3.9.3 \(\int \frac {81-12920 x+16038 x^2}{81 x} \, dx\)

Optimal. Leaf size=19 \[ -x^2+4 \left (-\frac {323}{81}+5 x\right )^2+\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.68, 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*} 99 x^2-\frac {12920 x}{81}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(81 - 12920*x + 16038*x^2)/(81*x),x]

[Out]

(-12920*x)/81 + 99*x^2 + 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}{81} \int \frac {81-12920 x+16038 x^2}{x} \, dx\\ &=\frac {1}{81} \int \left (-12920+\frac {81}{x}+16038 x\right ) \, dx\\ &=-\frac {12920 x}{81}+99 x^2+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.68 \begin {gather*} -\frac {12920 x}{81}+99 x^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81 - 12920*x + 16038*x^2)/(81*x),x]

[Out]

(-12920*x)/81 + 99*x^2 + Log[x]

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fricas [A]  time = 0.80, size = 11, normalized size = 0.58 \begin {gather*} 99 \, x^{2} - \frac {12920}{81} \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(16038*x^2-12920*x+81)/x,x, algorithm="fricas")

[Out]

99*x^2 - 12920/81*x + log(x)

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giac [A]  time = 0.43, size = 12, normalized size = 0.63 \begin {gather*} 99 \, x^{2} - \frac {12920}{81} \, x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(16038*x^2-12920*x+81)/x,x, algorithm="giac")

[Out]

99*x^2 - 12920/81*x + log(abs(x))

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maple [A]  time = 0.02, size = 12, normalized size = 0.63

method	result	size
default	\(99 x^{2}-\frac {12920 x}{81}+\ln \relax (x )\)	\(12\)
norman	\(99 x^{2}-\frac {12920 x}{81}+\ln \relax (x )\)	\(12\)
risch	\(99 x^{2}-\frac {12920 x}{81}+\ln \relax (x )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*(16038*x^2-12920*x+81)/x,x,method=_RETURNVERBOSE)

[Out]

99*x^2-12920/81*x+ln(x)

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maxima [A]  time = 0.55, size = 11, normalized size = 0.58 \begin {gather*} 99 \, x^{2} - \frac {12920}{81} \, x + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(16038*x^2-12920*x+81)/x,x, algorithm="maxima")

[Out]

99*x^2 - 12920/81*x + log(x)

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mupad [B]  time = 0.02, size = 11, normalized size = 0.58 \begin {gather*} \ln \relax (x)-\frac {12920\,x}{81}+99\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((198*x^2 - (12920*x)/81 + 1)/x,x)

[Out]

log(x) - (12920*x)/81 + 99*x^2

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sympy [A]  time = 0.07, size = 12, normalized size = 0.63 \begin {gather*} 99 x^{2} - \frac {12920 x}{81} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(16038*x**2-12920*x+81)/x,x)

[Out]

99*x**2 - 12920*x/81 + log(x)

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