3.22.26 \(\int \frac {32-3 x-16 x^4-4 x \log (x)}{4 x} \, dx\)

Optimal. Leaf size=22 \[ -x^4-4 \log \left (\frac {1}{x^2}\right )+x \left (\frac {1}{4}-\log (x)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2295} \begin {gather*} -x^4+\frac {x}{4}-x \log (x)+8 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 - 3*x - 16*x^4 - 4*x*Log[x])/(4*x),x]

[Out]

x/4 - x^4 + 8*Log[x] - 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 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} &=\frac {1}{4} \int \frac {32-3 x-16 x^4-4 x \log (x)}{x} \, dx\\ &=\frac {1}{4} \int \left (\frac {32-3 x-16 x^4}{x}-4 \log (x)\right ) \, dx\\ &=\frac {1}{4} \int \frac {32-3 x-16 x^4}{x} \, dx-\int \log (x) \, dx\\ &=x-x \log (x)+\frac {1}{4} \int \left (-3+\frac {32}{x}-16 x^3\right ) \, dx\\ &=\frac {x}{4}-x^4+8 \log (x)-x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.91 \begin {gather*} \frac {x}{4}-x^4+8 \log (x)-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 - 3*x - 16*x^4 - 4*x*Log[x])/(4*x),x]

[Out]

x/4 - x^4 + 8*Log[x] - x*Log[x]

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fricas [A]  time = 0.56, size = 16, normalized size = 0.73 \begin {gather*} -x^{4} - {\left (x - 8\right )} \log \relax (x) + \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)-16*x^4-3*x+32)/x,x, algorithm="fricas")

[Out]

-x^4 - (x - 8)*log(x) + 1/4*x

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giac [A]  time = 0.21, size = 18, normalized size = 0.82 \begin {gather*} -x^{4} - x \log \relax (x) + \frac {1}{4} \, x + 8 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)-16*x^4-3*x+32)/x,x, algorithm="giac")

[Out]

-x^4 - x*log(x) + 1/4*x + 8*log(x)

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maple [A]  time = 0.02, size = 19, normalized size = 0.86

method	result	size
default	\(-x^{4}-x \ln \relax (x )+\frac {x}{4}+8 \ln \relax (x )\)	\(19\)
norman	\(-x^{4}-x \ln \relax (x )+\frac {x}{4}+8 \ln \relax (x )\)	\(19\)
risch	\(-x^{4}-x \ln \relax (x )+\frac {x}{4}+8 \ln \relax (x )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*x*ln(x)-16*x^4-3*x+32)/x,x,method=_RETURNVERBOSE)

[Out]

-x^4-x*ln(x)+1/4*x+8*ln(x)

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maxima [A]  time = 0.51, size = 18, normalized size = 0.82 \begin {gather*} -x^{4} - x \log \relax (x) + \frac {1}{4} \, x + 8 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)-16*x^4-3*x+32)/x,x, algorithm="maxima")

[Out]

-x^4 - x*log(x) + 1/4*x + 8*log(x)

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mupad [B]  time = 1.17, size = 18, normalized size = 0.82 \begin {gather*} \frac {x}{4}+8\,\ln \relax (x)-x\,\ln \relax (x)-x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x)/4 + x*log(x) + 4*x^4 - 8)/x,x)

[Out]

x/4 + 8*log(x) - x*log(x) - x^4

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sympy [A]  time = 0.11, size = 15, normalized size = 0.68 \begin {gather*} - x^{4} - x \log {\relax (x )} + \frac {x}{4} + 8 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*ln(x)-16*x**4-3*x+32)/x,x)

[Out]

-x**4 - x*log(x) + x/4 + 8*log(x)

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