3.95.33 \(\int \frac {-18+9 x+e^x x+6 x \log (x)}{x} \, dx\)

Optimal. Leaf size=18 \[ e^x+3 \left (-\frac {4}{9}+x+2 (-3+x) \log (x)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 2194, 43, 2295} \begin {gather*} 3 x+e^x+6 x \log (x)-18 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 + 9*x + E^x*x + 6*x*Log[x])/x,x]

[Out]

E^x + 3*x - 18*Log[x] + 6*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 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 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 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 (e^x+\frac {3 (-6+3 x+2 x \log (x))}{x}\right ) \, dx\\ &=3 \int \frac {-6+3 x+2 x \log (x)}{x} \, dx+\int e^x \, dx\\ &=e^x+3 \int \left (\frac {3 (-2+x)}{x}+2 \log (x)\right ) \, dx\\ &=e^x+6 \int \log (x) \, dx+9 \int \frac {-2+x}{x} \, dx\\ &=e^x-6 x+6 x \log (x)+9 \int \left (1-\frac {2}{x}\right ) \, dx\\ &=e^x+3 x-18 \log (x)+6 x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.89 \begin {gather*} e^x+3 x-18 \log (x)+6 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 + 9*x + E^x*x + 6*x*Log[x])/x,x]

[Out]

E^x + 3*x - 18*Log[x] + 6*x*Log[x]

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fricas [A]  time = 0.58, size = 13, normalized size = 0.72 \begin {gather*} 6 \, {\left (x - 3\right )} \log \relax (x) + 3 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)+exp(x)*x+9*x-18)/x,x, algorithm="fricas")

[Out]

6*(x - 3)*log(x) + 3*x + e^x

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giac [A]  time = 0.18, size = 15, normalized size = 0.83 \begin {gather*} 6 \, x \log \relax (x) + 3 \, x + e^{x} - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)+exp(x)*x+9*x-18)/x,x, algorithm="giac")

[Out]

6*x*log(x) + 3*x + e^x - 18*log(x)

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

method	result	size
default	\(3 x -18 \ln \relax (x )+6 x \ln \relax (x )+{\mathrm e}^{x}\)	\(16\)
norman	\(3 x -18 \ln \relax (x )+6 x \ln \relax (x )+{\mathrm e}^{x}\)	\(16\)
risch	\(3 x -18 \ln \relax (x )+6 x \ln \relax (x )+{\mathrm e}^{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*ln(x)+exp(x)*x+9*x-18)/x,x,method=_RETURNVERBOSE)

[Out]

3*x-18*ln(x)+6*x*ln(x)+exp(x)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.83 \begin {gather*} 6 \, x \log \relax (x) + 3 \, x + e^{x} - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)+exp(x)*x+9*x-18)/x,x, algorithm="maxima")

[Out]

6*x*log(x) + 3*x + e^x - 18*log(x)

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mupad [B]  time = 5.48, size = 15, normalized size = 0.83 \begin {gather*} 3\,x+{\mathrm {e}}^x-18\,\ln \relax (x)+6\,x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x + x*exp(x) + 6*x*log(x) - 18)/x,x)

[Out]

3*x + exp(x) - 18*log(x) + 6*x*log(x)

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sympy [A]  time = 0.27, size = 17, normalized size = 0.94 \begin {gather*} 6 x \log {\relax (x )} + 3 x + e^{x} - 18 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*ln(x)+exp(x)*x+9*x-18)/x,x)

[Out]

6*x*log(x) + 3*x + exp(x) - 18*log(x)

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