\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{f+g x} \, dx\) [62]

Optimal result

Integrand size = 30, antiderivative size = 147 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx=-\frac {B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}+\frac {B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B n \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B n \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.83 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx=\frac {\left (A-B n \log \left (\frac {g (a+b x)}{-b f+a g}\right )+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)-B n \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )+B n \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2945, 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.

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

\(\Big \downarrow \) 2945

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[f + g*x])/g - (b*B*n*((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*n*((Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/ 
d + PolyLog[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 2945

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

Maple [F]

\[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{g x +f}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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