3.4.18 \(\int \frac {18-2 x-4 x^2}{-6+x+4 x^2+x^3} \, dx\) [318]

Optimal. Leaf size=19 \[ \log (1-x)-2 \log (2+x)-3 \log (3+x) \]

[Out]

ln(1-x)-2*ln(2+x)-3*ln(3+x)

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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2099} \begin {gather*} \log (1-x)-2 \log (x+2)-3 \log (x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18 - 2*x - 4*x^2)/(-6 + x + 4*x^2 + x^3),x]

[Out]

Log[1 - x] - 2*Log[2 + x] - 3*Log[3 + x]

Rule 2099

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 {align*} \int \frac {18-2 x-4 x^2}{-6+x+4 x^2+x^3} \, dx &=\int \left (\frac {1}{-1+x}-\frac {2}{2+x}-\frac {3}{3+x}\right ) \, dx\\ &=\log (1-x)-2 \log (2+x)-3 \log (3+x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.32 \begin {gather*} -2 \left (-\frac {1}{2} \log (1-x)+\log (2+x)+\frac {3}{2} \log (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 - 2*x - 4*x^2)/(-6 + x + 4*x^2 + x^3),x]

[Out]

-2*(-1/2*Log[1 - x] + Log[2 + x] + (3*Log[3 + x])/2)

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Maple [A]
time = 0.02, size = 18, normalized size = 0.95

method	result	size
default	\(-2 \ln \left (x +2\right )+\ln \left (-1+x \right )-3 \ln \left (3+x \right )\)	\(18\)
norman	\(-2 \ln \left (x +2\right )+\ln \left (-1+x \right )-3 \ln \left (3+x \right )\)	\(18\)
risch	\(-2 \ln \left (x +2\right )+\ln \left (-1+x \right )-3 \ln \left (3+x \right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-2*x+18)/(x^3+4*x^2+x-6),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x+2)+ln(-1+x)-3*ln(3+x)

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Maxima [A]
time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} -3 \, \log \left (x + 3\right ) - 2 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-2*x+18)/(x^3+4*x^2+x-6),x, algorithm="maxima")

[Out]

-3*log(x + 3) - 2*log(x + 2) + log(x - 1)

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Fricas [A]
time = 0.43, size = 17, normalized size = 0.89 \begin {gather*} -3 \, \log \left (x + 3\right ) - 2 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-2*x+18)/(x^3+4*x^2+x-6),x, algorithm="fricas")

[Out]

-3*log(x + 3) - 2*log(x + 2) + log(x - 1)

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Sympy [A]
time = 0.04, size = 17, normalized size = 0.89 \begin {gather*} \log {\left (x - 1 \right )} - 2 \log {\left (x + 2 \right )} - 3 \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-2*x+18)/(x**3+4*x**2+x-6),x)

[Out]

log(x - 1) - 2*log(x + 2) - 3*log(x + 3)

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Giac [A]
time = 4.01, size = 20, normalized size = 1.05 \begin {gather*} -3 \, \log \left ({\left | x + 3 \right |}\right ) - 2 \, \log \left ({\left | x + 2 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-2*x+18)/(x^3+4*x^2+x-6),x, algorithm="giac")

[Out]

-3*log(abs(x + 3)) - 2*log(abs(x + 2)) + log(abs(x - 1))

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Mupad [B]
time = 2.12, size = 17, normalized size = 0.89 \begin {gather*} \ln \left (x-1\right )-2\,\ln \left (x+2\right )-3\,\ln \left (x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 4*x^2 - 18)/(x + 4*x^2 + x^3 - 6),x)

[Out]

log(x - 1) - 2*log(x + 2) - 3*log(x + 3)

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