3.14.28 \(\int \frac {-20+360 x-270 x^2+15 x \log (x)}{-18 x^2+x \log (x)} \, dx\)

Optimal. Leaf size=27 \[ 5 \left (-4+x-2 \left (5-x+\log \left (\left (-\frac {9 x}{2}+\frac {\log (x)}{4}\right )^2\right )\right )\right ) \]

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Rubi [A]  time = 0.43, antiderivative size = 15, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2561, 6741, 12, 6742, 6684} \begin {gather*} 15 x-20 \log (18 x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20 + 360*x - 270*x^2 + 15*x*Log[x])/(-18*x^2 + x*Log[x]),x]

[Out]

15*x - 20*Log[18*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 {-20+360 x-270 x^2+15 x \log (x)}{x (-18 x+\log (x))} \, dx\\ &=\int \frac {5 \left (4-72 x+54 x^2-3 x \log (x)\right )}{x (18 x-\log (x))} \, dx\\ &=5 \int \frac {4-72 x+54 x^2-3 x \log (x)}{x (18 x-\log (x))} \, dx\\ &=5 \int \left (3-\frac {4 (-1+18 x)}{x (18 x-\log (x))}\right ) \, dx\\ &=15 x-20 \int \frac {-1+18 x}{x (18 x-\log (x))} \, dx\\ &=15 x-20 \log (18 x-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 17, normalized size = 0.63 \begin {gather*} 5 (3 x-4 \log (18 x-\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 360*x - 270*x^2 + 15*x*Log[x])/(-18*x^2 + x*Log[x]),x]

[Out]

5*(3*x - 4*Log[18*x - Log[x]])

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fricas [A]  time = 0.68, size = 13, normalized size = 0.48 \begin {gather*} 15 \, x - 20 \, \log \left (-18 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x*log(x)-270*x^2+360*x-20)/(x*log(x)-18*x^2),x, algorithm="fricas")

[Out]

15*x - 20*log(-18*x + log(x))

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giac [A]  time = 0.24, size = 15, normalized size = 0.56 \begin {gather*} 15 \, x - 20 \, \log \left (18 \, x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x*log(x)-270*x^2+360*x-20)/(x*log(x)-18*x^2),x, algorithm="giac")

[Out]

15*x - 20*log(18*x - log(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.52

method	result	size
risch	\(15 x -20 \ln \left (\ln \relax (x )-18 x \right )\)	\(14\)
norman	\(15 x -20 \ln \left (18 x -\ln \relax (x )\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*x*ln(x)-270*x^2+360*x-20)/(x*ln(x)-18*x^2),x,method=_RETURNVERBOSE)

[Out]

15*x-20*ln(ln(x)-18*x)

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maxima [A]  time = 0.45, size = 13, normalized size = 0.48 \begin {gather*} 15 \, x - 20 \, \log \left (-18 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x*log(x)-270*x^2+360*x-20)/(x*log(x)-18*x^2),x, algorithm="maxima")

[Out]

15*x - 20*log(-18*x + log(x))

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mupad [B]  time = 0.97, size = 15, normalized size = 0.56 \begin {gather*} 15\,x-20\,\ln \left (18\,x-\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((360*x + 15*x*log(x) - 270*x^2 - 20)/(x*log(x) - 18*x^2),x)

[Out]

15*x - 20*log(18*x - log(x))

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sympy [A]  time = 0.12, size = 12, normalized size = 0.44 \begin {gather*} 15 x - 20 \log {\left (- 18 x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x*ln(x)-270*x**2+360*x-20)/(x*ln(x)-18*x**2),x)

[Out]

15*x - 20*log(-18*x + log(x))

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