3.90.33 \(\int e^{-x} (160 x+1360 x^2+2400 x^3-720 x^4+(240 x^2+880 x^3-240 x^4) \log (3)+(80 x^3-20 x^4) \log ^2(3)) \, dx\)

Optimal. Leaf size=21 \[ 20 e^{-x} x^2 (-2-x (6+\log (3)))^2 \]

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Rubi [B]  time = 0.45, antiderivative size = 69, normalized size of antiderivative = 3.29, number of steps used = 41, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2196, 2176, 2194} \begin {gather*} 720 e^{-x} x^4+20 e^{-x} x^4 \log ^2(3)+240 e^{-x} x^4 \log (3)+480 e^{-x} x^3+80 e^{-x} x^3 \log (3)+80 e^{-x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(160*x + 1360*x^2 + 2400*x^3 - 720*x^4 + (240*x^2 + 880*x^3 - 240*x^4)*Log[3] + (80*x^3 - 20*x^4)*Log[3]^2
)/E^x,x]

[Out]

(80*x^2)/E^x + (480*x^3)/E^x + (720*x^4)/E^x + (80*x^3*Log[3])/E^x + (240*x^4*Log[3])/E^x + (20*x^4*Log[3]^2)/
E^x

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (160 e^{-x} x+1360 e^{-x} x^2+2400 e^{-x} x^3-720 e^{-x} x^4-80 e^{-x} x^2 \left (-3-11 x+3 x^2\right ) \log (3)-20 e^{-x} (-4+x) x^3 \log ^2(3)\right ) \, dx\\ &=160 \int e^{-x} x \, dx-720 \int e^{-x} x^4 \, dx+1360 \int e^{-x} x^2 \, dx+2400 \int e^{-x} x^3 \, dx-(80 \log (3)) \int e^{-x} x^2 \left (-3-11 x+3 x^2\right ) \, dx-\left (20 \log ^2(3)\right ) \int e^{-x} (-4+x) x^3 \, dx\\ &=-160 e^{-x} x-1360 e^{-x} x^2-2400 e^{-x} x^3+720 e^{-x} x^4+160 \int e^{-x} \, dx+2720 \int e^{-x} x \, dx-2880 \int e^{-x} x^3 \, dx+7200 \int e^{-x} x^2 \, dx-(80 \log (3)) \int \left (-3 e^{-x} x^2-11 e^{-x} x^3+3 e^{-x} x^4\right ) \, dx-\left (20 \log ^2(3)\right ) \int \left (-4 e^{-x} x^3+e^{-x} x^4\right ) \, dx\\ &=-160 e^{-x}-2880 e^{-x} x-8560 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4+2720 \int e^{-x} \, dx-8640 \int e^{-x} x^2 \, dx+14400 \int e^{-x} x \, dx+(240 \log (3)) \int e^{-x} x^2 \, dx-(240 \log (3)) \int e^{-x} x^4 \, dx+(880 \log (3)) \int e^{-x} x^3 \, dx-\left (20 \log ^2(3)\right ) \int e^{-x} x^4 \, dx+\left (80 \log ^2(3)\right ) \int e^{-x} x^3 \, dx\\ &=-2880 e^{-x}-17280 e^{-x} x+80 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4-240 e^{-x} x^2 \log (3)-880 e^{-x} x^3 \log (3)+240 e^{-x} x^4 \log (3)-80 e^{-x} x^3 \log ^2(3)+20 e^{-x} x^4 \log ^2(3)+14400 \int e^{-x} \, dx-17280 \int e^{-x} x \, dx+(480 \log (3)) \int e^{-x} x \, dx-(960 \log (3)) \int e^{-x} x^3 \, dx+(2640 \log (3)) \int e^{-x} x^2 \, dx-\left (80 \log ^2(3)\right ) \int e^{-x} x^3 \, dx+\left (240 \log ^2(3)\right ) \int e^{-x} x^2 \, dx\\ &=-17280 e^{-x}+80 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4-480 e^{-x} x \log (3)-2880 e^{-x} x^2 \log (3)+80 e^{-x} x^3 \log (3)+240 e^{-x} x^4 \log (3)-240 e^{-x} x^2 \log ^2(3)+20 e^{-x} x^4 \log ^2(3)-17280 \int e^{-x} \, dx+(480 \log (3)) \int e^{-x} \, dx-(2880 \log (3)) \int e^{-x} x^2 \, dx+(5280 \log (3)) \int e^{-x} x \, dx-\left (240 \log ^2(3)\right ) \int e^{-x} x^2 \, dx+\left (480 \log ^2(3)\right ) \int e^{-x} x \, dx\\ &=80 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4-480 e^{-x} \log (3)-5760 e^{-x} x \log (3)+80 e^{-x} x^3 \log (3)+240 e^{-x} x^4 \log (3)-480 e^{-x} x \log ^2(3)+20 e^{-x} x^4 \log ^2(3)+(5280 \log (3)) \int e^{-x} \, dx-(5760 \log (3)) \int e^{-x} x \, dx+\left (480 \log ^2(3)\right ) \int e^{-x} \, dx-\left (480 \log ^2(3)\right ) \int e^{-x} x \, dx\\ &=80 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4-5760 e^{-x} \log (3)+80 e^{-x} x^3 \log (3)+240 e^{-x} x^4 \log (3)-480 e^{-x} \log ^2(3)+20 e^{-x} x^4 \log ^2(3)-(5760 \log (3)) \int e^{-x} \, dx-\left (480 \log ^2(3)\right ) \int e^{-x} \, dx\\ &=80 e^{-x} x^2+480 e^{-x} x^3+720 e^{-x} x^4+80 e^{-x} x^3 \log (3)+240 e^{-x} x^4 \log (3)+20 e^{-x} x^4 \log ^2(3)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 30, normalized size = 1.43 \begin {gather*} 20 e^{-x} x^2 \left (4+x (24+\log (81))+x^2 \left (36+\log ^2(3)+\log (531441)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(160*x + 1360*x^2 + 2400*x^3 - 720*x^4 + (240*x^2 + 880*x^3 - 240*x^4)*Log[3] + (80*x^3 - 20*x^4)*Lo
g[3]^2)/E^x,x]

