3.81.35 \(\int \frac {-1+5 x+x \log (\frac {e^5}{x})}{x} \, dx\)

Optimal. Leaf size=26 \[ -x^2+x \left (6+x+\log \left (\frac {e^5}{x}\right )-\frac {\log (x)}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.54, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 43, 2295} \begin {gather*} 11 x+x \log \left (\frac {1}{x}\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 5*x + x*Log[E^5/x])/x,x]

[Out]

11*x + x*Log[x^(-1)] - 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 (5+\frac {-1+5 x}{x}+\log \left (\frac {1}{x}\right )\right ) \, dx\\ &=5 x+\int \frac {-1+5 x}{x} \, dx+\int \log \left (\frac {1}{x}\right ) \, dx\\ &=6 x+x \log \left (\frac {1}{x}\right )+\int \left (5-\frac {1}{x}\right ) \, dx\\ &=11 x+x \log \left (\frac {1}{x}\right )-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.54 \begin {gather*} 11 x+x \log \left (\frac {1}{x}\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

11*x + x*Log[x^(-1)] - Log[x]

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fricas [A]  time = 0.64, size = 15, normalized size = 0.58 \begin {gather*} {\left (x + 1\right )} \log \left (\frac {e^{5}}{x}\right ) + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*log(e^5/x) + 6*x

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giac [A]  time = 0.51, size = 32, normalized size = 1.23 \begin {gather*} {\left (e^{15} \log \left (\frac {e^{5}}{x}\right ) + \frac {e^{15} \log \left (\frac {e^{5}}{x}\right )}{x} + 6 \, e^{15}\right )} x e^{\left (-15\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^15*log(e^5/x) + e^15*log(e^5/x)/x + 6*e^15)*x*e^(-15)

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maple [A]  time = 0.07, size = 18, normalized size = 0.69

method	result	size
risch	\(x \ln \left (\frac {{\mathrm e}^{5}}{x}\right )+6 x -\ln \relax (x )\)	\(18\)
norman	\(x \ln \left (\frac {{\mathrm e}^{5}}{x}\right )+\ln \left (\frac {{\mathrm e}^{5}}{x}\right )+6 x\)	\(21\)
derivativedivides	\(-{\mathrm e}^{5} \left (-x \,{\mathrm e}^{-5} \ln \left (\frac {{\mathrm e}^{5}}{x}\right )-x \,{\mathrm e}^{-5}\right )+5 x +\ln \left (\frac {{\mathrm e}^{5}}{x}\right )\)	\(38\)
default	\(-{\mathrm e}^{5} \left (-x \,{\mathrm e}^{-5} \ln \left (\frac {{\mathrm e}^{5}}{x}\right )-x \,{\mathrm e}^{-5}\right )+5 x +\ln \left (\frac {{\mathrm e}^{5}}{x}\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(exp(5)/x)+6*x-ln(x)

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maxima [A]  time = 0.36, size = 17, normalized size = 0.65 \begin {gather*} x \log \left (\frac {e^{5}}{x}\right ) + 6 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(e^5/x) + 6*x - log(x)

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mupad [B]  time = 5.63, size = 14, normalized size = 0.54 \begin {gather*} 11\,x+\ln \left (\frac {1}{x}\right )+x\,\ln \left (\frac {1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + x*log(exp(5)/x) - 1)/x,x)

[Out]

11*x + log(1/x) + x*log(1/x)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.54 \begin {gather*} x \log {\left (\frac {e^{5}}{x} \right )} + 6 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(exp(5)/x) + 6*x - log(x)

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