3.186 \(\int \frac{-3+5 x+6 x^2}{-3 x+2 x^2+x^3} \, dx\)

Optimal. Leaf size=17 \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

[Out]

2*Log[1 - x] + Log[x] + 3*Log[3 + x]

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Rubi [A]  time = 0.0486953, antiderivative size = 17, 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 \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[(-3 + 5*x + 6*x^2)/(-3*x + 2*x^2 + x^3),x]

[Out]

2*Log[1 - x] + Log[x] + 3*Log[3 + x]

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Rubi in Sympy [A]  time = 5.72484, size = 15, normalized size = 0.88 \[ \log{\left (x \right )} + 2 \log{\left (- x + 1 \right )} + 3 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((6*x**2+5*x-3)/(x**3+2*x**2-3*x),x)

[Out]

log(x) + 2*log(-x + 1) + 3*log(x + 3)

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Mathematica [A]  time = 0.00926639, size = 17, normalized size = 1. \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(-3 + 5*x + 6*x^2)/(-3*x + 2*x^2 + x^3),x]

[Out]

2*Log[1 - x] + Log[x] + 3*Log[3 + x]

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Maple [A]  time = 0.01, size = 16, normalized size = 0.9 \[ \ln \left ( x \right ) +2\,\ln \left ( -1+x \right ) +3\,\ln \left ( 3+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((6*x^2+5*x-3)/(x^3+2*x^2-3*x),x)

[Out]

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

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Maxima [A]  time = 1.35308, size = 20, normalized size = 1.18 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((6*x^2 + 5*x - 3)/(x^3 + 2*x^2 - 3*x),x, algorithm="maxima")

[Out]

3*log(x + 3) + 2*log(x - 1) + log(x)

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Fricas [A]  time = 0.209134, size = 20, normalized size = 1.18 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((6*x^2 + 5*x - 3)/(x^3 + 2*x^2 - 3*x),x, algorithm="fricas")

[Out]

3*log(x + 3) + 2*log(x - 1) + log(x)

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Sympy [A]  time = 0.142826, size = 15, normalized size = 0.88 \[ \log{\left (x \right )} + 2 \log{\left (x - 1 \right )} + 3 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((6*x**2+5*x-3)/(x**3+2*x**2-3*x),x)

[Out]

log(x) + 2*log(x - 1) + 3*log(x + 3)

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GIAC/XCAS [A]  time = 0.202057, size = 24, normalized size = 1.41 \[ 3 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((6*x^2 + 5*x - 3)/(x^3 + 2*x^2 - 3*x),x, algorithm="giac")

[Out]

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