3.54.9 \(\int \frac {-72 x-36 x^2}{1+2 x+2 x^2+x^3} \, dx\)

Optimal. Leaf size=16 \[ 4-36 \log \left (x+\frac {x}{x+x^2}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1593, 2075, 628} \begin {gather*} 36 \log (x+1)-36 \log \left (x^2+x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-72*x - 36*x^2)/(1 + 2*x + 2*x^2 + x^3),x]

[Out]

36*Log[1 + x] - 36*Log[1 + 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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2075

Int[(P_)^(p_)*(Qm_), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProdu
ctQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-72-36 x) x}{1+2 x+2 x^2+x^3} \, dx\\ &=\int \left (\frac {36}{1+x}-\frac {36 (1+2 x)}{1+x+x^2}\right ) \, dx\\ &=36 \log (1+x)-36 \int \frac {1+2 x}{1+x+x^2} \, dx\\ &=36 \log (1+x)-36 \log \left (1+x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -36 \left (-\log (1+x)+\log \left (1+x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72*x - 36*x^2)/(1 + 2*x + 2*x^2 + x^3),x]

[Out]

-36*(-Log[1 + x] + Log[1 + x + x^2])

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fricas [A]  time = 0.87, size = 16, normalized size = 1.00 \begin {gather*} -36 \, \log \left (x^{2} + x + 1\right ) + 36 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^2-72*x)/(x^3+2*x^2+2*x+1),x, algorithm="fricas")

[Out]

-36*log(x^2 + x + 1) + 36*log(x + 1)

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giac [A]  time = 0.11, size = 17, normalized size = 1.06 \begin {gather*} -36 \, \log \left (x^{2} + x + 1\right ) + 36 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^2-72*x)/(x^3+2*x^2+2*x+1),x, algorithm="giac")

[Out]

-36*log(x^2 + x + 1) + 36*log(abs(x + 1))

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maple [A]  time = 0.02, size = 17, normalized size = 1.06

method	result	size
default	\(-36 \ln \left (x^{2}+x +1\right )+36 \ln \left (x +1\right )\)	\(17\)
norman	\(-36 \ln \left (x^{2}+x +1\right )+36 \ln \left (x +1\right )\)	\(17\)
risch	\(-36 \ln \left (x^{2}+x +1\right )+36 \ln \left (x +1\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-36*x^2-72*x)/(x^3+2*x^2+2*x+1),x,method=_RETURNVERBOSE)

[Out]

-36*ln(x^2+x+1)+36*ln(x+1)

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maxima [A]  time = 0.42, size = 16, normalized size = 1.00 \begin {gather*} -36 \, \log \left (x^{2} + x + 1\right ) + 36 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^2-72*x)/(x^3+2*x^2+2*x+1),x, algorithm="maxima")

[Out]

-36*log(x^2 + x + 1) + 36*log(x + 1)

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mupad [B]  time = 0.16, size = 16, normalized size = 1.00 \begin {gather*} 36\,\ln \left (x+1\right )-36\,\ln \left (x^2+x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(72*x + 36*x^2)/(2*x + 2*x^2 + x^3 + 1),x)

[Out]

36*log(x + 1) - 36*log(x + x^2 + 1)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.94 \begin {gather*} 36 \log {\left (x + 1 \right )} - 36 \log {\left (x^{2} + x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x**2-72*x)/(x**3+2*x**2+2*x+1),x)

[Out]

36*log(x + 1) - 36*log(x**2 + x + 1)

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