3.41.28 \(\int \frac {31-30 x-30 x^2-30 \log (x)}{30 x^2} \, dx\)

Optimal. Leaf size=21 \[ -\frac {1}{30 x}-x-\log (x)+\frac {\log (x)}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14, 2304} \begin {gather*} -x-\frac {1}{30 x}-\log (x)+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(31 - 30*x - 30*x^2 - 30*Log[x])/(30*x^2),x]

[Out]

-1/30*1/x - x - Log[x] + 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}{30} \int \frac {31-30 x-30 x^2-30 \log (x)}{x^2} \, dx\\ &=\frac {1}{30} \int \left (\frac {31-30 x-30 x^2}{x^2}-\frac {30 \log (x)}{x^2}\right ) \, dx\\ &=\frac {1}{30} \int \frac {31-30 x-30 x^2}{x^2} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+\frac {1}{30} \int \left (-30+\frac {31}{x^2}-\frac {30}{x}\right ) \, dx\\ &=-\frac {1}{30 x}-x-\log (x)+\frac {\log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{30 x}-x-\log (x)+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(31 - 30*x - 30*x^2 - 30*Log[x])/(30*x^2),x]

[Out]

-1/30*1/x - x - Log[x] + Log[x]/x

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fricas [A]  time = 0.88, size = 19, normalized size = 0.90 \begin {gather*} -\frac {30 \, x^{2} + 30 \, {\left (x - 1\right )} \log \relax (x) + 1}{30 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-30*log(x)-30*x^2-30*x+31)/x^2,x, algorithm="fricas")

[Out]

-1/30*(30*x^2 + 30*(x - 1)*log(x) + 1)/x

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giac [A]  time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} -x + \frac {\log \relax (x)}{x} - \frac {1}{30 \, x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-30*log(x)-30*x^2-30*x+31)/x^2,x, algorithm="giac")

[Out]

-x + log(x)/x - 1/30/x - log(x)

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maple [A]  time = 0.02, size = 19, normalized size = 0.90

method	result	size
norman	\(\frac {-\frac {1}{30}-x \ln \relax (x )-x^{2}+\ln \relax (x )}{x}\)	\(19\)
default	\(\frac {\ln \relax (x )}{x}-x -\frac {1}{30 x}-\ln \relax (x )\)	\(20\)
risch	\(\frac {\ln \relax (x )}{x}-\frac {30 x \ln \relax (x )+30 x^{2}+1}{30 x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/30*(-30*ln(x)-30*x^2-30*x+31)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-1/30-x*ln(x)-x^2+ln(x))/x

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maxima [A]  time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} -x + \frac {\log \relax (x)}{x} - \frac {1}{30 \, x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-30*log(x)-30*x^2-30*x+31)/x^2,x, algorithm="maxima")

[Out]

-x + log(x)/x - 1/30/x - log(x)

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mupad [B]  time = 3.18, size = 16, normalized size = 0.76 \begin {gather*} \frac {\ln \relax (x)-\frac {1}{30}}{x}-\ln \relax (x)-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + log(x) + x^2 - 31/30)/x^2,x)

[Out]

(log(x) - 1/30)/x - log(x) - x

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sympy [A]  time = 0.11, size = 14, normalized size = 0.67 \begin {gather*} - x - \log {\relax (x )} + \frac {\log {\relax (x )}}{x} - \frac {1}{30 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/30*(-30*ln(x)-30*x**2-30*x+31)/x**2,x)

[Out]

-x - log(x) + log(x)/x - 1/(30*x)

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