3.43.10 \(\int \frac {2+e^x x}{x} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\frac {1}{4} e^{\frac {2}{e^3}+e^x} x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.42, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {14, 2194} \begin {gather*} e^x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^x*x)/x,x]

[Out]

E^x + 2*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {2}{x}\right ) \, dx\\ &=2 \log (x)+\int e^x \, dx\\ &=e^x+2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.42 \begin {gather*} e^x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^x*x)/x,x]

[Out]

E^x + 2*Log[x]

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fricas [A]  time = 0.99, size = 7, normalized size = 0.37 \begin {gather*} e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+2)/x,x, algorithm="fricas")

[Out]

e^x + 2*log(x)

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giac [A]  time = 0.17, size = 7, normalized size = 0.37 \begin {gather*} e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+2)/x,x, algorithm="giac")

[Out]

e^x + 2*log(x)

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maple [A]  time = 0.02, size = 8, normalized size = 0.42

method	result	size
default	\(2 \ln \relax (x )+{\mathrm e}^{x}\)	\(8\)
norman	\(2 \ln \relax (x )+{\mathrm e}^{x}\)	\(8\)
risch	\(2 \ln \relax (x )+{\mathrm e}^{x}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

2*ln(x)+exp(x)

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maxima [A]  time = 0.34, size = 7, normalized size = 0.37 \begin {gather*} e^{x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+2)/x,x, algorithm="maxima")

[Out]

e^x + 2*log(x)

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mupad [B]  time = 0.06, size = 7, normalized size = 0.37 \begin {gather*} {\mathrm {e}}^x+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x) + 2)/x,x)

[Out]

exp(x) + 2*log(x)

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sympy [A]  time = 0.08, size = 7, normalized size = 0.37 \begin {gather*} e^{x} + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x+2)/x,x)

[Out]

exp(x) + 2*log(x)

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