3.1612 \(\int \frac{1}{(a+\frac{b}{x}) x} \, dx\)

Optimal. Leaf size=10 \[ \frac{\log (a x+b)}{a} \]

[Out]

Log[b + a*x]/a

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Rubi [A]  time = 0.0035947, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 31} \[ \frac{\log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x),x]

[Out]

Log[b + a*x]/a

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.0009955, size = 10, normalized size = 1. \[ \frac{\log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x),x]

[Out]

Log[b + a*x]/a

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( ax+b \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a*x+b)/a

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Maxima [A]  time = 0.992188, size = 14, normalized size = 1.4 \begin{align*} \frac{\log \left (a x + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b)/a

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Fricas [A]  time = 1.4874, size = 22, normalized size = 2.2 \begin{align*} \frac{\log \left (a x + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b)/a

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Sympy [A]  time = 0.062468, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (a x + b \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b)/a

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Giac [A]  time = 1.13985, size = 15, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | a x + b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*x + b))/a