3.37.6 \(\int \frac {1+5 x-3 e^{e^x+x} x+4 x^2+2 x \log (x)}{x} \, dx\)

Optimal. Leaf size=24 \[ -4-3 e^{e^x}+x-\log (x)+(2+2 x) (x+\log (x)) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {14, 2282, 2194, 2295} \begin {gather*} 2 x^2+3 x-3 e^{e^x}+2 x \log (x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 5*x - 3*E^(E^x + x)*x + 4*x^2 + 2*x*Log[x])/x,x]

[Out]

-3*E^E^x + 3*x + 2*x^2 + Log[x] + 2*x*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.96 \begin {gather*} -3 e^{e^x}+3 x+2 x^2+\log (x)+2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 5*x - 3*E^(E^x + x)*x + 4*x^2 + 2*x*Log[x])/x,x]

[Out]

-3*E^E^x + 3*x + 2*x^2 + Log[x] + 2*x*Log[x]

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fricas [A]  time = 0.77, size = 35, normalized size = 1.46 \begin {gather*} {\left ({\left (2 \, x + 1\right )} e^{x} \log \relax (x) + {\left (2 \, x^{2} + 3 \, x\right )} e^{x} - 3 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*exp(x)*exp(exp(x))+2*x*log(x)+4*x^2+5*x+1)/x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 37, normalized size = 1.54 \begin {gather*} {\left (2 \, x^{2} e^{x} + 2 \, x e^{x} \log \relax (x) + 3 \, x e^{x} + e^{x} \log \relax (x) - 3 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*exp(x)*exp(exp(x))+2*x*log(x)+4*x^2+5*x+1)/x,x, algorithm="giac")

[Out]

(2*x^2*e^x + 2*x*e^x*log(x) + 3*x*e^x + e^x*log(x) - 3*e^(x + e^x))*e^(-x)

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maple [A]  time = 0.02, size = 22, normalized size = 0.92

method	result	size
default	\(2 x^{2}+3 x +\ln \relax (x )-3 \,{\mathrm e}^{{\mathrm e}^{x}}+2 x \ln \relax (x )\)	\(22\)
risch	\(2 x^{2}+3 x +\ln \relax (x )-3 \,{\mathrm e}^{{\mathrm e}^{x}}+2 x \ln \relax (x )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x*exp(x)*exp(exp(x))+2*x*ln(x)+4*x^2+5*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

2*x^2+3*x+ln(x)-3*exp(exp(x))+2*x*ln(x)

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maxima [A]  time = 0.37, size = 21, normalized size = 0.88 \begin {gather*} 2 \, x^{2} + 2 \, x \log \relax (x) + 3 \, x - 3 \, e^{\left (e^{x}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*exp(x)*exp(exp(x))+2*x*log(x)+4*x^2+5*x+1)/x,x, algorithm="maxima")

[Out]

2*x^2 + 2*x*log(x) + 3*x - 3*e^(e^x) + log(x)

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mupad [B]  time = 2.27, size = 21, normalized size = 0.88 \begin {gather*} 3\,x-3\,{\mathrm {e}}^{{\mathrm {e}}^x}+\ln \relax (x)+2\,x\,\ln \relax (x)+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 2*x*log(x) + 4*x^2 - 3*x*exp(exp(x))*exp(x) + 1)/x,x)

[Out]

3*x - 3*exp(exp(x)) + log(x) + 2*x*log(x) + 2*x^2

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sympy [A]  time = 0.35, size = 24, normalized size = 1.00 \begin {gather*} 2 x^{2} + 2 x \log {\relax (x )} + 3 x - 3 e^{e^{x}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**2 + 2*x*log(x) + 3*x - 3*exp(exp(x)) + log(x)

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