3.67 \(\int \log (a+b x+c x) \, dx\)

Optimal. Leaf size=25 \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]

[Out]

-x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)

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Rubi [A]  time = 0.0147451, antiderivative size = 25, 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 = {2444, 2389, 2295} \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x + c*x],x]

[Out]

-x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)

Rule 2444

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

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

Mathematica [A]  time = 0.0068114, size = 25, normalized size = 1. \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x + c*x],x]

[Out]

-x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)

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Maple [B]  time = 0.057, size = 75, normalized size = 3. \begin{align*}{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xb}{b+c}}+{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xc}{b+c}}+{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) a}{b+c}}-{\frac{bx}{b+c}}-{\frac{cx}{b+c}}-{\frac{a}{b+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.24484, size = 46, normalized size = 1.84 \begin{align*} -\frac{b x + c x -{\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.00934, size = 80, normalized size = 3.2 \begin{align*} -\frac{{\left (b + c\right )} x -{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.357667, size = 36, normalized size = 1.44 \begin{align*} x \log{\left (a + b x + c x \right )} + \left (- b - c\right ) \left (- \frac{a \log{\left (a + x \left (b + c\right ) \right )}}{\left (b + c\right )^{2}} + \frac{x}{b + c}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19796, size = 46, normalized size = 1.84 \begin{align*} -\frac{b x + c x -{\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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