3.11.7 \(\int \frac {1-6 x^2+27 x^3+3 x \log (3)}{3 x} \, dx\)

Optimal. Leaf size=31 \[ -x \left (x-3 \left (\frac {e^{5/3}}{x}+x^2\right )\right )+x \log (3)+\frac {\log (x)}{3} \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 14} \begin {gather*} 3 x^3-x^2+\frac {1}{3} x \log (27)+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 6*x^2 + 27*x^3 + 3*x*Log[3])/(3*x),x]

[Out]

-x^2 + 3*x^3 + (x*Log[27])/3 + Log[x]/3

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {1-6 x^2+27 x^3+3 x \log (3)}{x} \, dx\\ &=\frac {1}{3} \int \left (\frac {1}{x}-6 x+27 x^2+\log (27)\right ) \, dx\\ &=-x^2+3 x^3+\frac {1}{3} x \log (27)+\frac {\log (x)}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.68 \begin {gather*} -x^2+3 x^3+x \log (3)+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 6*x^2 + 27*x^3 + 3*x*Log[3])/(3*x),x]

[Out]

-x^2 + 3*x^3 + x*Log[3] + Log[x]/3

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fricas [A]  time = 0.57, size = 19, normalized size = 0.61 \begin {gather*} 3 \, x^{3} - x^{2} + x \log \relax (3) + \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*log(3)+27*x^3-6*x^2+1)/x,x, algorithm="fricas")

[Out]

3*x^3 - x^2 + x*log(3) + 1/3*log(x)

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giac [A]  time = 0.45, size = 20, normalized size = 0.65 \begin {gather*} 3 \, x^{3} - x^{2} + x \log \relax (3) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*log(3)+27*x^3-6*x^2+1)/x,x, algorithm="giac")

[Out]

3*x^3 - x^2 + x*log(3) + 1/3*log(abs(x))

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maple [A]  time = 0.03, size = 20, normalized size = 0.65

method	result	size
default	\(3 x^{3}-x^{2}+x \ln \relax (3)+\frac {\ln \relax (x )}{3}\)	\(20\)
norman	\(3 x^{3}-x^{2}+x \ln \relax (3)+\frac {\ln \relax (x )}{3}\)	\(20\)
risch	\(3 x^{3}-x^{2}+x \ln \relax (3)+\frac {\ln \relax (x )}{3}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*x*ln(3)+27*x^3-6*x^2+1)/x,x,method=_RETURNVERBOSE)

[Out]

3*x^3-x^2+x*ln(3)+1/3*ln(x)

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maxima [A]  time = 0.40, size = 19, normalized size = 0.61 \begin {gather*} 3 \, x^{3} - x^{2} + x \log \relax (3) + \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*log(3)+27*x^3-6*x^2+1)/x,x, algorithm="maxima")

[Out]

3*x^3 - x^2 + x*log(3) + 1/3*log(x)

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mupad [B]  time = 0.04, size = 19, normalized size = 0.61 \begin {gather*} \frac {\ln \relax (x)}{3}+x\,\ln \relax (3)-x^2+3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(3) - 2*x^2 + 9*x^3 + 1/3)/x,x)

[Out]

log(x)/3 + x*log(3) - x^2 + 3*x^3

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sympy [A]  time = 0.08, size = 17, normalized size = 0.55 \begin {gather*} 3 x^{3} - x^{2} + x \log {\relax (3 )} + \frac {\log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*ln(3)+27*x**3-6*x**2+1)/x,x)

[Out]

3*x**3 - x**2 + x*log(3) + log(x)/3

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