3.96 \(\int \frac{-1+x+x^2}{-6 x+x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]

[Out]

Log[2 - x]/2 + Log[x]/6 + Log[3 + x]/3

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Rubi [A]  time = 0.0468113, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x + x^2)/(-6*x + x^2 + x^3),x]

[Out]

Log[2 - x]/2 + Log[x]/6 + Log[3 + x]/3

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Rubi in Sympy [A]  time = 4.90234, size = 17, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{6} + \frac{\log{\left (- x + 2 \right )}}{2} + \frac{\log{\left (x + 3 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+x-1)/(x**3+x**2-6*x),x)

[Out]

log(x)/6 + log(-x + 2)/2 + log(x + 3)/3

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Mathematica [A]  time = 0.00845203, size = 25, normalized size = 1. \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x + x^2)/(-6*x + x^2 + x^3),x]

[Out]

Log[2 - x]/2 + Log[x]/6 + Log[3 + x]/3

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Maple [A]  time = 0.01, size = 18, normalized size = 0.7 \[{\frac{\ln \left ( x \right ) }{6}}+{\frac{\ln \left ( -2+x \right ) }{2}}+{\frac{\ln \left ( 3+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+x-1)/(x^3+x^2-6*x),x)

[Out]

1/6*ln(x)+1/2*ln(-2+x)+1/3*ln(3+x)

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Maxima [A]  time = 1.34207, size = 23, normalized size = 0.92 \[ \frac{1}{3} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x - 2\right ) + \frac{1}{6} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x - 1)/(x^3 + x^2 - 6*x),x, algorithm="maxima")

[Out]

1/3*log(x + 3) + 1/2*log(x - 2) + 1/6*log(x)

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Fricas [A]  time = 0.201388, size = 23, normalized size = 0.92 \[ \frac{1}{3} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x - 2\right ) + \frac{1}{6} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x - 1)/(x^3 + x^2 - 6*x),x, algorithm="fricas")

[Out]

1/3*log(x + 3) + 1/2*log(x - 2) + 1/6*log(x)

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Sympy [A]  time = 0.140896, size = 17, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{6} + \frac{\log{\left (x - 2 \right )}}{2} + \frac{\log{\left (x + 3 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+x-1)/(x**3+x**2-6*x),x)

[Out]

log(x)/6 + log(x - 2)/2 + log(x + 3)/3

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GIAC/XCAS [A]  time = 0.200219, size = 27, normalized size = 1.08 \[ \frac{1}{3} \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x - 1)/(x^3 + x^2 - 6*x),x, algorithm="giac")

[Out]

1/3*ln(abs(x + 3)) + 1/2*ln(abs(x - 2)) + 1/6*ln(abs(x))