3.11.44 \(\int \frac {2+2044 x-1920 x^2-48 x^3-32 x^4-x \log (x)}{x} \, dx\)

Optimal. Leaf size=21 \[ (2-x) (3+4 x (256+2 x (4+x))+\log (x)) \]

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Rubi [A]  time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {14, 2295} \begin {gather*} -8 x^4-16 x^3-960 x^2+2045 x-x \log (x)+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 2044*x - 1920*x^2 - 48*x^3 - 32*x^4 - x*Log[x])/x,x]

[Out]

2045*x - 960*x^2 - 16*x^3 - 8*x^4 + 2*Log[x] - 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 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 (-\frac {2 \left (-1-1022 x+960 x^2+24 x^3+16 x^4\right )}{x}-\log (x)\right ) \, dx\\ &=-\left (2 \int \frac {-1-1022 x+960 x^2+24 x^3+16 x^4}{x} \, dx\right )-\int \log (x) \, dx\\ &=x-x \log (x)-2 \int \left (-1022-\frac {1}{x}+960 x+24 x^2+16 x^3\right ) \, dx\\ &=2045 x-960 x^2-16 x^3-8 x^4+2 \log (x)-x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.33 \begin {gather*} 2045 x-960 x^2-16 x^3-8 x^4+2 \log (x)-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2044*x - 1920*x^2 - 48*x^3 - 32*x^4 - x*Log[x])/x,x]

[Out]

2045*x - 960*x^2 - 16*x^3 - 8*x^4 + 2*Log[x] - x*Log[x]

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fricas [A]  time = 0.68, size = 26, normalized size = 1.24 \begin {gather*} -8 \, x^{4} - 16 \, x^{3} - 960 \, x^{2} - {\left (x - 2\right )} \log \relax (x) + 2045 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)-32*x^4-48*x^3-1920*x^2+2044*x+2)/x,x, algorithm="fricas")

[Out]

-8*x^4 - 16*x^3 - 960*x^2 - (x - 2)*log(x) + 2045*x

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giac [A]  time = 0.21, size = 28, normalized size = 1.33 \begin {gather*} -8 \, x^{4} - 16 \, x^{3} - 960 \, x^{2} - x \log \relax (x) + 2045 \, x + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)-32*x^4-48*x^3-1920*x^2+2044*x+2)/x,x, algorithm="giac")

[Out]

-8*x^4 - 16*x^3 - 960*x^2 - x*log(x) + 2045*x + 2*log(x)

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maple [A]  time = 0.02, size = 29, normalized size = 1.38

method	result	size
default	\(-8 x^{4}-16 x^{3}+2045 x -x \ln \relax (x )-960 x^{2}+2 \ln \relax (x )\)	\(29\)
norman	\(-8 x^{4}-16 x^{3}+2045 x -x \ln \relax (x )-960 x^{2}+2 \ln \relax (x )\)	\(29\)
risch	\(-8 x^{4}-16 x^{3}+2045 x -x \ln \relax (x )-960 x^{2}+2 \ln \relax (x )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x)-32*x^4-48*x^3-1920*x^2+2044*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

-8*x^4-16*x^3+2045*x-x*ln(x)-960*x^2+2*ln(x)

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maxima [A]  time = 0.42, size = 28, normalized size = 1.33 \begin {gather*} -8 \, x^{4} - 16 \, x^{3} - 960 \, x^{2} - x \log \relax (x) + 2045 \, x + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)-32*x^4-48*x^3-1920*x^2+2044*x+2)/x,x, algorithm="maxima")

[Out]

-8*x^4 - 16*x^3 - 960*x^2 - x*log(x) + 2045*x + 2*log(x)

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mupad [B]  time = 0.71, size = 28, normalized size = 1.33 \begin {gather*} 2045\,x+2\,\ln \relax (x)-x\,\ln \relax (x)-960\,x^2-16\,x^3-8\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*log(x) - 2044*x + 1920*x^2 + 48*x^3 + 32*x^4 - 2)/x,x)

[Out]

2045*x + 2*log(x) - x*log(x) - 960*x^2 - 16*x^3 - 8*x^4

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sympy [A]  time = 0.11, size = 27, normalized size = 1.29 \begin {gather*} - 8 x^{4} - 16 x^{3} - 960 x^{2} - x \log {\relax (x )} + 2045 x + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x)-32*x**4-48*x**3-1920*x**2+2044*x+2)/x,x)

[Out]

-8*x**4 - 16*x**3 - 960*x**2 - x*log(x) + 2045*x + 2*log(x)

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