3.19.71 \(\int \frac {e^{-\frac {x}{\log (3)}} (-x^4+4 x^3 \log (3)+e^{\frac {x}{\log (3)}} (15+6 x) \log (3))}{3 \log (3)} \, dx\)

Optimal. Leaf size=21 \[ x \left (5+x+\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.30, antiderivative size = 166, normalized size of antiderivative = 7.90, number of steps used = 12, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 6742, 2176, 2194} \begin {gather*} \frac {1}{3} x^4 e^{-\frac {x}{\log (3)}}-\frac {1}{3} x^3 \log (81) e^{-\frac {x}{\log (3)}}+\frac {4}{3} x^3 \log (3) e^{-\frac {x}{\log (3)}}+4 x^2 \log ^2(3) e^{-\frac {x}{\log (3)}}-x^2 \log (3) \log (81) e^{-\frac {x}{\log (3)}}+\frac {1}{4} (2 x+5)^2+8 \log ^4(3) e^{-\frac {x}{\log (3)}}+8 x \log ^3(3) e^{-\frac {x}{\log (3)}}-2 \log ^3(3) \log (81) e^{-\frac {x}{\log (3)}}-2 x \log ^2(3) \log (81) e^{-\frac {x}{\log (3)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^4 + 4*x^3*Log[3] + E^(x/Log[3])*(15 + 6*x)*Log[3])/(3*E^(x/Log[3])*Log[3]),x]

[Out]

x^4/(3*E^(x/Log[3])) + (5 + 2*x)^2/4 + (4*x^3*Log[3])/(3*E^(x/Log[3])) + (4*x^2*Log[3]^2)/E^(x/Log[3]) + (8*x*
Log[3]^3)/E^(x/Log[3]) + (8*Log[3]^4)/E^(x/Log[3]) - (x^3*Log[81])/(3*E^(x/Log[3])) - (x^2*Log[3]*Log[81])/E^(
x/Log[3]) - (2*x*Log[3]^2*Log[81])/E^(x/Log[3]) - (2*Log[3]^3*Log[81])/E^(x/Log[3])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-\frac {x}{\log (3)}} \left (-x^4+4 x^3 \log (3)+e^{\frac {x}{\log (3)}} (15+6 x) \log (3)\right ) \, dx}{3 \log (3)}\\ &=\frac {\int \left (-e^{-\frac {x}{\log (3)}} x^4+3 (5+2 x) \log (3)+e^{-\frac {x}{\log (3)}} x^3 \log (81)\right ) \, dx}{3 \log (3)}\\ &=\frac {1}{4} (5+2 x)^2-\frac {\int e^{-\frac {x}{\log (3)}} x^4 \, dx}{3 \log (3)}+\frac {\log (81) \int e^{-\frac {x}{\log (3)}} x^3 \, dx}{3 \log (3)}\\ &=\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4+\frac {1}{4} (5+2 x)^2-\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3 \log (81)-\frac {4}{3} \int e^{-\frac {x}{\log (3)}} x^3 \, dx+\log (81) \int e^{-\frac {x}{\log (3)}} x^2 \, dx\\ &=\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4+\frac {1}{4} (5+2 x)^2+\frac {4}{3} e^{-\frac {x}{\log (3)}} x^3 \log (3)-\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3 \log (81)-e^{-\frac {x}{\log (3)}} x^2 \log (3) \log (81)-(4 \log (3)) \int e^{-\frac {x}{\log (3)}} x^2 \, dx+(2 \log (3) \log (81)) \int e^{-\frac {x}{\log (3)}} x \, dx\\ &=\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4+\frac {1}{4} (5+2 x)^2+\frac {4}{3} e^{-\frac {x}{\log (3)}} x^3 \log (3)+4 e^{-\frac {x}{\log (3)}} x^2 \log ^2(3)-\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3 \log (81)-e^{-\frac {x}{\log (3)}} x^2 \log (3) \log (81)-2 e^{-\frac {x}{\log (3)}} x \log ^2(3) \log (81)-\left (8 \log ^2(3)\right ) \int e^{-\frac {x}{\log (3)}} x \, dx+\left (2 \log ^2(3) \log (81)\right ) \int e^{-\frac {x}{\log (3)}} \, dx\\ &=\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4+\frac {1}{4} (5+2 x)^2+\frac {4}{3} e^{-\frac {x}{\log (3)}} x^3 \log (3)+4 e^{-\frac {x}{\log (3)}} x^2 \log ^2(3)+8 e^{-\frac {x}{\log (3)}} x \log ^3(3)-\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3 \log (81)-e^{-\frac {x}{\log (3)}} x^2 \log (3) \log (81)-2 e^{-\frac {x}{\log (3)}} x \log ^2(3) \log (81)-2 e^{-\frac {x}{\log (3)}} \log ^3(3) \log (81)-\left (8 \log ^3(3)\right ) \int e^{-\frac {x}{\log (3)}} \, dx\\ &=\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4+\frac {1}{4} (5+2 x)^2+\frac {4}{3} e^{-\frac {x}{\log (3)}} x^3 \log (3)+4 e^{-\frac {x}{\log (3)}} x^2 \log ^2(3)+8 e^{-\frac {x}{\log (3)}} x \log ^3(3)+8 e^{-\frac {x}{\log (3)}} \log ^4(3)-\frac {1}{3} e^{-\frac {x}{\log (3)}} x^3 \log (81)-e^{-\frac {x}{\log (3)}} x^2 \log (3) \log (81)-2 e^{-\frac {x}{\log (3)}} x \log ^2(3) \log (81)-2 e^{-\frac {x}{\log (3)}} \log ^3(3) \log (81)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 23, normalized size = 1.10 \begin {gather*} 5 x+x^2+\frac {1}{3} e^{-\frac {x}{\log (3)}} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^4 + 4*x^3*Log[3] + E^(x/Log[3])*(15 + 6*x)*Log[3])/(3*E^(x/Log[3])*Log[3]),x]

