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

Optimal. Leaf size=19 \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

[Out]

-x^(-1) - b*Log[x] + b*Log[1 + b*x]

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Rubi [A]  time = 0.0080161, antiderivative size = 19, 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} \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

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 = 0.0041456, size = 19, normalized size = 1. \[ -b \log (x)+b \log (b x+1)-\frac{1}{x} \]

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 = 0.007, size = 20, normalized size = 1.1 \begin{align*} -{x}^{-1}-b\ln \left ( x \right ) +b\ln \left ( bx+1 \right ) \end{align*}

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)

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Maxima [A]  time = 1.09871, size = 26, normalized size = 1.37 \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 = 1.50508, size = 53, normalized size = 2.79 \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 = 0.333195, size = 14, normalized size = 0.74 \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 = 1.17108, size = 28, normalized size = 1.47 \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