3.24.66 \(\int \frac {2+9 x}{9 x^2} \, dx\)

Optimal. Leaf size=11 \[ 5-\frac {2}{9 x}+\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 43} \begin {gather*} \log (x)-\frac {2}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 9*x)/(9*x^2),x]

[Out]

-2/(9*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {2+9 x}{x^2} \, dx\\ &=\frac {1}{9} \int \left (\frac {2}{x^2}+\frac {9}{x}\right ) \, dx\\ &=-\frac {2}{9 x}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.27 \begin {gather*} \frac {1}{9} \left (-\frac {2}{x}+9 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 9*x)/(9*x^2),x]

[Out]

(-2/x + 9*Log[x])/9

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fricas [A]  time = 0.54, size = 12, normalized size = 1.09 \begin {gather*} \frac {9 \, x \log \relax (x) - 2}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(9*x+2)/x^2,x, algorithm="fricas")

[Out]

1/9*(9*x*log(x) - 2)/x

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giac [A]  time = 0.23, size = 9, normalized size = 0.82 \begin {gather*} -\frac {2}{9 \, x} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(9*x+2)/x^2,x, algorithm="giac")

[Out]

-2/9/x + log(abs(x))

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maple [A]  time = 0.03, size = 9, normalized size = 0.82

method	result	size
default	\(\ln \relax (x )-\frac {2}{9 x}\)	\(9\)
norman	\(\ln \relax (x )-\frac {2}{9 x}\)	\(9\)
risch	\(\ln \relax (x )-\frac {2}{9 x}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(9*x+2)/x^2,x,method=_RETURNVERBOSE)

[Out]

ln(x)-2/9/x

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maxima [A]  time = 0.49, size = 8, normalized size = 0.73 \begin {gather*} -\frac {2}{9 \, x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(9*x+2)/x^2,x, algorithm="maxima")

[Out]

-2/9/x + log(x)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.73 \begin {gather*} \ln \relax (x)-\frac {2}{9\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2/9)/x^2,x)

[Out]

log(x) - 2/(9*x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.64 \begin {gather*} \log {\relax (x )} - \frac {2}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(9*x+2)/x**2,x)

[Out]

log(x) - 2/(9*x)

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