3.441 \(\int \frac{1+x}{-6 x+x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{3}{10} \log (2-x)-\frac{\log (x)}{6}-\frac{2}{15} \log (x+3) \]

[Out]

(3*Log[2 - x])/10 - Log[x]/6 - (2*Log[3 + x])/15

_______________________________________________________________________________________

Rubi [A]  time = 0.0466046, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{3}{10} \log (2-x)-\frac{\log (x)}{6}-\frac{2}{15} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/(-6*x + x^2 + x^3),x]

[Out]

(3*Log[2 - x])/10 - Log[x]/6 - (2*Log[3 + x])/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.82311, size = 20, normalized size = 0.8 \[ - \frac{\log{\left (x \right )}}{6} + \frac{3 \log{\left (- x + 2 \right )}}{10} - \frac{2 \log{\left (x + 3 \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(x**3+x**2-6*x),x)

[Out]

-log(x)/6 + 3*log(-x + 2)/10 - 2*log(x + 3)/15

_______________________________________________________________________________________

Mathematica [A]  time = 0.00781111, size = 25, normalized size = 1. \[ \frac{3}{10} \log (2-x)-\frac{\log (x)}{6}-\frac{2}{15} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/(-6*x + x^2 + x^3),x]

[Out]

(3*Log[2 - x])/10 - Log[x]/6 - (2*Log[3 + x])/15

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 18, normalized size = 0.7 \[ -{\frac{\ln \left ( x \right ) }{6}}+{\frac{3\,\ln \left ( x-2 \right ) }{10}}-{\frac{2\,\ln \left ( 3+x \right ) }{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^3+x^2-6*x),x)

[Out]

-1/6*ln(x)+3/10*ln(x-2)-2/15*ln(3+x)

_______________________________________________________________________________________

Maxima [A]  time = 0.793163, size = 23, normalized size = 0.92 \[ -\frac{2}{15} \, \log \left (x + 3\right ) + \frac{3}{10} \, \log \left (x - 2\right ) - \frac{1}{6} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + x^2 - 6*x),x, algorithm="maxima")

[Out]

-2/15*log(x + 3) + 3/10*log(x - 2) - 1/6*log(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.25752, size = 23, normalized size = 0.92 \[ -\frac{2}{15} \, \log \left (x + 3\right ) + \frac{3}{10} \, \log \left (x - 2\right ) - \frac{1}{6} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + x^2 - 6*x),x, algorithm="fricas")

[Out]

-2/15*log(x + 3) + 3/10*log(x - 2) - 1/6*log(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.298799, size = 20, normalized size = 0.8 \[ - \frac{\log{\left (x \right )}}{6} + \frac{3 \log{\left (x - 2 \right )}}{10} - \frac{2 \log{\left (x + 3 \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**3+x**2-6*x),x)

[Out]

-log(x)/6 + 3*log(x - 2)/10 - 2*log(x + 3)/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263813, size = 27, normalized size = 1.08 \[ -\frac{2}{15} \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + \frac{3}{10} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + x^2 - 6*x),x, algorithm="giac")

[Out]

-2/15*ln(abs(x + 3)) + 3/10*ln(abs(x - 2)) - 1/6*ln(abs(x))