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]

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Rubi [A]  time = 0.0099948, 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, Rules used = {44} \[ -b \log (x)+b \log (1-b x)+\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(-1 + b*x)),x]

[Out]

x^(-1) - b*Log[x] + b*Log[1 - b*x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 (-1+b x)} \, dx &=\int \left (-\frac{1}{x^2}-\frac{b}{x}+\frac{b^2}{-1+b x}\right ) \, dx\\ &=\frac{1}{x}-b \log (x)+b \log (1-b x)\\ \end{align*}

Mathematica [A]  time = 0.0034923, 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]

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Maple [A]  time = 0.008, size = 18, normalized size = 1. \begin{align*} b\ln \left ( bx-1 \right ) +{x}^{-1}-b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x-1),x)

[Out]

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

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Maxima [A]  time = 1.09852, size = 23, normalized size = 1.28 \begin{align*} b \log \left (b x - 1\right ) - b \log \left (x\right ) + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(b*x - 1) - b*log(x) + 1/x

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Fricas [A]  time = 1.52477, size = 53, normalized size = 2.94 \begin{align*} \frac{b x \log \left (b x - 1\right ) - b x \log \left (x\right ) + 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*log(b*x - 1) - b*x*log(x) + 1)/x

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Sympy [A]  time = 0.165637, size = 14, normalized size = 0.78 \begin{align*} b \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{b} \right )}\right ) + \frac{1}{x} \end{align*}

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

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Giac [A]  time = 1.23621, size = 26, normalized size = 1.44 \begin{align*} b \log \left ({\left | b x - 1 \right |}\right ) - b \log \left ({\left | x \right |}\right ) + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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