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

Optimal. Leaf size=25 \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]

[Out]

Log[1 - 2*x]/10 + Log[x]/2 - Log[2 + x]/10

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Rubi [A]  time = 0.0551782, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 - 2*x]/10 + Log[x]/2 - Log[2 + x]/10

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Rubi in Sympy [A]  time = 6.00211, size = 19, normalized size = 0.76 \[ \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (- 2 x + 1 \right )}}{10} - \frac{\log{\left (x + 2 \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/2 + log(-2*x + 1)/10 - log(x + 2)/10

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Mathematica [A]  time = 0.0097486, size = 25, normalized size = 1. \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 - 2*x]/10 + Log[x]/2 - Log[2 + x]/10

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Maple [A]  time = 0.01, size = 20, normalized size = 0.8 \[ -{\frac{\ln \left ( 2+x \right ) }{10}}+{\frac{\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( 2\,x-1 \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*ln(2+x)+1/2*ln(x)+1/10*ln(2*x-1)

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Maxima [A]  time = 1.33618, size = 26, normalized size = 1.04 \[ \frac{1}{10} \, \log \left (2 \, x - 1\right ) - \frac{1}{10} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*log(2*x - 1) - 1/10*log(x + 2) + 1/2*log(x)

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Fricas [A]  time = 0.202873, size = 26, normalized size = 1.04 \[ \frac{1}{10} \, \log \left (2 \, x - 1\right ) - \frac{1}{10} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*log(2*x - 1) - 1/10*log(x + 2) + 1/2*log(x)

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Sympy [A]  time = 0.1562, size = 19, normalized size = 0.76 \[ \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (x - \frac{1}{2} \right )}}{10} - \frac{\log{\left (x + 2 \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/2 + log(x - 1/2)/10 - log(x + 2)/10

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GIAC/XCAS [A]  time = 0.208091, size = 30, normalized size = 1.2 \[ \frac{1}{10} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) - \frac{1}{10} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*ln(abs(2*x - 1)) - 1/10*ln(abs(x + 2)) + 1/2*ln(abs(x))