3.1613 \(\int \frac{1}{(a+\frac{b}{x}) x^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{\log \left (a+\frac{b}{x}\right )}{b} \]

[Out]

-(Log[a + b/x]/b)

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Rubi [A]  time = 0.0038023, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {260} \[ -\frac{\log \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^2),x]

[Out]

-(Log[a + b/x]/b)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0034102, size = 18, normalized size = 1.38 \[ \frac{\log (x)}{b}-\frac{\log (a x+b)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^2),x]

[Out]

Log[x]/b - Log[b + a*x]/b

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Maple [A]  time = 0.003, size = 19, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( ax+b \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.03593, size = 18, normalized size = 1.38 \begin{align*} -\frac{\log \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^2,x, algorithm="maxima")

[Out]

-log(a + b/x)/b

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Fricas [A]  time = 1.45474, size = 38, normalized size = 2.92 \begin{align*} -\frac{\log \left (a x + b\right ) - \log \left (x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^2,x, algorithm="fricas")

[Out]

-(log(a*x + b) - log(x))/b

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Sympy [A]  time = 0.129191, size = 10, normalized size = 0.77 \begin{align*} \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x**2,x)

[Out]

(log(x) - log(x + b/a))/b

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Giac [A]  time = 1.10156, size = 19, normalized size = 1.46 \begin{align*} -\frac{\log \left ({\left | a + \frac{b}{x} \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^2,x, algorithm="giac")

[Out]

-log(abs(a + b/x))/b