3.20.33 \(\int \frac {-18+8 x-x^2+(8 x-2 x^2) \log (x)}{x} \, dx\)

Optimal. Leaf size=14 \[ (-18+4 x-(-4+x) x) \log (x) \]

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Rubi [B]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 2.71, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {14, 2313, 9} \begin {gather*} -\frac {x^2}{2}+\left (8 x-x^2\right ) \log (x)+\frac {1}{2} (8-x)^2+8 x-18 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 + 8*x - x^2 + (8*x - 2*x^2)*Log[x])/x,x]

[Out]

(8 - x)^2/2 + 8*x - x^2/2 - 18*Log[x] + (8*x - x^2)*Log[x]

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, 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 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-18+8 x-x^2}{x}-2 (-4+x) \log (x)\right ) \, dx\\ &=-(2 \int (-4+x) \log (x) \, dx)+\int \frac {-18+8 x-x^2}{x} \, dx\\ &=\left (8 x-x^2\right ) \log (x)+2 \int \frac {1}{2} (-8+x) \, dx+\int \left (8-\frac {18}{x}-x\right ) \, dx\\ &=\frac {1}{2} (8-x)^2+8 x-\frac {x^2}{2}-18 \log (x)+\left (8 x-x^2\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.21 \begin {gather*} -18 \log (x)+8 x \log (x)-x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 + 8*x - x^2 + (8*x - 2*x^2)*Log[x])/x,x]

[Out]

-18*Log[x] + 8*x*Log[x] - x^2*Log[x]

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fricas [A]  time = 0.70, size = 12, normalized size = 0.86 \begin {gather*} -{\left (x^{2} - 8 \, x + 18\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+8*x)*log(x)-x^2+8*x-18)/x,x, algorithm="fricas")

[Out]

-(x^2 - 8*x + 18)*log(x)

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giac [A]  time = 0.38, size = 16, normalized size = 1.14 \begin {gather*} -{\left (x^{2} - 8 \, x\right )} \log \relax (x) - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+8*x)*log(x)-x^2+8*x-18)/x,x, algorithm="giac")

[Out]

-(x^2 - 8*x)*log(x) - 18*log(x)

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maple [A]  time = 0.02, size = 18, normalized size = 1.29

method	result	size
default	\(-x^{2} \ln \relax (x )+8 x \ln \relax (x )-18 \ln \relax (x )\)	\(18\)
norman	\(-x^{2} \ln \relax (x )+8 x \ln \relax (x )-18 \ln \relax (x )\)	\(18\)
risch	\(\left (-x^{2}+8 x \right ) \ln \relax (x )-18 \ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+8*x)*ln(x)-x^2+8*x-18)/x,x,method=_RETURNVERBOSE)

[Out]

-x^2*ln(x)+8*x*ln(x)-18*ln(x)

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maxima [A]  time = 0.66, size = 17, normalized size = 1.21 \begin {gather*} -x^{2} \log \relax (x) + 8 \, x \log \relax (x) - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+8*x)*log(x)-x^2+8*x-18)/x,x, algorithm="maxima")

[Out]

-x^2*log(x) + 8*x*log(x) - 18*log(x)

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mupad [B]  time = 1.13, size = 12, normalized size = 0.86 \begin {gather*} -\ln \relax (x)\,\left (x^2-8\,x+18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + log(x)*(8*x - 2*x^2) - x^2 - 18)/x,x)

[Out]

-log(x)*(x^2 - 8*x + 18)

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sympy [A]  time = 0.10, size = 14, normalized size = 1.00 \begin {gather*} \left (- x^{2} + 8 x\right ) \log {\relax (x )} - 18 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+8*x)*ln(x)-x**2+8*x-18)/x,x)

[Out]

(-x**2 + 8*x)*log(x) - 18*log(x)

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