3.188 \(\int \frac{1}{(-1+x)^2 (4+x)} \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{5 (1-x)}-\frac{1}{25} \log (1-x)+\frac{1}{25} \log (x+4) \]

[Out]

1/(5*(1 - x)) - Log[1 - x]/25 + Log[4 + x]/25

_______________________________________________________________________________________

Rubi [A]  time = 0.0243878, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{5 (1-x)}-\frac{1}{25} \log (1-x)+\frac{1}{25} \log (x+4) \]

Antiderivative was successfully verified.

[In]  Int[1/((-1 + x)^2*(4 + x)),x]

[Out]

1/(5*(1 - x)) - Log[1 - x]/25 + Log[4 + x]/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.78079, size = 19, normalized size = 0.63 \[ - \frac{\log{\left (- x + 1 \right )}}{25} + \frac{\log{\left (x + 4 \right )}}{25} + \frac{1}{5 \left (- x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1+x)**2/(4+x),x)

[Out]

-log(-x + 1)/25 + log(x + 4)/25 + 1/(5*(-x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0120141, size = 22, normalized size = 0.73 \[ \frac{1}{25} \left (-\frac{5}{x-1}-\log (x-1)+\log (x+4)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((-1 + x)^2*(4 + x)),x]

[Out]

(-5/(-1 + x) - Log[-1 + x] + Log[4 + x])/25

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 21, normalized size = 0.7 \[{\frac{\ln \left ( 4+x \right ) }{25}}-{\frac{1}{-5+5\,x}}-{\frac{\ln \left ( -1+x \right ) }{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+x)^2/(4+x),x)

[Out]

1/25*ln(4+x)-1/5/(-1+x)-1/25*ln(-1+x)

_______________________________________________________________________________________

Maxima [A]  time = 1.36408, size = 27, normalized size = 0.9 \[ -\frac{1}{5 \,{\left (x - 1\right )}} + \frac{1}{25} \, \log \left (x + 4\right ) - \frac{1}{25} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 4)*(x - 1)^2),x, algorithm="maxima")

[Out]

-1/5/(x - 1) + 1/25*log(x + 4) - 1/25*log(x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.204318, size = 35, normalized size = 1.17 \[ \frac{{\left (x - 1\right )} \log \left (x + 4\right ) -{\left (x - 1\right )} \log \left (x - 1\right ) - 5}{25 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 4)*(x - 1)^2),x, algorithm="fricas")

[Out]

1/25*((x - 1)*log(x + 4) - (x - 1)*log(x - 1) - 5)/(x - 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.118058, size = 19, normalized size = 0.63 \[ - \frac{\log{\left (x - 1 \right )}}{25} + \frac{\log{\left (x + 4 \right )}}{25} - \frac{1}{5 x - 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+x)**2/(4+x),x)

[Out]

-log(x - 1)/25 + log(x + 4)/25 - 1/(5*x - 5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20437, size = 28, normalized size = 0.93 \[ -\frac{1}{5 \,{\left (x - 1\right )}} + \frac{1}{25} \,{\rm ln}\left ({\left | -\frac{5}{x - 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 4)*(x - 1)^2),x, algorithm="giac")

[Out]

-1/5/(x - 1) + 1/25*ln(abs(-5/(x - 1) - 1))