[Out]

(20*x^2*(4 + x*(24 + Log[81]) + x^2*(36 + Log[3]^2 + Log[531441])))/E^x

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fricas [B]  time = 0.59, size = 43, normalized size = 2.05 \begin {gather*} 20 \, {\left (x^{4} \log \relax (3)^{2} + 36 \, x^{4} + 24 \, x^{3} + 4 \, x^{2} + 4 \, {\left (3 \, x^{4} + x^{3}\right )} \log \relax (3)\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^4+80*x^3)*log(3)^2+(-240*x^4+880*x^3+240*x^2)*log(3)-720*x^4+2400*x^3+1360*x^2+160*x)/exp(x)
,x, algorithm="fricas")

[Out]

20*(x^4*log(3)^2 + 36*x^4 + 24*x^3 + 4*x^2 + 4*(3*x^4 + x^3)*log(3))*e^(-x)

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giac [B]  time = 0.12, size = 44, normalized size = 2.10 \begin {gather*} 20 \, {\left (x^{4} \log \relax (3)^{2} + 12 \, x^{4} \log \relax (3) + 36 \, x^{4} + 4 \, x^{3} \log \relax (3) + 24 \, x^{3} + 4 \, x^{2}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^4+80*x^3)*log(3)^2+(-240*x^4+880*x^3+240*x^2)*log(3)-720*x^4+2400*x^3+1360*x^2+160*x)/exp(x)
,x, algorithm="giac")

[Out]

20*(x^4*log(3)^2 + 12*x^4*log(3) + 36*x^4 + 4*x^3*log(3) + 24*x^3 + 4*x^2)*e^(-x)

