3.39.40 \(\int \frac {-8+9 x-49 x^2}{9 x^2} \, dx\)

Optimal. Leaf size=25 \[ \log \left (e^{-6+\frac {40}{9} \left (\frac {1}{5 x}-x\right )-x} x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 14} \begin {gather*} -\frac {49 x}{9}+\frac {8}{9 x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 + 9*x - 49*x^2)/(9*x^2),x]

[Out]

8/(9*x) - (49*x)/9 + 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 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}{9} \int \frac {-8+9 x-49 x^2}{x^2} \, dx\\ &=\frac {1}{9} \int \left (-49-\frac {8}{x^2}+\frac {9}{x}\right ) \, dx\\ &=\frac {8}{9 x}-\frac {49 x}{9}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.60 \begin {gather*} \frac {8}{9 x}-\frac {49 x}{9}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 9*x - 49*x^2)/(9*x^2),x]

[Out]

8/(9*x) - (49*x)/9 + Log[x]

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fricas [A]  time = 0.82, size = 17, normalized size = 0.68 \begin {gather*} -\frac {49 \, x^{2} - 9 \, x \log \relax (x) - 8}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-49*x^2+9*x-8)/x^2,x, algorithm="fricas")

[Out]

-1/9*(49*x^2 - 9*x*log(x) - 8)/x

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giac [A]  time = 0.17, size = 12, normalized size = 0.48 \begin {gather*} -\frac {49}{9} \, x + \frac {8}{9 \, x} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-49*x^2+9*x-8)/x^2,x, algorithm="giac")

[Out]

-49/9*x + 8/9/x + log(abs(x))

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maple [A]  time = 0.02, size = 12, normalized size = 0.48

method	result	size
default	\(-\frac {49 x}{9}+\ln \relax (x )+\frac {8}{9 x}\)	\(12\)
risch	\(-\frac {49 x}{9}+\ln \relax (x )+\frac {8}{9 x}\)	\(12\)
norman	\(\frac {\frac {8}{9}-\frac {49 x^{2}}{9}}{x}+\ln \relax (x )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(-49*x^2+9*x-8)/x^2,x,method=_RETURNVERBOSE)

[Out]

-49/9*x+ln(x)+8/9/x

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maxima [A]  time = 0.34, size = 11, normalized size = 0.44 \begin {gather*} -\frac {49}{9} \, x + \frac {8}{9 \, x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-49*x^2+9*x-8)/x^2,x, algorithm="maxima")

[Out]

-49/9*x + 8/9/x + log(x)

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mupad [B]  time = 0.03, size = 11, normalized size = 0.44 \begin {gather*} \ln \relax (x)-\frac {49\,x}{9}+\frac {8}{9\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((49*x^2)/9 - x + 8/9)/x^2,x)

[Out]

log(x) - (49*x)/9 + 8/(9*x)

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sympy [A]  time = 0.07, size = 12, normalized size = 0.48 \begin {gather*} - \frac {49 x}{9} + \log {\relax (x )} + \frac {8}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-49*x**2+9*x-8)/x**2,x)

[Out]

-49*x/9 + log(x) + 8/(9*x)

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