3.1553 \(\int \frac{a+\frac{b}{x}}{x} \, dx\)

Optimal. Leaf size=11 \[ a \log (x)-\frac{b}{x} \]

[Out]

-(b/x) + a*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0046539, antiderivative size = 11, 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 = {14} \[ a \log (x)-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x,x]

[Out]

-(b/x) + a*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0018222, size = 11, normalized size = 1. \[ a \log (x)-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x,x]

[Out]

-(b/x) + a*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*} -{\frac{b}{x}}+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/x+a*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.957746, size = 15, normalized size = 1.36 \begin{align*} a \log \left (x\right ) - \frac{b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - b/x

________________________________________________________________________________________

Fricas [A]  time = 1.42082, size = 27, normalized size = 2.45 \begin{align*} \frac{a x \log \left (x\right ) - b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*log(x) - b)/x

________________________________________________________________________________________

Sympy [A]  time = 0.239214, size = 7, normalized size = 0.64 \begin{align*} a \log{\left (x \right )} - \frac{b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - b/x

________________________________________________________________________________________

Giac [A]  time = 1.14366, size = 16, normalized size = 1.45 \begin{align*} a \log \left ({\left | x \right |}\right ) - \frac{b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(abs(x)) - b/x