3.282 \(\int \frac{1}{x^2 (-1+b x)} \, dx\)

Optimal. Leaf size=18 \[ -b \log (x)+b \log (1-b x)+\frac{1}{x} \]

[Out]

x^(-1) - b*Log[x] + b*Log[1 - b*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0231498, antiderivative size = 18, 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 \[ -b \log (x)+b \log (1-b x)+\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(-1 + b*x)),x]

[Out]

x^(-1) - b*Log[x] + b*Log[1 - b*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.78813, size = 15, normalized size = 0.83 \[ - b \log{\left (x \right )} + b \log{\left (- b x + 1 \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*log(x) + b*log(-b*x + 1) + 1/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.00434249, size = 18, normalized size = 1. \[ -b \log (x)+b \log (1-b x)+\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(-1 + b*x)),x]

[Out]

x^(-1) - b*Log[x] + b*Log[1 - b*x]

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 18, normalized size = 1. \[{x}^{-1}-b\ln \left ( x \right ) +b\ln \left ( bx-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x-1),x)

[Out]

1/x-b*ln(x)+b*ln(b*x-1)

_______________________________________________________________________________________

Maxima [A]  time = 1.33982, size = 23, normalized size = 1.28 \[ b \log \left (b x - 1\right ) - b \log \left (x\right ) + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*log(b*x - 1) - b*log(x) + 1/x

_______________________________________________________________________________________

Fricas [A]  time = 0.218869, size = 28, normalized size = 1.56 \[ \frac{b x \log \left (b x - 1\right ) - b x \log \left (x\right ) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*log(b*x - 1) - b*x*log(x) + 1)/x

_______________________________________________________________________________________

Sympy [A]  time = 0.314871, size = 14, normalized size = 0.78 \[ b \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{b} \right )}\right ) + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x-1),x)

[Out]

b*(-log(x) + log(x - 1/b)) + 1/x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207142, size = 26, normalized size = 1.44 \[ b{\rm ln}\left ({\left | b x - 1 \right |}\right ) - b{\rm ln}\left ({\left | x \right |}\right ) + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*ln(abs(b*x - 1)) - b*ln(abs(x)) + 1/x