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

Optimal. Leaf size=21 \[ -\frac{3 \log (x)}{2}+4 \log (x+1)-\frac{5}{2} \log (x+2) \]

[Out]

(-3*Log[x])/2 + 4*Log[1 + x] - (5*Log[2 + x])/2

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Rubi [A]  time = 0.0476858, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{3 \log (x)}{2}+4 \log (x+1)-\frac{5}{2} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Log[x])/2 + 4*Log[1 + x] - (5*Log[2 + x])/2

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Rubi in Sympy [A]  time = 12.6167, size = 20, normalized size = 0.95 \[ - \frac{3 \log{\left (x \right )}}{2} + 4 \log{\left (x + 1 \right )} - \frac{5 \log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x)/2 + 4*log(x + 1) - 5*log(x + 2)/2

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Mathematica [A]  time = 0.00860018, size = 21, normalized size = 1. \[ -\frac{3 \log (x)}{2}+4 \log (x+1)-\frac{5}{2} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Log[x])/2 + 4*Log[1 + x] - (5*Log[2 + x])/2

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Maple [A]  time = 0.01, size = 18, normalized size = 0.9 \[ -{\frac{3\,\ln \left ( x \right ) }{2}}+4\,\ln \left ( 1+x \right ) -{\frac{5\,\ln \left ( 2+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3+x)/(x^3+3*x^2+2*x),x)

[Out]

-3/2*ln(x)+4*ln(1+x)-5/2*ln(2+x)

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Maxima [A]  time = 0.785245, size = 23, normalized size = 1.1 \[ -\frac{5}{2} \, \log \left (x + 2\right ) + 4 \, \log \left (x + 1\right ) - \frac{3}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*log(x + 2) + 4*log(x + 1) - 3/2*log(x)

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Fricas [A]  time = 0.271498, size = 23, normalized size = 1.1 \[ -\frac{5}{2} \, \log \left (x + 2\right ) + 4 \, \log \left (x + 1\right ) - \frac{3}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*log(x + 2) + 4*log(x + 1) - 3/2*log(x)

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Sympy [A]  time = 0.275557, size = 20, normalized size = 0.95 \[ - \frac{3 \log{\left (x \right )}}{2} + 4 \log{\left (x + 1 \right )} - \frac{5 \log{\left (x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x)/2 + 4*log(x + 1) - 5*log(x + 2)/2

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GIAC/XCAS [A]  time = 0.259774, size = 27, normalized size = 1.29 \[ -\frac{5}{2} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + 4 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{3}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*ln(abs(x + 2)) + 4*ln(abs(x + 1)) - 3/2*ln(abs(x))