\(\int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx\) [4949]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=x-4 \left (-6+7 x+x^2+x^4-\log (x)\right ) \]

[Out]

-27*x-4*x^4-4*x^2+24+4*ln(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {14} \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=-4 x^4-4 x^2-27 x+4 \log (x) \]

[In]

Int[(4 - 27*x - 8*x^2 - 16*x^4)/x,x]

[Out]

-27*x - 4*x^2 - 4*x^4 + 4*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{align*} \text {integral}& = \int \left (-27+\frac {4}{x}-8 x-16 x^3\right ) \, dx \\ & = -27 x-4 x^2-4 x^4+4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=-27 x-4 x^2-4 x^4+4 \log (x) \]

[In]

Integrate[(4 - 27*x - 8*x^2 - 16*x^4)/x,x]

[Out]

-27*x - 4*x^2 - 4*x^4 + 4*Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

method	result	size
default	\(-4 x^{4}-4 x^{2}-27 x +4 \ln \left (x \right )\)	\(19\)
norman	\(-4 x^{4}-4 x^{2}-27 x +4 \ln \left (x \right )\)	\(19\)
risch	\(-4 x^{4}-4 x^{2}-27 x +4 \ln \left (x \right )\)	\(19\)
parallelrisch	\(-4 x^{4}-4 x^{2}-27 x +4 \ln \left (x \right )\)	\(19\)

[In]

int((-16*x^4-8*x^2-27*x+4)/x,x,method=_RETURNVERBOSE)

[Out]

-4*x^4-4*x^2-27*x+4*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=-4 \, x^{4} - 4 \, x^{2} - 27 \, x + 4 \, \log \left (x\right ) \]

[In]

integrate((-16*x^4-8*x^2-27*x+4)/x,x, algorithm="fricas")

[Out]

-4*x^4 - 4*x^2 - 27*x + 4*log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=- 4 x^{4} - 4 x^{2} - 27 x + 4 \log {\left (x \right )} \]

[In]

integrate((-16*x**4-8*x**2-27*x+4)/x,x)

[Out]

-4*x**4 - 4*x**2 - 27*x + 4*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=-4 \, x^{4} - 4 \, x^{2} - 27 \, x + 4 \, \log \left (x\right ) \]

[In]

integrate((-16*x^4-8*x^2-27*x+4)/x,x, algorithm="maxima")

[Out]

-4*x^4 - 4*x^2 - 27*x + 4*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=-4 \, x^{4} - 4 \, x^{2} - 27 \, x + 4 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-16*x^4-8*x^2-27*x+4)/x,x, algorithm="giac")

[Out]

-4*x^4 - 4*x^2 - 27*x + 4*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {4-27 x-8 x^2-16 x^4}{x} \, dx=4\,\ln \left (x\right )-27\,x-4\,x^2-4\,x^4 \]

[In]

int(-(27*x + 8*x^2 + 16*x^4 - 4)/x,x)

[Out]

4*log(x) - 27*x - 4*x^2 - 4*x^4