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 + 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

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Rubi [A]  time = 0.241771, 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, Rules used = {2176, 2194} \[ e^x x^{20}-20 e^x x^{19}+380 e^x x^{18}-6840 e^x x^{17}+116280 e^x x^{16}-1860480 e^x x^{15}[phone] e^x x^{14}-390700800 e^x x^{13}[phone] e^x x^{12}-60949324800 e^x x^{11}[phone] e^x x^{10}-6704425728000 e^x x^9+60339831552000 e^x x^8-482718652416000 e^x x^7+3379030566912000 e^x x^6-20274183401472000 e^x x^5+101370917007360000 e^x x^4-405483668029440000 e^x x^3+1216451004088320000 e^x x^2-2432902008176640000 e^x x+2432902008176640000 e^x \]

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 + 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

Rule 2176

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] - Dist[(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] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^x x^{20} \, dx &=e^x x^{20}-20 \int e^x x^{19} \, dx\\ &=-20 e^x x^{19}+e^x x^{20}+380 \int e^x x^{18} \, dx\\ &=380 e^x x^{18}-20 e^x x^{19}+e^x x^{20}-6840 \int e^x x^{17} \, dx\\ &=-6840 e^x x^{17}+380 e^x x^{18}-20 e^x x^{19}+e^x x^{20}+116280 \int e^x x^{16} \, dx\\ &=116280 e^x x^{16}-6840 e^x x^{17}+380 e^x x^{18}-20 e^x x^{19}+e^x x^{20}-1860480 \int e^x x^{15} \, dx\\ &=-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}[phone] \int e^x x^{14} \, dx\\ &=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}-390700800 \int e^x x^{13} \, dx\\ &=-390700800 e^x x^{13}[phone] 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}[phone] \int e^x x^{12} \, dx\\ &=5079110400 e^x x^{12}-390700800 e^x x^{13}[phone] 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}-60949324800 \int e^x x^{11} \, dx\\ &=-60949324800 e^x x^{11}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}[phone] \int e^x x^{10} \, dx\\ &=670442572800 e^x x^{10}-60949324800 e^x x^{11}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}-6704425728000 \int e^x x^9 \, dx\\ &=-6704425728000 e^x x^9+670442572800 e^x x^{10}-60949324800 e^x x^{11}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}[phone] \int e^x x^8 \, dx\\ &=60339831552000 e^x x^8-6704425728000 e^x x^9+670442572800 e^x x^{10}-60949324800 e^x x^{11}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}-482718652416000 \int e^x x^7 \, dx\\ &=-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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}+3379030566912000 \int e^x x^6 \, dx\\ &=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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}-20274183401472000 \int e^x x^5 \, dx\\ &=-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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}+101370917007360000 \int e^x x^4 \, dx\\ &=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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}-405483668029440000 \int e^x x^3 \, dx\\ &=-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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}+1216451004088320000 \int e^x x^2 \, dx\\ &=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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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 \int e^x x \, dx\\ &=-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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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 \int e^x \, 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}[phone] e^x x^{12}-390700800 e^x x^{13}[phone] 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}\\ \end{align*}

Mathematica [A]  time = 0.0100452, 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 - 405483668029440000*x^3 + 10137091
7007360000*x^4 - 20274183401472000*x^5 + 3379030566912000*x^6 - 482718652416000*x^7 + 60339831552000*x^8 - 670
4425728000*x^9 + 670442572800*x^10 - 60949324800*x^11 + 5079110400*x^12 - 390700800*x^13 + 27907200*x^14 - 186
0480*x^15 + 116280*x^16 - 6840*x^17 + 380*x^18 - 20*x^19 + x^20)

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Maple [A]  time = 0.003, size = 102, normalized size = 0.6 \begin{align*} \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}} \end{align*}

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^14-390700800*x^13+5079110400*x^12-6094932
4800*x^11+670442572800*x^10-6704425728000*x^9+60339831552000*x^8-482718652416000*x^7+3379030566912000*x^6-2027
4183401472000*x^5+101370917007360000*x^4-405483668029440000*x^3+1216451004088320000*x^2-2432902008176640000*x+
2432902008176640000)*exp(x)

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Maxima [A]  time = 0.948502, size = 136, normalized size = 0.83 \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^20,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 + 5079110
400*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 + 1216451004
088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x

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Fricas [A]  time = 1.5842, size = 535, normalized size = 3.28 \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^20,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 + 5079110
400*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 + 1216451004
088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x

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Sympy [A]  time = 0.127938, size = 102, normalized size = 0.63 \begin{align*} \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} \end{align*}

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 + 27907200*x**14 - 390700800*x**13 +
5079110400*x**12 - 60949324800*x**11 + 670442572800*x**10 - 6704425728000*x**9 + 60339831552000*x**8 - 482718
652416000*x**7 + 3379030566912000*x**6 - 20274183401472000*x**5 + 101370917007360000*x**4 - 405483668029440000
*x**3 + 1216451004088320000*x**2 - 2432902008176640000*x + 2432902008176640000)*exp(x)

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Giac [A]  time = 1.08302, size = 136, normalized size = 0.83 \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^20,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 + 5079110
400*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 + 1216451004
088320000*x^2 - 2432902008176640000*x + 2432902008176640000)*e^x