3.9.25 \(\int \frac {-432-144 x-864 x^2}{x^3+6 x^4+9 x^5+(6 x^2+18 x^3) \log (x)+9 x \log ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {144}{x \left (1+3 \left (x+\frac {\log (x)}{x}\right )\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6688, 12, 6686} \begin {gather*} \frac {144}{3 x^2+x+3 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-432 - 144*x - 864*x^2)/(x^3 + 6*x^4 + 9*x^5 + (6*x^2 + 18*x^3)*Log[x] + 9*x*Log[x]^2),x]

[Out]

144/(x + 3*x^2 + 3*Log[x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {144 \left (-3-x-6 x^2\right )}{x \left (x+3 x^2+3 \log (x)\right )^2} \, dx\\ &=144 \int \frac {-3-x-6 x^2}{x \left (x+3 x^2+3 \log (x)\right )^2} \, dx\\ &=\frac {144}{x+3 x^2+3 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} \frac {144}{x+3 x^2+3 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-432 - 144*x - 864*x^2)/(x^3 + 6*x^4 + 9*x^5 + (6*x^2 + 18*x^3)*Log[x] + 9*x*Log[x]^2),x]

[Out]

144/(x + 3*x^2 + 3*Log[x])

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fricas [A]  time = 0.99, size = 15, normalized size = 0.79 \begin {gather*} \frac {144}{3 \, x^{2} + x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-864*x^2-144*x-432)/(9*x*log(x)^2+(18*x^3+6*x^2)*log(x)+9*x^5+6*x^4+x^3),x, algorithm="fricas")

[Out]

144/(3*x^2 + x + 3*log(x))

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giac [A]  time = 0.26, size = 15, normalized size = 0.79 \begin {gather*} \frac {144}{3 \, x^{2} + x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-864*x^2-144*x-432)/(9*x*log(x)^2+(18*x^3+6*x^2)*log(x)+9*x^5+6*x^4+x^3),x, algorithm="giac")

[Out]

144/(3*x^2 + x + 3*log(x))

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maple [A]  time = 0.04, size = 16, normalized size = 0.84

method	result	size
norman	\(\frac {144}{3 x^{2}+3 \ln \relax (x )+x}\)	\(16\)
risch	\(\frac {144}{3 x^{2}+3 \ln \relax (x )+x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-864*x^2-144*x-432)/(9*x*ln(x)^2+(18*x^3+6*x^2)*ln(x)+9*x^5+6*x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

144/(3*x^2+3*ln(x)+x)

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maxima [A]  time = 0.45, size = 15, normalized size = 0.79 \begin {gather*} \frac {144}{3 \, x^{2} + x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-864*x^2-144*x-432)/(9*x*log(x)^2+(18*x^3+6*x^2)*log(x)+9*x^5+6*x^4+x^3),x, algorithm="maxima")

[Out]

144/(3*x^2 + x + 3*log(x))

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mupad [B]  time = 0.81, size = 15, normalized size = 0.79 \begin {gather*} \frac {144}{x+3\,\ln \relax (x)+3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(144*x + 864*x^2 + 432)/(log(x)*(6*x^2 + 18*x^3) + 9*x*log(x)^2 + x^3 + 6*x^4 + 9*x^5),x)

[Out]

144/(x + 3*log(x) + 3*x^2)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.63 \begin {gather*} \frac {144}{3 x^{2} + x + 3 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-864*x**2-144*x-432)/(9*x*ln(x)**2+(18*x**3+6*x**2)*ln(x)+9*x**5+6*x**4+x**3),x)

[Out]

144/(3*x**2 + x + 3*log(x))

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