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

Optimal. Leaf size=23 \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]

[Out]

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

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Rubi [A]  time = 0.0552012, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.4245, size = 17, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{2} - \log{\left (- x + 1 \right )} + \frac{3 \log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00895568, size = 23, normalized size = 1. \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 18, normalized size = 0.8 \[{\frac{3\,\ln \left ( 2+x \right ) }{2}}-\ln \left ( -1+x \right ) +{\frac{\ln \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.806418, size = 23, normalized size = 1. \[ \frac{3}{2} \, \log \left (x + 2\right ) - \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253737, size = 23, normalized size = 1. \[ \frac{3}{2} \, \log \left (x + 2\right ) - \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.302288, size = 17, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{2} - \log{\left (x - 1 \right )} + \frac{3 \log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261631, size = 27, normalized size = 1.17 \[ \frac{3}{2} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) -{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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