3.103.33 \(\int \frac {38 x+2 x^2+36 \log (x)}{x^3+x^2 \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (e^{-\frac {2 \left (16+\frac {2 \left (x-x^2\right )}{x}\right )}{x}} (x+\log (x))^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.29, antiderivative size = 13, normalized size of antiderivative = 0.45, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2561, 6741, 12, 6742, 6684} \begin {gather*} 2 \log (x+\log (x))-\frac {36}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(38*x + 2*x^2 + 36*Log[x])/(x^3 + x^2*Log[x]),x]

[Out]

-36/x + 2*Log[x + Log[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

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 {38 x+2 x^2+36 \log (x)}{x^2 (x+\log (x))} \, dx\\ &=\int \frac {2 \left (19 x+x^2+18 \log (x)\right )}{x^2 (x+\log (x))} \, dx\\ &=2 \int \frac {19 x+x^2+18 \log (x)}{x^2 (x+\log (x))} \, dx\\ &=2 \int \left (\frac {18}{x^2}+\frac {1+x}{x (x+\log (x))}\right ) \, dx\\ &=-\frac {36}{x}+2 \int \frac {1+x}{x (x+\log (x))} \, dx\\ &=-\frac {36}{x}+2 \log (x+\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 13, normalized size = 0.45 \begin {gather*} -\frac {36}{x}+2 \log (x+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(38*x + 2*x^2 + 36*Log[x])/(x^3 + x^2*Log[x]),x]

[Out]

-36/x + 2*Log[x + Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 14, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (x \log \left (x + \log \relax (x)\right ) - 18\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*log(x)+2*x^2+38*x)/(x^2*log(x)+x^3),x, algorithm="fricas")

[Out]

2*(x*log(x + log(x)) - 18)/x

________________________________________________________________________________________

giac [A]  time = 0.14, size = 13, normalized size = 0.45 \begin {gather*} -\frac {36}{x} + 2 \, \log \left (x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*log(x)+2*x^2+38*x)/(x^2*log(x)+x^3),x, algorithm="giac")

[Out]

-36/x + 2*log(x + log(x))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 14, normalized size = 0.48

method	result	size
norman	\(-\frac {36}{x}+2 \ln \left (x +\ln \relax (x )\right )\)	\(14\)
risch	\(-\frac {36}{x}+2 \ln \left (x +\ln \relax (x )\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*ln(x)+2*x^2+38*x)/(x^2*ln(x)+x^3),x,method=_RETURNVERBOSE)

[Out]

-36/x+2*ln(x+ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 13, normalized size = 0.45 \begin {gather*} -\frac {36}{x} + 2 \, \log \left (x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*log(x)+2*x^2+38*x)/(x^2*log(x)+x^3),x, algorithm="maxima")

[Out]

-36/x + 2*log(x + log(x))

________________________________________________________________________________________

mupad [B]  time = 5.96, size = 13, normalized size = 0.45 \begin {gather*} 2\,\ln \left (x+\ln \relax (x)\right )-\frac {36}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((38*x + 36*log(x) + 2*x^2)/(x^2*log(x) + x^3),x)

[Out]

2*log(x + log(x)) - 36/x

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 10, normalized size = 0.34 \begin {gather*} 2 \log {\left (x + \log {\relax (x )} \right )} - \frac {36}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*ln(x)+2*x**2+38*x)/(x**2*ln(x)+x**3),x)

[Out]

2*log(x + log(x)) - 36/x

________________________________________________________________________________________