\(\int \frac {\log (\frac {a+b x}{x})}{c+d x} \, dx\) [396]

Optimal result

Integrand size = 18, antiderivative size = 105 \[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{d} \] Output:

ln(b+a/x)*ln(d*x+c)/d+ln(-d*x/c)*ln(d*x+c)/d-ln(-d*(b*x+a)/(-a*d+b*c))*ln( 
d*x+c)/d-polylog(2,b*(d*x+c)/(-a*d+b*c))/d+polylog(2,1+d*x/c)/d
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)+\log (x) \log (c+d x)-\log \left (\frac {a}{b}+x\right ) \log (c+d x)+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-\log (x) \log \left (1+\frac {d x}{c}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+\operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{d} \] Input:

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

Output:

(Log[b + a/x]*Log[c + d*x] + Log[x]*Log[c + d*x] - Log[a/b + x]*Log[c + d* 
x] + Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] - Log[x]*Log[1 + (d*x)/c] 
 - PolyLog[2, -((d*x)/c)] + PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/d
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2915, 2912, 2005, 2863, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx\)

\(\Big \downarrow \) 2915

\(\displaystyle \int \frac {\log \left (\frac {a}{x}+b\right )}{c+d x}dx\)

\(\Big \downarrow \) 2912

\(\displaystyle \frac {a \int \frac {\log (c+d x)}{\left (\frac {a}{x}+b\right ) x^2}dx}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2005

\(\displaystyle \frac {a \int \frac {\log (c+d x)}{x (a+b x)}dx}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {a \int \left (\frac {\log (c+d x)}{a x}-\frac {b \log (c+d x)}{a (a+b x)}\right )dx}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \left (-\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{a}-\frac {\log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}\right )}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}\)

Input:

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

Output:

(Log[b + a/x]*Log[c + d*x])/d + (a*((Log[-((d*x)/c)]*Log[c + d*x])/a - (Lo 
g[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/a - PolyLog[2, (b*(c + d*x)) 
/(b*c - a*d)]/a + PolyLog[2, 1 + (d*x)/c]/a))/d
 

Defintions of rubi rules used

rule 2005

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2912

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_. 
)*(x_)), x_Symbol] :> Simp[Log[f + g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x 
] - Simp[b*e*n*(p/g)   Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x] /; 
FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]
 

rule 2915

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

Maple [A] (verified)

Time = 10.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {\operatorname {dilog}\left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{d}+\frac {\ln \left (b +\frac {a}{x}\right ) \ln \left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{d}-\frac {\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )}{d}-\frac {\operatorname {dilog}\left (-\frac {a}{b x}\right )}{d}\)	\(114\)
derivativedivides	\(-a \left (-\frac {\left (\frac {\operatorname {dilog}\left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{c}+\frac {\ln \left (b +\frac {a}{x}\right ) \ln \left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{c}\right ) c}{d a}+\frac {\operatorname {dilog}\left (-\frac {a}{b x}\right )+\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )}{d a}\right )\)	\(126\)
default	\(-a \left (-\frac {\left (\frac {\operatorname {dilog}\left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{c}+\frac {\ln \left (b +\frac {a}{x}\right ) \ln \left (\frac {d a -b c +c \left (b +\frac {a}{x}\right )}{d a -b c}\right )}{c}\right ) c}{d a}+\frac {\operatorname {dilog}\left (-\frac {a}{b x}\right )+\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )}{d a}\right )\)	\(126\)
parts	\(\frac {\ln \left (\frac {b x +a}{x}\right ) \ln \left (d x +c \right )}{d}-\frac {-\left (\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )\right ) d^{2}+\left (\frac {\operatorname {dilog}\left (\frac {d a -b c +b \left (d x +c \right )}{d a -b c}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {d a -b c +b \left (d x +c \right )}{d a -b c}\right )}{b}\right ) d^{2} b}{d^{3}}\)	\(131\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x + a}{x}\right )}{d x + c} \,d x } \] Input:

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

Output:

integral(log((b*x + a)/x)/(d*x + c), x)
 

Sympy [F]

\[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\int \frac {\log {\left (\frac {a}{x} + b \right )}}{c + d x}\, dx \] Input:

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

Output:

Integral(log(a/x + b)/(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18 \[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=-\frac {{\left (\log \left (b x + a\right ) - \log \left (x\right )\right )} \log \left (d x + c\right )}{d} + \frac {\log \left (d x + c\right ) \log \left (\frac {b x + a}{x}\right )}{d} - \frac {\log \left (\frac {d x}{c} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{c}\right )}{d} + \frac {\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )}{d} \] Input:

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

Output:

-(log(b*x + a) - log(x))*log(d*x + c)/d + log(d*x + c)*log((b*x + a)/x)/d 
- (log(d*x/c + 1)*log(x) + dilog(-d*x/c))/d + (log(b*x + a)*log((b*d*x + a 
*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))/d
 

Giac [F]

\[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x + a}{x}\right )}{d x + c} \,d x } \] Input:

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

Output:

integrate(log((b*x + a)/x)/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\int \frac {\ln \left (\frac {a+b\,x}{x}\right )}{c+d\,x} \,d x \] Input:

int(log((a + b*x)/x)/(c + d*x),x)
 

Output:

int(log((a + b*x)/x)/(c + d*x), x)
 

Reduce [F]

\[ \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx=\int \frac {\mathrm {log}\left (\frac {b x +a}{x}\right )}{d x +c}d x \] Input:

int(log((b*x+a)/x)/(d*x+c),x)
 

Output:

int(log((a + b*x)/x)/(c + d*x),x)