3.3.55 \(\int \frac {1}{x (2+3 x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {\log (x)}{2}-\frac {1}{2} \log (3 x+2) \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {36, 29, 31} \begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 + 3*x)),x]

[Out]

Log[x]/2 - Log[2 + 3*x]/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{x (2+3 x)} \, dx &=\frac {1}{2} \int \frac {1}{x} \, dx-\frac {3}{2} \int \frac {1}{2+3 x} \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(2 + 3*x)),x]

[Out]

Log[x]/2 - Log[2 + 3*x]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (2+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*(2 + 3*x)),x]

[Out]

IntegrateAlgebraic[1/(x*(2 + 3*x)), x]

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fricas [A]  time = 0.80, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*x),x, algorithm="fricas")

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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giac [A]  time = 1.15, size = 15, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} \frac {\ln \relax (x )}{2}-\frac {\ln \left (3 x +2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(2+3*x),x)

[Out]

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

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maxima [A]  time = 1.34, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*x),x, algorithm="maxima")

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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mupad [B]  time = 0.17, size = 10, normalized size = 0.59 \begin {gather*} -\frac {\ln \left (\frac {2}{x}+3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(3*x + 2)),x)

[Out]

-log(2/x + 3)/2

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sympy [A]  time = 0.12, size = 12, normalized size = 0.71 \begin {gather*} \frac {\log {\relax (x )}}{2} - \frac {\log {\left (x + \frac {2}{3} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+3*x),x)

[Out]

log(x)/2 - log(x + 2/3)/2

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