\(\int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx\) [4665]

Optimal result

Integrand size = 19, antiderivative size = 20 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=\frac {4 \left (-x-\frac {x^2}{2}\right ) (x+\log (x))}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {14, 2332} \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-2 x^2-4 x-2 x \log (x)-4 \log (x) \]

[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 2332

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-4 x-2 x^2-4 \log (x)-2 x \log (x) \]

[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]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
default	\(-4 x -2 x \ln \left (x \right )-2 x^{2}-4 \ln \left (x \right )\)	\(19\)
norman	\(-4 x -2 x \ln \left (x \right )-2 x^{2}-4 \ln \left (x \right )\)	\(19\)
risch	\(-4 x -2 x \ln \left (x \right )-2 x^{2}-4 \ln \left (x \right )\)	\(19\)
parallelrisch	\(-4 x -2 x \ln \left (x \right )-2 x^{2}-4 \ln \left (x \right )\)	\(19\)
parts	\(-4 x -2 x \ln \left (x \right )-2 x^{2}-4 \ln \left (x \right )\)	\(19\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-2 \, x^{2} - 2 \, {\left (x + 2\right )} \log \left (x\right ) - 4 \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=- 2 x^{2} - 2 x \log {\left (x \right )} - 4 x - 4 \log {\left (x \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-2 \, x^{2} - 2 \, x \log \left (x\right ) - 4 \, x - 4 \, \log \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-2 \, x^{2} - 2 \, x \log \left (x\right ) - 4 \, x - 4 \, \log \left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 9.94 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.45 \[ \int \frac {-4-6 x-4 x^2-2 x \log (x)}{x} \, dx=-2\,\left (x+\ln \left (x\right )\right )\,\left (x+2\right ) \]

[In]

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

[Out]

-2*(x + log(x))*(x + 2)