3.28.15 \(\int \frac {-2+2 x+4 x^2}{x} \, dx\)

Optimal. Leaf size=16 \[ 2+2 x+2 x^2-\log \left (x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} 2 x^2+2 x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 2*x + 4*x^2)/x,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} 2 x+2 x^2-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 2*x + 4*x^2)/x,x]

[Out]

2*x + 2*x^2 - 2*Log[x]

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fricas [A]  time = 0.64, size = 13, normalized size = 0.81 \begin {gather*} 2 \, x^{2} + 2 \, x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2 + 2*x - 2*log(x)

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giac [A]  time = 0.24, size = 14, normalized size = 0.88 \begin {gather*} 2 \, x^{2} + 2 \, x - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2 + 2*x - 2*log(abs(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.88

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2-2*ln(x)+2*x

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maxima [A]  time = 0.42, size = 13, normalized size = 0.81 \begin {gather*} 2 \, x^{2} + 2 \, x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2 + 2*x - 2*log(x)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.81 \begin {gather*} 2\,x-2\,\ln \relax (x)+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 4*x^2 - 2)/x,x)

[Out]

2*x - 2*log(x) + 2*x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.75 \begin {gather*} 2 x^{2} + 2 x - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**2 + 2*x - 2*log(x)

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