3.93 \(\int \log (\frac {c (b+a x)}{x}) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2453, 2448, 263, 31} \[ x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (a x+b)}{a} \]

Antiderivative was successfully verified.

[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 263

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 2448

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 2453

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*} \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx &=\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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.12 \[ \frac {(a x+b) \log \left (\frac {c (a x+b)}{x}\right )}{a}+\frac {b \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(c*(b + a*x))/x],x]

[Out]

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

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fricas [A]  time = 0.71, size = 29, normalized size = 1.16 \[ \frac {a x \log \left (\frac {a c x + b c}{x}\right ) + b \log \left (a x + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.18, size = 153, normalized size = 6.12 \[ \frac {b^{2} c^{2} {\left (\frac {\log \left (\frac {{\left | a c x + b c \right |}}{{\left | x \right |}}\right )}{a c} - \frac {\log \left ({\left | -a c + \frac {a c x + b c}{x} \right |}\right )}{a c}\right )} - \frac {b^{2} c^{2} \log \left (-{\left (b - \frac {a}{\frac {a}{b} - \frac {a c x + b c}{b c x}}\right )} c {\left (\frac {a}{b} - \frac {a c x + b c}{b c x}\right )}\right )}{a c - \frac {a c x + b c}{x}}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.12, size = 44, normalized size = 1.76 \[ x \ln \left (a c +\frac {b c}{x}\right )-\frac {b \ln \left (\frac {b c}{x}\right )}{a}+\frac {b \ln \left (a c +\frac {b c}{x}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 25, normalized size = 1.00 \[ x \log \left (\frac {{\left (a x + b\right )} c}{x}\right ) + \frac {b \log \left (a x + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.08, size = 25, normalized size = 1.00 \[ x\,\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )+\frac {b\,\ln \left (b+a\,x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((c*(b + a*x))/x),x)

[Out]

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

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sympy [A]  time = 0.27, size = 20, normalized size = 0.80 \[ x \log {\left (\frac {c \left (a x + b\right )}{x} \right )} + \frac {b \log {\left (a x + b \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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