3.109 \(\int \frac{-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0489408, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ 2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + 5*x + 2*x^2)/(-2*x + x^2 + x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.9607, size = 17, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{2} + 2 \log{\left (- x + 1 \right )} - \frac{\log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+5*x-1)/(x**3+x**2-2*x),x)

[Out]

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

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Mathematica [A]  time = 0.00915951, size = 23, normalized size = 1. \[ 2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + 5*x + 2*x^2)/(-2*x + x^2 + x^3),x]

[Out]

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

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Maple [A]  time = 0.012, size = 18, normalized size = 0.8 \[ -{\frac{\ln \left ( 2+x \right ) }{2}}+{\frac{\ln \left ( x \right ) }{2}}+2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+5*x-1)/(x^3+x^2-2*x),x)

[Out]

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

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Maxima [A]  time = 1.44809, size = 23, normalized size = 1. \[ -\frac{1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5*x - 1)/(x^3 + x^2 - 2*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.20072, size = 23, normalized size = 1. \[ -\frac{1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5*x - 1)/(x^3 + x^2 - 2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.141812, size = 17, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{2} + 2 \log{\left (x - 1 \right )} - \frac{\log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+5*x-1)/(x**3+x**2-2*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218559, size = 27, normalized size = 1.17 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 5*x - 1)/(x^3 + x^2 - 2*x),x, algorithm="giac")

[Out]

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