\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{f+g x} \, dx\) [235]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=-\frac {B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \] Output:

-B*ln(-g*(b*x+a)/(-a*g+b*f))*ln(g*x+f)/g+(A+B*ln(e*(b*x+a)/(d*x+c)))*ln(g* 
x+f)/g+B*ln(-g*(d*x+c)/(-c*g+d*f))*ln(g*x+f)/g-B*polylog(2,b*(g*x+f)/(-a*g 
+b*f))/g+B*polylog(2,d*(g*x+f)/(-c*g+d*f))/g
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\frac {\left (A-B \log \left (\frac {g (a+b x)}{-b f+a g}\right )+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)-B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )+B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \] Input:

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x),x]
 

Output:

((A - B*Log[(g*(a + b*x))/(-(b*f) + a*g)] + B*Log[(e*(a + b*x))/(c + d*x)] 
 + B*Log[(g*(c + d*x))/(-(d*f) + c*g)])*Log[f + g*x] - B*PolyLog[2, (b*(f 
+ g*x))/(b*f - a*g)] + B*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2946, 2841, 2840, 2838}

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.

\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{f+g x} \, dx\)

\(\Big \downarrow \) 2946

\(\displaystyle -\frac {b B \int \frac {\log (f+g x)}{a+b x}dx}{g}+\frac {B d \int \frac {\log (f+g x)}{c+d x}dx}{g}+\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}\)

\(\Big \downarrow \) 2841

\(\displaystyle -\frac {b B \left (\frac {\log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{b}-\frac {g \int \frac {\log \left (-\frac {g (a+b x)}{b f-a g}\right )}{f+g x}dx}{b}\right )}{g}+\frac {B d \left (\frac {\log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{d}-\frac {g \int \frac {\log \left (-\frac {g (c+d x)}{d f-c g}\right )}{f+g x}dx}{d}\right )}{g}+\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}\)

\(\Big \downarrow \) 2840

\(\displaystyle -\frac {b B \left (\frac {\log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{b}-\frac {\int \frac {\log \left (1-\frac {b (f+g x)}{b f-a g}\right )}{f+g x}d(f+g x)}{b}\right )}{g}+\frac {B d \left (\frac {\log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{d}-\frac {\int \frac {\log \left (1-\frac {d (f+g x)}{d f-c g}\right )}{f+g x}d(f+g x)}{d}\right )}{g}+\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {b B \left (\frac {\operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{b}+\frac {\log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{b}\right )}{g}+\frac {B d \left (\frac {\operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{d}+\frac {\log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{d}\right )}{g}\)

Input:

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x),x]
 

Output:

((A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x])/g - (b*B*((Log[-((g*(a 
 + b*x))/(b*f - a*g))]*Log[f + g*x])/b + PolyLog[2, (b*(f + g*x))/(b*f - a 
*g)]/b))/g + (B*d*((Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/d + Po 
lyLog[2, (d*(f + g*x))/(d*f - c*g)]/d))/g
 

Defintions of rubi rules used

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[1/g   Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* 
x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c 
*(e*f - d*g), 0]
 

rule 2841

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ 
)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x 
)^n])/g), x] - Simp[b*e*(n/g)   Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x 
), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
 

rule 2946

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f + g*x]*((A + B*Log[ 
e*((a + b*x)^n/(c + d*x)^n)])/g), x] + (-Simp[b*B*(n/g)   Int[Log[f + g*x]/ 
(a + b*x), x], x] + Simp[B*d*(n/g)   Int[Log[f + g*x]/(c + d*x), x], x]) /; 
 FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a* 
d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(140)=280\).

Time = 5.72 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.66

method	result	size
parts	\(\frac {A \ln \left (g x +f \right )}{g}-\frac {B \left (d a -b c \right ) e \left (-\frac {\left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right ) d^{2} \left (c g -d f \right )}{e g \left (d a -b c \right )}+\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{3}}{e g \left (d a -b c \right )}\right )}{d^{2}}\)	\(372\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-d^{2} A \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (d a -b c \right )}-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (d a -b c \right ) \left (-c g +d f \right )}\right )-d^{2} B \left (-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{e g \left (d a -b c \right )}+\frac {\left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right ) \left (c g -d f \right )}{e g \left (d a -b c \right )}\right )\right )}{d^{2}}\)	\(529\)
default	\(-\frac {e \left (d a -b c \right ) \left (-d^{2} A \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (d a -b c \right )}-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (d a -b c \right ) \left (-c g +d f \right )}\right )-d^{2} B \left (-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{e g \left (d a -b c \right )}+\frac {\left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right ) \left (c g -d f \right )}{e g \left (d a -b c \right )}\right )\right )}{d^{2}}\)	\(529\)
risch	\(\text {Expression too large to display}\)	\(1127\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f),x,method=_RETURNVERBOSE)
 

Output:

A*ln(g*x+f)/g-B/d^2*(a*d-b*c)*e*(-(dilog(((c*g-d*f)*(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)+ln(b*e/d+(a*d-b*c)*e/d/(d*x 
+c))*ln(((c*g-d*f)*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)/(-a*e*g+b*e* 
f))/(c*g-d*f))*d^2*(c*g-d*f)/e/g/(a*d-b*c)+(dilog(-((b*e/d+(a*d-b*c)*e/d/( 
d*x+c))*d-b*e)/b/e)/d+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-((b*e/d+(a*d-b*c 
)*e/d/(d*x+c))*d-b*e)/b/e)/d)*d^3/e/g/(a*d-b*c))
 

Fricas [F]

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

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f),x, algorithm="fricas")
 

Output:

integral((B*log((b*e*x + a*e)/(d*x + c)) + A)/(g*x + f), x)
 

Sympy [F]

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

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f),x)
 

Output:

Integral((A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x)))/(f + g*x), x)
 

Maxima [F]

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

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f),x, algorithm="maxima")
 

Output:

-B*integrate(-(log(b*x + a) - log(d*x + c) + log(e))/(g*x + f), x) + A*log 
(g*x + f)/g
 

Giac [F]

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

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f),x, algorithm="giac")
 

Output:

integrate((B*log((b*x + a)*e/(d*x + c)) + A)/(g*x + f), x)
 

Mupad [F(-1)]

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

int((A + B*log((e*(a + b*x))/(c + d*x)))/(f + g*x),x)
 

Output:

int((A + B*log((e*(a + b*x))/(c + d*x)))/(f + g*x), x)
 

Reduce [F]

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

int((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f),x)
                                                                                    
                                                                                    
 

Output:

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