3.7.72 \(\int \frac {-x^2-8 x^3+3 \log (x)}{3 x^2} \, dx\)

Optimal. Leaf size=27 \[ 5-\frac {1+\frac {1}{3} x^3 \left (4+\frac {2+x}{x^2}\right )+\log (x)}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2304} \begin {gather*} -\frac {4 x^2}{3}-\frac {x}{3}-\frac {1}{x}-\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 - 8*x^3 + 3*Log[x])/(3*x^2),x]

[Out]

-x^(-1) - x/3 - (4*x^2)/3 - Log[x]/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 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-x^2-8 x^3+3 \log (x)}{x^2} \, dx\\ &=\frac {1}{3} \int \left (-1-8 x+\frac {3 \log (x)}{x^2}\right ) \, dx\\ &=-\frac {x}{3}-\frac {4 x^2}{3}+\int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {1}{x}-\frac {x}{3}-\frac {4 x^2}{3}-\frac {\log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 0.93 \begin {gather*} -\frac {1}{x}-\frac {x}{3}-\frac {4 x^2}{3}-\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 - 8*x^3 + 3*Log[x])/(3*x^2),x]

[Out]

-x^(-1) - x/3 - (4*x^2)/3 - Log[x]/x

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fricas [A]  time = 0.59, size = 19, normalized size = 0.70 \begin {gather*} -\frac {4 \, x^{3} + x^{2} + 3 \, \log \relax (x) + 3}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)-8*x^3-x^2)/x^2,x, algorithm="fricas")

[Out]

-1/3*(4*x^3 + x^2 + 3*log(x) + 3)/x

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giac [A]  time = 0.34, size = 21, normalized size = 0.78 \begin {gather*} -\frac {4}{3} \, x^{2} - \frac {1}{3} \, x - \frac {\log \relax (x)}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)-8*x^3-x^2)/x^2,x, algorithm="giac")

[Out]

-4/3*x^2 - 1/3*x - log(x)/x - 1/x

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maple [A]  time = 0.02, size = 21, normalized size = 0.78

method	result	size
norman	\(\frac {-1-\frac {x^{2}}{3}-\frac {4 x^{3}}{3}-\ln \relax (x )}{x}\)	\(21\)
default	\(-\frac {\ln \relax (x )}{x}-\frac {1}{x}-\frac {4 x^{2}}{3}-\frac {x}{3}\)	\(22\)
risch	\(-\frac {\ln \relax (x )}{x}-\frac {4 x^{3}+x^{2}+3}{3 x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*ln(x)-8*x^3-x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-1-1/3*x^2-4/3*x^3-ln(x))/x

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maxima [A]  time = 0.60, size = 21, normalized size = 0.78 \begin {gather*} -\frac {4}{3} \, x^{2} - \frac {1}{3} \, x - \frac {\log \relax (x)}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*log(x)-8*x^3-x^2)/x^2,x, algorithm="maxima")

[Out]

-4/3*x^2 - 1/3*x - log(x)/x - 1/x

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mupad [B]  time = 0.61, size = 18, normalized size = 0.67 \begin {gather*} -\frac {x}{3}-\frac {\ln \relax (x)+1}{x}-\frac {4\,x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2/3 - log(x) + (8*x^3)/3)/x^2,x)

[Out]

- x/3 - (log(x) + 1)/x - (4*x^2)/3

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sympy [A]  time = 0.10, size = 19, normalized size = 0.70 \begin {gather*} - \frac {4 x^{2}}{3} - \frac {x}{3} - \frac {\log {\relax (x )}}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*ln(x)-8*x**3-x**2)/x**2,x)

[Out]

-4*x**2/3 - x/3 - log(x)/x - 1/x

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