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

Optimal. Leaf size=18 \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

[Out]

x/b - (a*Log[a + b*x])/b^2

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Rubi [A]  time = 0.0099383, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x),x]

[Out]

x/b - (a*Log[a + b*x])/b^2

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{a+b x} \, dx &=\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx\\ &=\frac{x}{b}-\frac{a \log (a+b x)}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0023684, size = 18, normalized size = 1. \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x),x]

[Out]

x/b - (a*Log[a + b*x])/b^2

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Maple [A]  time = 0.002, size = 19, normalized size = 1.1 \begin{align*}{\frac{x}{b}}-{\frac{a\ln \left ( bx+a \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b-a*ln(b*x+a)/b^2

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Maxima [A]  time = 0.950947, size = 24, normalized size = 1.33 \begin{align*} \frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b - a*log(b*x + a)/b^2

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Fricas [A]  time = 1.88243, size = 38, normalized size = 2.11 \begin{align*} \frac{b x - a \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x - a*log(b*x + a))/b^2

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Sympy [A]  time = 0.260511, size = 14, normalized size = 0.78 \begin{align*} - \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(a + b*x)/b**2 + x/b

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Giac [A]  time = 1.08058, size = 26, normalized size = 1.44 \begin{align*} \frac{x}{b} - \frac{a \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b - a*log(abs(b*x + a))/b^2