\(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{a g+b g x} \, dx\) [123]

Optimal result

Integrand size = 32, antiderivative size = 83 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g}+\frac {2 B \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b g} \] Output:

-ln(-(-a*d+b*c)/d/(b*x+a))*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/g+2*B*polylog 
(2,1+(-a*d+b*c)/d/(b*x+a))/b/g
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.06 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\frac {\log (a+b x) \left (A-B \log (a+b x)+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 B \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2942, 2858, 27, 2778, 2005, 2752}

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.

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

\(\Big \downarrow \) 2942

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

\(\Big \downarrow \) 2858

\(\displaystyle \frac {2 B (b c-a d) \int \frac {b \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}d(a+b x)}{b^2 g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (b c-a d+d (a+b x))}d(a+b x)}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g}\)

\(\Big \downarrow \) 2778

\(\displaystyle -\frac {2 B (b c-a d) \int \frac {(a+b x) \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{b c-a d+d (a+b x)}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g}\)

\(\Big \downarrow \) 2005

\(\displaystyle -\frac {2 B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{d+\frac {b c-a d}{a+b x}}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g}\)

\(\Big \downarrow \) 2752

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

Input:

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

Output:

-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^ 
2]))/(b*g)) + (2*B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2778

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), 
x_Symbol] :> Simp[1/n   Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], 
 x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2942

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(83)=166\).

Time = 1.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.18

method	result	size
parts	\(\frac {A \ln \left (b x +a \right )}{g b}+\frac {B \left (-\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 d a -2 b c \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}\right )}{b}-\frac {\left (\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{d a -b c}-\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right )^{2}}{d a -b c}\right ) \left (-d a +b c \right )}{b}\right )}{g}\)	\(264\)
derivativedivides	\(-\frac {\frac {d A \left (\frac {\left (-d a +b c \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (d a -b c \right )}+\frac {\ln \left (\frac {1}{d x +c}\right )}{b}\right )}{g}+\frac {d B \left (\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 d a -2 b c \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}\right )}{b}+\frac {\left (\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{d a -b c}-\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right )^{2}}{d a -b c}\right ) \left (-d a +b c \right )}{b}\right )}{g}}{d}\)	\(319\)
default	\(-\frac {\frac {d A \left (\frac {\left (-d a +b c \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (d a -b c \right )}+\frac {\ln \left (\frac {1}{d x +c}\right )}{b}\right )}{g}+\frac {d B \left (\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 d a -2 b c \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right )}{d a -b c}\right )}{b}+\frac {\left (\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{d a -b c}-\frac {\ln \left (\frac {d a -b c}{d x +c}+b \right )^{2}}{d a -b c}\right ) \left (-d a +b c \right )}{b}\right )}{g}}{d}\)	\(319\)
risch	\(\frac {A \ln \left (b x +a \right )}{g b}-\frac {B \ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g b}+\frac {2 B \operatorname {dilog}\left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right ) d a}{g b \left (d a -b c \right )}-\frac {2 B \operatorname {dilog}\left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right ) c}{g \left (d a -b c \right )}+\frac {2 B \ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right ) d a}{g b \left (d a -b c \right )}-\frac {2 B \ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {d a -b c}{d x +c}+b}{b}\right ) c}{g \left (d a -b c \right )}+\frac {B \ln \left (\frac {d a -b c}{d x +c}+b \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right ) d a}{g b \left (d a -b c \right )}-\frac {B \ln \left (\frac {d a -b c}{d x +c}+b \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right ) c}{g \left (d a -b c \right )}-\frac {B \ln \left (\frac {d a -b c}{d x +c}+b \right )^{2} d a}{g b \left (d a -b c \right )}+\frac {B \ln \left (\frac {d a -b c}{d x +c}+b \right )^{2} c}{g \left (d a -b c \right )}\)	\(464\)

Input:

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g),x,method=_RETURNVERBOSE)
 

Output:

A/g*ln(b*x+a)/b+B/g*(-(ln(1/(d*x+c))*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^ 
2)-(2*a*d-2*b*c)*(dilog(((a*d-b*c)/(d*x+c)+b)/b)/(a*d-b*c)+ln(1/(d*x+c))*l 
n(((a*d-b*c)/(d*x+c)+b)/b)/(a*d-b*c)))/b-(ln((a*d-b*c)/(d*x+c)+b)/(a*d-b*c 
)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)-1/(a*d-b*c)*ln((a*d-b*c)/(d*x+c) 
+b)^2)*(-a*d+b*c)/b)
 

Fricas [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{b g x + a g} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g),x, algorithm="frica 
s")
 

Output:

integral((B*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)) 
 + A)/(b*g*x + a*g), x)
 

Sympy [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\frac {\int \frac {A}{a + b x}\, dx + \int \frac {B \log {\left (\frac {a^{2} e}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {2 a b e x}{c^{2} + 2 c d x + d^{2} x^{2}} + \frac {b^{2} e x^{2}}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}}{a + b x}\, dx}{g} \] Input:

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

Output:

(Integral(A/(a + b*x), x) + Integral(B*log(a**2*e/(c**2 + 2*c*d*x + d**2*x 
**2) + 2*a*b*e*x/(c**2 + 2*c*d*x + d**2*x**2) + b**2*e*x**2/(c**2 + 2*c*d* 
x + d**2*x**2))/(a + b*x), x))/g
 

Maxima [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{b g x + a g} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g),x, algorithm="maxim 
a")
 

Output:

-B*(2*log(b*x + a)*log(d*x + c)/(b*g) - integrate((b*d*x*log(e) + b*c*log( 
e) + 2*(2*b*d*x + b*c + a*d)*log(b*x + a))/(b^2*d*g*x^2 + a*b*c*g + (b^2*c 
*g + a*b*d*g)*x), x)) + A*log(b*g*x + a*g)/(b*g)
 

Giac [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{b g x + a g} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g),x, algorithm="giac" 
)
 

Output:

integrate((B*log((b*x + a)^2*e/(d*x + c)^2) + A)/(b*g*x + a*g), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{a\,g+b\,g\,x} \,d x \] Input:

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(a*g + b*g*x),x)
 

Output:

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(a*g + b*g*x), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a g+b g x} \, dx=\frac {\left (\int \frac {\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right )}{b x +a}d x \right ) b^{2}+\mathrm {log}\left (b x +a \right ) a}{b g} \] Input:

int((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g),x)
 

Output:

(int(log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))/ 
(a + b*x),x)*b**2 + log(a + b*x)*a)/(b*g)