^20*e^x,x, algorithm="fricas")

[Out]

(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)*e^x

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Sympy [A]  time = 0.119834, size = 102, normalized size = 0.63 \[ \left (x^{20} - 20 x^{19} + 380 x^{18} - 6840 x^{17} + 116280 x^{16} - 1860480 x^{15} + 27907200 x^{14} - 390700800 x^{13} + 5079110400 x^{12} - 60949324800 x^{11} + 670442572800 x^{10} - 6704425728000 x^{9} + 60339831552000 x^{8} - 482718652416000 x^{7} + 3379030566912000 x^{6} - 20274183401472000 x^{5} + 101370917007360000 x^{4} - 405483668029440000 x^{3} + 1216451004088320000 x^{2} - 2432902008176640000 x + 2432902008176640000\right ) e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*x**20,x)

[Out]

(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 + 6704425728
00*x**10 - 6704425728000*x**9 + 60339831552000*x**8 - 482718652416000*x**7 + 337
9030566912000*x**6 - 20274183401472000*x**5 + 101370917007360000*x**4 - 40548366
8029440000*x**3 + 1216451004088320000*x**2 - 2432902008176640000*x + 24329020081
76640000)*exp(x)

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GIAC/XCAS [A]  time = 0.200304, size = 136, normalized size = 0.83 \[{\left (x^{20} - 20 \, x^{19} + 380 \, x^{18} - 6840 \, x^{17} + 116280 \, x^{16} - 1860480 \, x^{15} + 27907200 \, x^{14} - 390700800 \, x^{13} + 5079110400 \, x^{12} - 60949324800 \, x^{11} + 670442572800 \, x^{10} - 6704425728000 \, x^{9} + 60339831552000 \, x^{8} - 482718652416000 \, x^{7} + 3379030566912000 \, x^{6} - 20274183401472000 \, x^{5} + 101370917007360000 \, x^{4} - 405483668029440000 \, x^{3} + 1216451004088320000 \, x^{2} - 2432902008176640000 \, x + 2432902008176640000\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^20*e^x,x, algorithm="giac")

[Out]

(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)*e^x