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

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

[Out]

ln(a*x+b)/a

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x),x]

[Out]

Log[b + a*x]/a

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fricas [A]  time = 0.72, size = 10, normalized size = 1.00 \[ \frac {\log \left (a x + b\right )}{a} \]

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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giac [A]  time = 0.15, size = 11, normalized size = 1.10 \[ \frac {\log \left ({\left | a x + b \right |}\right )}{a} \]

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

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maple [A]  time = 0.00, size = 11, normalized size = 1.10 \[ \frac {\ln \left (a x +b \right )}{a} \]

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 = 0.99, size = 10, normalized size = 1.00 \[ \frac {\log \left (a x + b\right )}{a} \]

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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mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \[ \frac {\ln \left (b+a\,x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/x)),x)

[Out]

log(b + a*x)/a

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sympy [A]  time = 0.07, size = 7, normalized size = 0.70 \[ \frac {\log {\left (a x + b \right )}}{a} \]

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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