3.74.40 \(\int \frac {10+3 x+(3+x) \log (x)}{3 x+x \log (x)} \, dx\)

Optimal. Leaf size=15 \[ x-\log \left (\frac {1}{x^3 (3+\log (x))}\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 11, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2561, 6742, 43, 2302, 29} \begin {gather*} x+3 \log (x)+\log (\log (x)+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10 + 3*x + (3 + x)*Log[x])/(3*x + x*Log[x]),x]

[Out]

x + 3*Log[x] + Log[3 + Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10+3 x+(3+x) \log (x)}{x (3+\log (x))} \, dx\\ &=\int \left (\frac {3+x}{x}+\frac {1}{x (3+\log (x))}\right ) \, dx\\ &=\int \frac {3+x}{x} \, dx+\int \frac {1}{x (3+\log (x))} \, dx\\ &=\int \left (1+\frac {3}{x}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,3+\log (x)\right )\\ &=x+3 \log (x)+\log (3+\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 11, normalized size = 0.73 \begin {gather*} x+3 \log (x)+\log (3+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 + 3*x + (3 + x)*Log[x])/(3*x + x*Log[x]),x]

[Out]

x + 3*Log[x] + Log[3 + Log[x]]

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fricas [A]  time = 0.95, size = 11, normalized size = 0.73 \begin {gather*} x + 3 \, \log \relax (x) + \log \left (\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(x)+3*x+10)/(x*log(x)+3*x),x, algorithm="fricas")

[Out]

x + 3*log(x) + log(log(x) + 3)

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giac [A]  time = 0.19, size = 11, normalized size = 0.73 \begin {gather*} x + 3 \, \log \relax (x) + \log \left (\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(x)+3*x+10)/(x*log(x)+3*x),x, algorithm="giac")

[Out]

x + 3*log(x) + log(log(x) + 3)

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maple [A]  time = 0.02, size = 12, normalized size = 0.80

method	result	size
norman	\(x +3 \ln \relax (x )+\ln \left (3+\ln \relax (x )\right )\)	\(12\)
risch	\(x +3 \ln \relax (x )+\ln \left (3+\ln \relax (x )\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+x)*ln(x)+3*x+10)/(x*ln(x)+3*x),x,method=_RETURNVERBOSE)

[Out]

x+3*ln(x)+ln(3+ln(x))

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maxima [A]  time = 0.39, size = 11, normalized size = 0.73 \begin {gather*} x + 3 \, \log \relax (x) + \log \left (\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(x)+3*x+10)/(x*log(x)+3*x),x, algorithm="maxima")

[Out]

x + 3*log(x) + log(log(x) + 3)

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mupad [B]  time = 4.74, size = 11, normalized size = 0.73 \begin {gather*} x+\ln \left (\ln \relax (x)+3\right )+3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + log(x)*(x + 3) + 10)/(3*x + x*log(x)),x)

[Out]

x + log(log(x) + 3) + 3*log(x)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} x + 3 \log {\relax (x )} + \log {\left (\log {\relax (x )} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*ln(x)+3*x+10)/(x*ln(x)+3*x),x)

[Out]

x + 3*log(x) + log(log(x) + 3)

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