\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{a g+b g x} \, dx\) [101]

Optimal result

Integrand size = 32, antiderivative size = 128 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \] Output:

-(A+B*ln(e*(b*x+a)/(d*x+c)))^2*ln(1-b*(d*x+c)/d/(b*x+a))/b/g+2*B*(A+B*ln(e 
*(b*x+a)/(d*x+c)))*polylog(2,b*(d*x+c)/d/(b*x+a))/b/g+2*B^2*polylog(3,b*(d 
*x+c)/d/(b*x+a))/b/g
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(458\) vs. \(2(128)=256\).

Time = 1.47 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.58 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=\frac {3 A^2 \log (a+b x)+3 A B \left (\log ^2\left (\frac {a}{b}+x\right )-2 \log (a+b x) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B^2 \left (\log ^3\left (\frac {a}{b}+x\right )+3 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+3 \log (a+b x) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 \log ^2\left (\frac {a}{b}+x\right ) \left (-\log \left (\frac {c}{d}+x\right )+\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+6 \log \left (\frac {a}{b}+x\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+6 \log \left (\frac {c}{d}+x\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )-3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \left (\log ^2\left (\frac {a}{b}+x\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )-6 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-6 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )\right )}{3 b g} \] Input:

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

Output:

(3*A^2*Log[a + b*x] + 3*A*B*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x] 
 - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)]) - 2*(Log[c/d + x]*Log[(d*( 
a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B^2* 
(Log[a/b + x]^3 + 3*Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 3*L 
og[a + b*x]*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^ 
2 + 3*Log[a/b + x]^2*(-Log[c/d + x] + Log[(b*(c + d*x))/(b*c - a*d)]) + 6* 
Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 6*Log[c/d + x]*Pol 
yLog[2, (b*(c + d*x))/(b*c - a*d)] - 3*(Log[a/b + x] - Log[c/d + x] - Log[ 
(e*(a + b*x))/(c + d*x)])*(Log[a/b + x]^2 - 2*(Log[c/d + x]*Log[(d*(a + b* 
x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - 6*PolyLog[ 
3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] 
))/(3*b*g)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2950, 2779, 2821, 7143}

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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {2 B \left (\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+B \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b}}{g}\)

Input:

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

Output:

(-(((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b 
*x))])/b) + (2*B*((A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + 
d*x))/(d*(a + b*x))] + B*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))]))/b)/g
 

Defintions of rubi rules used

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] 
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(128)=256\).

Time = 2.16 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.62

method	result	size
parts	\(\frac {A^{2} \ln \left (b x +a \right )}{g b}-\frac {B^{2} e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{b e}\right )}{g}-\frac {2 A B \left (d a -b c \right ) e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{2}}{2 \left (d a -b c \right ) b e}+\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{3}}{\left (d a -b c \right ) b e}\right )}{g \,d^{2}}\)	\(464\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e}\right )}{g \left (d a -b c \right )}-\frac {d^{2} B^{2} \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 b e}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{b e}\right )}{g \left (d a -b c \right )}-\frac {2 d^{2} A B \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b e}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{b e}\right )}{g \left (d a -b c \right )}\right )}{d^{2}}\)	\(552\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e}\right )}{g \left (d a -b c \right )}-\frac {d^{2} B^{2} \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 b e}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{b e}\right )}{g \left (d a -b c \right )}-\frac {2 d^{2} A B \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b e}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{b e}\right )}{g \left (d a -b c \right )}\right )}{d^{2}}\)	\(552\)
risch	\(\frac {A^{2} \ln \left (b x +a \right )}{g b}+\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g b}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{g b}-\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{g b}+\frac {2 B^{2} \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{g b}+\frac {A B d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} a}{g \left (d a -b c \right ) b}-\frac {A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} c}{g \left (d a -b c \right )}-\frac {2 A B d \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 ) a}{g \left (d a -b c \right ) b}+\frac {2 A B \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 ) c}{g \left (d a -b c \right )}-\frac {2 A B d \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 ) a}{g \left (d a -b c \right ) b}+\frac {2 A B \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 ) c}{g \left (d a -b c \right )}\)	\(670\)

Input:

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

Output:

1/g*A^2*ln(b*x+a)/b-B^2/g*e*(-1/3/b/e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3+1/ 
b/e*(ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(1-1/b/e*d*(b*e/d+(a*d-b*c)*e/d/( 
d*x+c)))+2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*polylog(2,1/b/e*d*(b*e/d+(a*d-b 
*c)*e/d/(d*x+c)))-2*polylog(3,1/b/e*d*(b*e/d+(a*d-b*c)*e/d/(d*x+c)))))-2*A 
*B/g/d^2*(a*d-b*c)*e*(-1/2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(a*d-b*c)*d^2 
/b/e+(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)/(a*d-b*c 
)*d^3/b/e)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=\frac {\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2}}{b x +a}d x \right ) b^{3}+2 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )}{b x +a}d x \right ) a \,b^{2}+\mathrm {log}\left (b x +a \right ) a^{2}}{b g} \] Input:

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

Output:

(int(log((a*e + b*e*x)/(c + d*x))**2/(a + b*x),x)*b**3 + 2*int(log((a*e + 
b*e*x)/(c + d*x))/(a + b*x),x)*a*b**2 + log(a + b*x)*a**2)/(b*g)