3.191 \(\int \frac{18-2 x-4 x^2}{-6+x+4 x^2+x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0477299, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \log (1-x)-2 \log (x+2)-3 \log (x+3) \]

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]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{- 4 x^{2} - 2 x + 18}{x^{3} + 4 x^{2} + x - 6}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-4*x**2-2*x+18)/(x**3+4*x**2+x-6),x)

[Out]

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

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Mathematica [A]  time = 0.0108794, size = 25, normalized size = 1.32 \[ -2 \left (-\frac{1}{2} \log (1-x)+\log (x+2)+\frac{3}{2} \log (x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 18, normalized size = 1. \[ -2\,\ln \left ( 2+x \right ) +\ln \left ( -1+x \right ) -3\,\ln \left ( 3+x \right ) \]

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)

[Out]

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

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Maxima [A]  time = 1.35688, size = 23, normalized size = 1.21 \[ -3 \, \log \left (x + 3\right ) - 2 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-2*(2*x^2 + x - 9)/(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 = 0.217169, size = 23, normalized size = 1.21 \[ -3 \, \log \left (x + 3\right ) - 2 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-2*(2*x^2 + x - 9)/(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 = 0.138196, size = 17, normalized size = 0.89 \[ \log{\left (x - 1 \right )} - 2 \log{\left (x + 2 \right )} - 3 \log{\left (x + 3 \right )} \]

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/XCAS [A]  time = 0.203251, size = 27, normalized size = 1.42 \[ -3 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) - 2 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-2*(2*x^2 + x - 9)/(x^3 + 4*x^2 + x - 6),x, algorithm="giac")

[Out]

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