3.32.89 \(\int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\frac {4}{x (2+x) \left (2+x+x^2\right )^2 (-3+\log (5))}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2074, 628} \begin {gather*} -2 \log \left (x^2+x+2\right )-\log (x)-\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 10*x - 14*x^2 - 6*x^3)/(4*x + 4*x^2 + 3*x^3 + x^4),x]

[Out]

-Log[x] - Log[2 + x] - 2*Log[2 + x + x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.96 \begin {gather*} -2 \left (\frac {\log (x)}{2}+\frac {1}{2} \log (2+x)+\log \left (2+x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 10*x - 14*x^2 - 6*x^3)/(4*x + 4*x^2 + 3*x^3 + x^4),x]

[Out]

-2*(Log[x]/2 + Log[2 + x]/2 + Log[2 + x + x^2])

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fricas [A]  time = 0.62, size = 20, normalized size = 0.80 \begin {gather*} -\log \left (x^{2} + 2 \, x\right ) - 2 \, \log \left (x^{2} + x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3-14*x^2-10*x-4)/(x^4+3*x^3+4*x^2+4*x),x, algorithm="fricas")

[Out]

-log(x^2 + 2*x) - 2*log(x^2 + x + 2)

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giac [A]  time = 0.26, size = 22, normalized size = 0.88 \begin {gather*} -2 \, \log \left (x^{2} + x + 2\right ) - \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3-14*x^2-10*x-4)/(x^4+3*x^3+4*x^2+4*x),x, algorithm="giac")

[Out]

-2*log(x^2 + x + 2) - log(abs(x + 2)) - log(abs(x))

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maple [A]  time = 0.03, size = 21, normalized size = 0.84

method	result	size
default	\(-\ln \left (2+x \right )-\ln \relax (x )-2 \ln \left (x^{2}+x +2\right )\)	\(21\)
norman	\(-\ln \left (2+x \right )-\ln \relax (x )-2 \ln \left (x^{2}+x +2\right )\)	\(21\)
risch	\(-2 \ln \left (x^{2}+x +2\right )-\ln \left (x^{2}+2 x \right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^3-14*x^2-10*x-4)/(x^4+3*x^3+4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.81, size = 20, normalized size = 0.80 \begin {gather*} -2 \, \log \left (x^{2} + x + 2\right ) - \log \left (x + 2\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3-14*x^2-10*x-4)/(x^4+3*x^3+4*x^2+4*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 18, normalized size = 0.72 \begin {gather*} -\ln \left (x\,\left (x+2\right )\right )-2\,\ln \left (x^2+x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x + 14*x^2 + 6*x^3 + 4)/(4*x + 4*x^2 + 3*x^3 + x^4),x)

[Out]

- log(x*(x + 2)) - 2*log(x + x^2 + 2)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.76 \begin {gather*} - \log {\left (x^{2} + 2 x \right )} - 2 \log {\left (x^{2} + x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**3-14*x**2-10*x-4)/(x**4+3*x**3+4*x**2+4*x),x)

[Out]

-log(x**2 + 2*x) - 2*log(x**2 + x + 2)

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