3.84.41 \(\int (4+5 e^{-8+4 x}+2 x+e^{-8+4 x} (5+20 x) \log (x)) \, dx\)

Optimal. Leaf size=22 \[ 2+x-x \left (-3-x-5 e^{-8+4 x} \log (x)\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.73, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2194, 2176, 2554, 12} \begin {gather*} x^2+4 x-\frac {5}{4} e^{4 x-8} \log (x)+\frac {5}{4} e^{4 x-8} (4 x+1) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 + 5*E^(-8 + 4*x) + 2*x + E^(-8 + 4*x)*(5 + 20*x)*Log[x],x]

[Out]

4*x + x^2 - (5*E^(-8 + 4*x)*Log[x])/4 + (5*E^(-8 + 4*x)*(1 + 4*x)*Log[x])/4

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 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x+x^2+5 \int e^{-8+4 x} \, dx+\int e^{-8+4 x} (5+20 x) \log (x) \, dx\\ &=\frac {5}{4} e^{-8+4 x}+4 x+x^2-\frac {5}{4} e^{-8+4 x} \log (x)+\frac {5}{4} e^{-8+4 x} (1+4 x) \log (x)-\int 5 e^{-8+4 x} \, dx\\ &=\frac {5}{4} e^{-8+4 x}+4 x+x^2-\frac {5}{4} e^{-8+4 x} \log (x)+\frac {5}{4} e^{-8+4 x} (1+4 x) \log (x)-5 \int e^{-8+4 x} \, dx\\ &=4 x+x^2-\frac {5}{4} e^{-8+4 x} \log (x)+\frac {5}{4} e^{-8+4 x} (1+4 x) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 19, normalized size = 0.86 \begin {gather*} 4 x+x^2+5 e^{-8+4 x} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 + 5*E^(-8 + 4*x) + 2*x + E^(-8 + 4*x)*(5 + 20*x)*Log[x],x]

[Out]

4*x + x^2 + 5*E^(-8 + 4*x)*x*Log[x]

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fricas [A]  time = 0.61, size = 18, normalized size = 0.82 \begin {gather*} 5 \, x e^{\left (4 \, x - 8\right )} \log \relax (x) + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+5)*exp(x-2)^4*log(x)+5*exp(x-2)^4+2*x+4,x, algorithm="fricas")

[Out]

5*x*e^(4*x - 8)*log(x) + x^2 + 4*x

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giac [A]  time = 0.17, size = 18, normalized size = 0.82 \begin {gather*} 5 \, x e^{\left (4 \, x - 8\right )} \log \relax (x) + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+5)*exp(x-2)^4*log(x)+5*exp(x-2)^4+2*x+4,x, algorithm="giac")

[Out]

5*x*e^(4*x - 8)*log(x) + x^2 + 4*x

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maple [A]  time = 0.05, size = 19, normalized size = 0.86

method	result	size
default	\(4 x +5 x \,{\mathrm e}^{4 x -8} \ln \relax (x )+x^{2}\)	\(19\)
risch	\(4 x +5 x \,{\mathrm e}^{4 x -8} \ln \relax (x )+x^{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x+5)*exp(x-2)^4*ln(x)+5*exp(x-2)^4+2*x+4,x,method=_RETURNVERBOSE)

[Out]

4*x+5*x*exp(4*x-8)*ln(x)+x^2

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maxima [A]  time = 0.36, size = 18, normalized size = 0.82 \begin {gather*} 5 \, x e^{\left (4 \, x - 8\right )} \log \relax (x) + x^{2} + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+5)*exp(x-2)^4*log(x)+5*exp(x-2)^4+2*x+4,x, algorithm="maxima")

[Out]

5*x*e^(4*x - 8)*log(x) + x^2 + 4*x

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mupad [B]  time = 5.15, size = 18, normalized size = 0.82 \begin {gather*} 4\,x+x^2+5\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-8}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 5*exp(4*x - 8) + exp(4*x - 8)*log(x)*(20*x + 5) + 4,x)

[Out]

4*x + x^2 + 5*x*exp(4*x)*exp(-8)*log(x)

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sympy [A]  time = 0.28, size = 19, normalized size = 0.86 \begin {gather*} x^{2} + 5 x e^{4 x - 8} \log {\relax (x )} + 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+5)*exp(x-2)**4*ln(x)+5*exp(x-2)**4+2*x+4,x)

[Out]

x**2 + 5*x*exp(4*x - 8)*log(x) + 4*x

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