3.185 \(\int \frac{1}{x (1+x) (3+2 x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0272626, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x)*(3 + 2*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.89823, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x \right )}}{3} - \log{\left (x + 1 \right )} + \frac{2 \log{\left (2 x + 3 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00727353, size = 23, normalized size = 1. \[ \frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 20, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{3}}-\ln \left ( 1+x \right ) +{\frac{2\,\ln \left ( 3+2\,x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+x)/(3+2*x),x)

[Out]

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

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Maxima [A]  time = 1.33825, size = 26, normalized size = 1.13 \[ \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210755, size = 26, normalized size = 1.13 \[ \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.14694, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x \right )}}{3} - \log{\left (x + 1 \right )} + \frac{2 \log{\left (x + \frac{3}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(1+x)/(3+2*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21507, size = 30, normalized size = 1.3 \[ \frac{2}{3} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) -{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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