3.64 \(\int \log (a+b x) \, dx\)

Optimal. Leaf size=19 \[ \frac{(a+b x) \log (a+b x)}{b}-x \]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

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Rubi [A]  time = 0.0058197, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2389, 2295} \[ \frac{(a+b x) \log (a+b x)}{b}-x \]

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0044641, size = 19, normalized size = 1. \[ \frac{(a+b x) \log (a+b x)}{b}-x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

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Maple [A]  time = 0.056, size = 30, normalized size = 1.6 \begin{align*} \ln \left ( bx+a \right ) x+{\frac{\ln \left ( bx+a \right ) a}{b}}-x-{\frac{a}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(b*x+a)*x+1/b*ln(b*x+a)*a-x-a/b

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Maxima [A]  time = 1.15998, size = 31, normalized size = 1.63 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b

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Fricas [A]  time = 1.92999, size = 47, normalized size = 2.47 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a))/b

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Sympy [A]  time = 0.302881, size = 24, normalized size = 1.26 \begin{align*} - b \left (- \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b}\right ) + x \log{\left (a + b x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(-a*log(a + b*x)/b**2 + x/b) + x*log(a + b*x)

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Giac [A]  time = 1.21122, size = 31, normalized size = 1.63 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b