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

Optimal. Leaf size=8 \[ \frac{\log (x)}{a+b} \]

[Out]

Log[x]/(a + b)

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Rubi [A]  time = 0.0031854, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6, 12, 29} \[ \frac{\log (x)}{a+b} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x)^(-1),x]

[Out]

Log[x]/(a + b)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0025627, size = 14, normalized size = 1.75 \[ \frac{\log (a x+b x)}{a+b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x)^(-1),x]

[Out]

Log[a*x + b*x]/(a + b)

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Maple [A]  time = 0., size = 9, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{a+b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)/(a+b)

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Maxima [A]  time = 1.02556, size = 19, normalized size = 2.38 \begin{align*} \frac{\log \left (a x + b x\right )}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x)/(a + b)

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Fricas [A]  time = 0.84249, size = 22, normalized size = 2.75 \begin{align*} \frac{\log \left (x\right )}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/(a + b)

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Sympy [A]  time = 0.093372, size = 5, normalized size = 0.62 \begin{align*} \frac{\log{\left (x \right )}}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/(a + b)

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Giac [A]  time = 1.18924, size = 20, normalized size = 2.5 \begin{align*} \frac{\log \left ({\left | a x + b x \right |}\right )}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*x + b*x))/(a + b)