\(\int \log (\frac {c (b+a x)}{x}) \, dx\) [93]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (b+a x)}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2503, 2498, 269, 31} \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (a x+b)}{a} \]

[In]

Int[Log[(c*(b + a*x))/x],x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2498

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

Rule 2503

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.), x_Symbol] :> Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /;
FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int \log \left (a c+\frac {b c}{x}\right ) \, dx \\ & = x \log \left (a c+\frac {b c}{x}\right )+(b c) \int \frac {1}{\left (a c+\frac {b c}{x}\right ) x} \, dx \\ & = x \log \left (a c+\frac {b c}{x}\right )+(b c) \int \frac {1}{b c+a c x} \, dx \\ & = x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (b+a x)}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=\frac {b \log (x)}{a}+\frac {(b+a x) \log \left (\frac {c (b+a x)}{x}\right )}{a} \]

[In]

Integrate[Log[(c*(b + a*x))/x],x]

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04

method	result	size
risch	\(x \ln \left (\frac {c \left (a x +b \right )}{x}\right )+\frac {b \ln \left (a x +b \right )}{a}\)	\(26\)
parts	\(x \ln \left (\frac {c \left (a x +b \right )}{x}\right )+\frac {b \ln \left (a x +b \right )}{a}\)	\(26\)
parallelrisch	\(-\frac {-\ln \left (\frac {c \left (a x +b \right )}{x}\right ) x a b -\ln \left (x \right ) b^{2}-b^{2} \ln \left (\frac {c \left (a x +b \right )}{x}\right )}{a b}\)	\(49\)
derivativedivides	\(-c b \left (\frac {\ln \left (-\frac {b c}{x}\right )}{a c}-\frac {\ln \left (c a +\frac {b c}{x}\right ) \left (c a +\frac {b c}{x}\right ) x}{a \,c^{2} b}\right )\)	\(54\)
default	\(-c b \left (\frac {\ln \left (-\frac {b c}{x}\right )}{a c}-\frac {\ln \left (c a +\frac {b c}{x}\right ) \left (c a +\frac {b c}{x}\right ) x}{a \,c^{2} b}\right )\)	\(54\)

[In]

int(ln(c*(a*x+b)/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=\frac {a x \log \left (\frac {a c x + b c}{x}\right ) + b \log \left (a x + b\right )}{a} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=x \log {\left (\frac {c \left (a x + b\right )}{x} \right )} + \frac {b \log {\left (a x + b \right )}}{a} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=x \log \left (\frac {{\left (a x + b\right )} c}{x}\right ) + \frac {b \log \left (a x + b\right )}{a} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (25) = 50\).

Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.12 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=\frac {b^{2} c^{2} {\left (\frac {\log \left (\frac {{\left | a c x + b c \right |}}{{\left | x \right |}}\right )}{a c} - \frac {\log \left ({\left | -a c + \frac {a c x + b c}{x} \right |}\right )}{a c}\right )} - \frac {b^{2} c^{2} \log \left (-{\left (b - \frac {a}{\frac {a}{b} - \frac {a c x + b c}{b c x}}\right )} c {\left (\frac {a}{b} - \frac {a c x + b c}{b c x}\right )}\right )}{a c - \frac {a c x + b c}{x}}}{b c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx=x\,\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )+\frac {b\,\ln \left (b+a\,x\right )}{a} \]

[In]

int(log((c*(b + a*x))/x),x)

[Out]

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