3.47.65 \(\int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx\)

Optimal. Leaf size=20 \[ \frac {4 \left (-x-\frac {x^2}{2}\right ) (x+\log (x))}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {14, 2295} \begin {gather*} -2 x^2-4 x-2 x \log (x)-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 6*x - 4*x^2 - 2*x*Log[x])/x,x]

[Out]

-4*x - 2*x^2 - 4*Log[x] - 2*x*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]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (2+3 x+2 x^2\right )}{x}-2 \log (x)\right ) \, dx\\ &=-\left (2 \int \frac {2+3 x+2 x^2}{x} \, dx\right )-2 \int \log (x) \, dx\\ &=2 x-2 x \log (x)-2 \int \left (3+\frac {2}{x}+2 x\right ) \, dx\\ &=-4 x-2 x^2-4 \log (x)-2 x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} -4 x-2 x^2-4 \log (x)-2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 6*x - 4*x^2 - 2*x*Log[x])/x,x]

[Out]

-4*x - 2*x^2 - 4*Log[x] - 2*x*Log[x]

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fricas [A]  time = 0.86, size = 16, normalized size = 0.80 \begin {gather*} -2 \, x^{2} - 2 \, {\left (x + 2\right )} \log \relax (x) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-4*x^2-6*x-4)/x,x, algorithm="fricas")

[Out]

-2*x^2 - 2*(x + 2)*log(x) - 4*x

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giac [A]  time = 0.13, size = 18, normalized size = 0.90 \begin {gather*} -2 \, x^{2} - 2 \, x \log \relax (x) - 4 \, x - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-4*x^2-6*x-4)/x,x, algorithm="giac")

[Out]

-2*x^2 - 2*x*log(x) - 4*x - 4*log(x)

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maple [A]  time = 0.02, size = 19, normalized size = 0.95

method	result	size
default	\(-4 x -2 x \ln \relax (x )-2 x^{2}-4 \ln \relax (x )\)	\(19\)
norman	\(-4 x -2 x \ln \relax (x )-2 x^{2}-4 \ln \relax (x )\)	\(19\)
risch	\(-4 x -2 x \ln \relax (x )-2 x^{2}-4 \ln \relax (x )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)-4*x^2-6*x-4)/x,x,method=_RETURNVERBOSE)

[Out]

-4*x-2*x*ln(x)-2*x^2-4*ln(x)

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maxima [A]  time = 0.37, size = 18, normalized size = 0.90 \begin {gather*} -2 \, x^{2} - 2 \, x \log \relax (x) - 4 \, x - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-4*x^2-6*x-4)/x,x, algorithm="maxima")

[Out]

-2*x^2 - 2*x*log(x) - 4*x - 4*log(x)

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mupad [B]  time = 3.13, size = 9, normalized size = 0.45 \begin {gather*} -2\,\left (x+\ln \relax (x)\right )\,\left (x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + 2*x*log(x) + 4*x^2 + 4)/x,x)

[Out]

-2*(x + log(x))*(x + 2)

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sympy [A]  time = 0.10, size = 20, normalized size = 1.00 \begin {gather*} - 2 x^{2} - 2 x \log {\relax (x )} - 4 x - 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x)-4*x**2-6*x-4)/x,x)

[Out]

-2*x**2 - 2*x*log(x) - 4*x - 4*log(x)

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