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]

-ln(a+b/x)/b

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.96, size = 16, normalized size = 1.23 \[ -\frac {\log \left (a x + b\right ) - \log \relax (x)}{b} \]

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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giac [A]  time = 0.23, size = 14, normalized size = 1.08 \[ -\frac {\log \left ({\left | a + \frac {b}{x} \right |}\right )}{b} \]

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

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maple [A]  time = 0.00, size = 19, normalized size = 1.46 \[ \frac {\ln \relax (x )}{b}-\frac {\ln \left (a x +b \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.10, size = 13, normalized size = 1.00 \[ -\frac {\log \left (a + \frac {b}{x}\right )}{b} \]

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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mupad [B]  time = 0.04, size = 15, normalized size = 1.15 \[ -\frac {2\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/x)),x)

[Out]

-(2*atanh((2*a*x)/b + 1))/b

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sympy [A]  time = 0.15, size = 10, normalized size = 0.77 \[ \frac {\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}}{b} \]

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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