3.91.89 \(\int \frac {-1+8 x-8 x^2}{4 x} \, dx\)

Optimal. Leaf size=17 \[ -3+e+2 x-x^2-\frac {\log (x)}{4} \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.88, 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*} -x^2+2 x-\frac {\log (x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 8*x - 8*x^2)/(4*x),x]

[Out]

2*x - x^2 - Log[x]/4

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}{4} \int \frac {-1+8 x-8 x^2}{x} \, dx\\ &=\frac {1}{4} \int \left (8-\frac {1}{x}-8 x\right ) \, dx\\ &=2 x-x^2-\frac {\log (x)}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.88 \begin {gather*} 2 x-x^2-\frac {\log (x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 8*x - 8*x^2)/(4*x),x]

[Out]

2*x - x^2 - Log[x]/4

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fricas [A]  time = 0.55, size = 13, normalized size = 0.76 \begin {gather*} -x^{2} + 2 \, x - \frac {1}{4} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-8*x^2+8*x-1)/x,x, algorithm="fricas")

[Out]

-x^2 + 2*x - 1/4*log(x)

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giac [A]  time = 0.15, size = 14, normalized size = 0.82 \begin {gather*} -x^{2} + 2 \, x - \frac {1}{4} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-8*x^2+8*x-1)/x,x, algorithm="giac")

[Out]

-x^2 + 2*x - 1/4*log(abs(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(-x^{2}+2 x -\frac {\ln \relax (x )}{4}\)	\(14\)
norman	\(-x^{2}+2 x -\frac {\ln \relax (x )}{4}\)	\(14\)
risch	\(-x^{2}+2 x -\frac {\ln \relax (x )}{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-8*x^2+8*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x-1/4*ln(x)

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maxima [A]  time = 0.35, size = 13, normalized size = 0.76 \begin {gather*} -x^{2} + 2 \, x - \frac {1}{4} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-8*x^2+8*x-1)/x,x, algorithm="maxima")

[Out]

-x^2 + 2*x - 1/4*log(x)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \begin {gather*} 2\,x-\frac {\ln \relax (x)}{4}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - 2*x + 1/4)/x,x)

[Out]

2*x - log(x)/4 - x^2

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sympy [A]  time = 0.07, size = 10, normalized size = 0.59 \begin {gather*} - x^{2} + 2 x - \frac {\log {\relax (x )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-8*x**2+8*x-1)/x,x)

[Out]

-x**2 + 2*x - log(x)/4

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