[Out]

5*x + x^2 + x^4/(3*E^(x/Log[3]))

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 30, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 3 \, {\left (x^{2} + 5 \, x\right )} e^{\frac {x}{\log \relax (3)}}\right )} e^{\left (-\frac {x}{\log \relax (3)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x+15)*log(3)*exp(x/log(3))+4*x^3*log(3)-x^4)/log(3)/exp(x/log(3)),x, algorithm="fricas")

[Out]

1/3*(x^4 + 3*(x^2 + 5*x)*e^(x/log(3)))*e^(-x/log(3))

________________________________________________________________________________________

giac [A]  time = 0.25, size = 33, normalized size = 1.57 \begin {gather*} \frac {x^{4} e^{\left (-\frac {x}{\log \relax (3)}\right )} \log \relax (3) + 3 \, x^{2} \log \relax (3) + 15 \, x \log \relax (3)}{3 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x+15)*log(3)*exp(x/log(3))+4*x^3*log(3)-x^4)/log(3)/exp(x/log(3)),x, algorithm="giac")

[Out]

1/3*(x^4*e^(-x/log(3))*log(3) + 3*x^2*log(3) + 15*x*log(3))/log(3)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 21, normalized size = 1.00

method	result	size
risch	\(x^{2}+\frac {{\mathrm e}^{-\frac {x}{\ln \relax (3)}} x^{4}}{3}+5 x\)	\(21\)
derivativedivides	\(x^{2}+\frac {{\mathrm e}^{-\frac {x}{\ln \relax (3)}} x^{4}}{3}+5 x\)	\(22\)
default	\(x^{2}+\frac {{\mathrm e}^{-\frac {x}{\ln \relax (3)}} x^{4}}{3}+5 x\)	\(22\)
norman	\(\left (x^{2} {\mathrm e}^{\frac {x}{\ln \relax (3)}}+\frac {x^{4}}{3}+5 x \,{\mathrm e}^{\frac {x}{\ln \relax (3)}}\right ) {\mathrm e}^{-\frac {x}{\ln \relax (3)}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((6*x+15)*ln(3)*exp(x/ln(3))+4*x^3*ln(3)-x^4)/ln(3)/exp(x/ln(3)),x,method=_RETURNVERBOSE)

[Out]

x^2+1/3*exp(-x/ln(3))*x^4+5*x

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 107, normalized size = 5.10 \begin {gather*} \frac {3 \, x^{2} \log \relax (3) - 4 \, {\left (x^{3} \log \relax (3) + 3 \, x^{2} \log \relax (3)^{2} + 6 \, x \log \relax (3)^{3} + 6 \, \log \relax (3)^{4}\right )} e^{\left (-\frac {x}{\log \relax (3)}\right )} \log \relax (3) + {\left (x^{4} \log \relax (3) + 4 \, x^{3} \log \relax (3)^{2} + 12 \, x^{2} \log \relax (3)^{3} + 24 \, x \log \relax (3)^{4} + 24 \, \log \relax (3)^{5}\right )} e^{\left (-\frac {x}{\log \relax (3)}\right )} + 15 \, x \log \relax (3)}{3 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x+15)*log(3)*exp(x/log(3))+4*x^3*log(3)-x^4)/log(3)/exp(x/log(3)),x, algorithm="maxima")

[Out]

1/3*(3*x^2*log(3) - 4*(x^3*log(3) + 3*x^2*log(3)^2 + 6*x*log(3)^3 + 6*log(3)^4)*e^(-x/log(3))*log(3) + (x^4*lo
g(3) + 4*x^3*log(3)^2 + 12*x^2*log(3)^3 + 24*x*log(3)^4 + 24*log(3)^5)*e^(-x/log(3)) + 15*x*log(3))/log(3)

________________________________________________________________________________________

mupad [B]  time = 1.15, size = 20, normalized size = 0.95 \begin {gather*} 5\,x+\frac {x^4\,{\mathrm {e}}^{-\frac {x}{\ln \relax (3)}}}{3}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x/log(3))*((4*x^3*log(3))/3 - x^4/3 + (exp(x/log(3))*log(3)*(6*x + 15))/3))/log(3),x)

[Out]

5*x + (x^4*exp(-x/log(3)))/3 + x^2

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 17, normalized size = 0.81 \begin {gather*} \frac {x^{4} e^{- \frac {x}{\log {\relax (3 )}}}}{3} + x^{2} + 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x+15)*ln(3)*exp(x/ln(3))+4*x**3*ln(3)-x**4)/ln(3)/exp(x/ln(3)),x)

[Out]

x**4*exp(-x/log(3))/3 + x**2 + 5*x

________________________________________________________________________________________