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maple [A]  time = 0.05, size = 38, normalized size = 1.81

method	result	size
norman	\(\left (\left (480+80 \ln \relax (3)\right ) x^{3}+\left (20 \ln \relax (3)^{2}+240 \ln \relax (3)+720\right ) x^{4}+80 x^{2}\right ) {\mathrm e}^{-x}\)	\(38\)
gosper	\(20 \left (x^{2} \ln \relax (3)^{2}+12 x^{2} \ln \relax (3)+4 x \ln \relax (3)+36 x^{2}+24 x +4\right ) x^{2} {\mathrm e}^{-x}\)	\(40\)
risch	\(\left (20 x^{4} \ln \relax (3)^{2}+240 x^{4} \ln \relax (3)+80 x^{3} \ln \relax (3)+720 x^{4}+480 x^{3}+80 x^{2}\right ) {\mathrm e}^{-x}\)	\(45\)
default	\(80 x^{2} {\mathrm e}^{-x}+480 x^{3} {\mathrm e}^{-x}+720 x^{4} {\mathrm e}^{-x}+80 \,{\mathrm e}^{-x} x^{3} \ln \relax (3)+240 \,{\mathrm e}^{-x} x^{4} \ln \relax (3)+20 \,{\mathrm e}^{-x} x^{4} \ln \relax (3)^{2}\)	\(64\)
meijerg	\(\left (240 \ln \relax (3)+1360\right ) \left (2-\frac {\left (3 x^{2}+6 x +6\right ) {\mathrm e}^{-x}}{3}\right )+160-80 \left (2 x +2\right ) {\mathrm e}^{-x}+\left (-20 \ln \relax (3)^{2}-240 \ln \relax (3)-720\right ) \left (24-\frac {\left (5 x^{4}+20 x^{3}+60 x^{2}+120 x +120\right ) {\mathrm e}^{-x}}{5}\right )+\left (80 \ln \relax (3)^{2}+880 \ln \relax (3)+2400\right ) \left (6-\frac {\left (4 x^{3}+12 x^{2}+24 x +24\right ) {\mathrm e}^{-x}}{4}\right )\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*x^4+80*x^3)*ln(3)^2+(-240*x^4+880*x^3+240*x^2)*ln(3)-720*x^4+2400*x^3+1360*x^2+160*x)/exp(x),x,metho
d=_RETURNVERBOSE)

[Out]

((480+80*ln(3))*x^3+(20*ln(3)^2+240*ln(3)+720)*x^4+80*x^2)/exp(x)

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maxima [B]  time = 0.46, size = 181, normalized size = 8.62 \begin {gather*} 20 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \log \relax (3)^{2} - 80 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} \log \relax (3)^{2} + 240 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} \log \relax (3) - 880 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} \log \relax (3) - 240 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \log \relax (3) + 720 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} - 2400 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} - 1360 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - 160 \, {\left (x + 1\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^4+80*x^3)*log(3)^2+(-240*x^4+880*x^3+240*x^2)*log(3)-720*x^4+2400*x^3+1360*x^2+160*x)/exp(x)
,x, algorithm="maxima")

[Out]

20*(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x)*log(3)^2 - 80*(x^3 + 3*x^2 + 6*x + 6)*e^(-x)*log(3)^2 + 240*(x^4
+ 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x)*log(3) - 880*(x^3 + 3*x^2 + 6*x + 6)*e^(-x)*log(3) - 240*(x^2 + 2*x + 2)*
e^(-x)*log(3) + 720*(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x) - 2400*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) - 1360*(x^
2 + 2*x + 2)*e^(-x) - 160*(x + 1)*e^(-x)

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mupad [B]  time = 0.10, size = 20, normalized size = 0.95 \begin {gather*} 20\,x^2\,{\mathrm {e}}^{-x}\,{\left (6\,x+x\,\ln \relax (3)+2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(160*x + log(3)*(240*x^2 + 880*x^3 - 240*x^4) + 1360*x^2 + 2400*x^3 - 720*x^4 + log(3)^2*(80*x^3 -
20*x^4)),x)

[Out]

20*x^2*exp(-x)*(6*x + x*log(3) + 2)^2

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sympy [B]  time = 0.16, size = 44, normalized size = 2.10 \begin {gather*} \left (20 x^{4} \log {\relax (3 )}^{2} + 240 x^{4} \log {\relax (3 )} + 720 x^{4} + 80 x^{3} \log {\relax (3 )} + 480 x^{3} + 80 x^{2}\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x**4+80*x**3)*ln(3)**2+(-240*x**4+880*x**3+240*x**2)*ln(3)-720*x**4+2400*x**3+1360*x**2+160*x)
/exp(x),x)

[Out]

(20*x**4*log(3)**2 + 240*x**4*log(3) + 720*x**4 + 80*x**3*log(3) + 480*x**3 + 80*x**2)*exp(-x)

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