\(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^2} \, dx\) [133]

Optimal result

Integrand size = 34, antiderivative size = 130 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {8 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {4 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \] Output:

-8*B^2*(d*x+c)/(-a*d+b*c)/g^2/(b*x+a)-4*B*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x 
+c)^2))/(-a*d+b*c)/g^2/(b*x+a)-(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/( 
-a*d+b*c)/g^2/(b*x+a)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.46 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.47 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {4 B \left ((b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-d (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{b g^2 (a+b x)} \] Input:

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

Output:

-(((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (4*B*((b*c - a*d)*(A + B*L 
og[(e*(a + b*x)^2)/(c + d*x)^2]) + d*(a + b*x)*Log[a + b*x]*(A + B*Log[(e* 
(a + b*x)^2)/(c + d*x)^2]) - d*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d 
*x)^2])*Log[c + d*x] + 2*B*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + 
b*x)*Log[c + d*x]) - B*d*(a + b*x)*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b* 
(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + B 
*d*(a + b*x)*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + 
 d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d))/(b*g^2*(a 
+ b*x)))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 2742, 2741}

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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2742

\(\displaystyle \frac {4 B \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}}{g^2 (b c-a d)}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {4 B \left (-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}-\frac {2 B (c+d x)}{a+b x}\right )-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}}{g^2 (b c-a d)}\)

Input:

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

Output:

(-(((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(a + b*x)) + 4*B 
*((-2*B*(c + d*x))/(a + b*x) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + 
d*x)^2]))/(a + b*x)))/((b*c - a*d)*g^2)
 

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* 
(p/(m + 1))   Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
 

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])
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.45

method	result	size
norman	\(\frac {\frac {\left (A^{2}+4 B A +8 B^{2}\right ) x}{g a}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {2 c B \left (A +2 B \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}+\frac {2 B d \left (A +2 B \right ) x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}\)	\(188\)
parallelrisch	\(-\frac {2 A^{2} a \,b^{2} d^{2}-2 A^{2} b^{3} c d +16 B^{2} a \,b^{2} d^{2}-16 B^{2} b^{3} c d -4 A B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{2}-4 A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c d -8 A B \,b^{3} c d +8 A B a \,b^{2} d^{2}-2 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{3} d^{2}-8 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{2}-2 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{3} c d -8 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c d}{2 g^{2} \left (b x +a \right ) b^{3} d \left (d a -b c \right )}\)	\(264\)
risch	\(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {4 B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}+\frac {4 B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}+\frac {\frac {4 B A x}{a g}+\frac {2 c A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}+\frac {2 A B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}\)	\(286\)
parts	\(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (d a -b c \right )}+\frac {4 B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}+\frac {4 B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}+\frac {\frac {4 B A x}{a g}+\frac {2 c A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}+\frac {2 A B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}\)	\(286\)
derivativedivides	\(-\frac {-\frac {d^{2} A^{2}}{g^{2} \left (\frac {d a -b c}{d x +c}+b \right ) \left (d a -b c \right )}+\frac {\frac {8 d^{2} B^{2}}{b g \left (d x +c \right )}-\frac {4 d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {\frac {4 d^{2} A B}{b g \left (d x +c \right )}-\frac {2 d^{2} A B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\)	\(306\)
default	\(-\frac {-\frac {d^{2} A^{2}}{g^{2} \left (\frac {d a -b c}{d x +c}+b \right ) \left (d a -b c \right )}+\frac {\frac {8 d^{2} B^{2}}{b g \left (d x +c \right )}-\frac {4 d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {\frac {4 d^{2} A B}{b g \left (d x +c \right )}-\frac {2 d^{2} A B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\)	\(306\)
orering	\(-\frac {\left (b x +a \right ) \left (8 b \,d^{2} x^{2}+a \,d^{2} x +15 b c d x +a c d +7 b \,c^{2}\right ) {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2}}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b g x +a g \right )^{2}}-\frac {\left (b x +a \right )^{2} \left (d x +c \right ) \left (7 b d x +d a +6 b c \right ) \left (\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )^{2}}{\left (b g x +a g \right )^{2} e \left (b x +a \right )^{2}}-\frac {2 {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2} b g}{\left (b g x +a g \right )^{3}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}-\frac {\left (d x +c \right )^{2} \left (b x +a \right )^{3} \left (\frac {2 B^{2} \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right )^{2} \left (d x +c \right )^{4}}{e^{2} \left (b x +a \right )^{4} \left (b g x +a g \right )^{2}}-\frac {8 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )^{2} b g}{\left (b g x +a g \right )^{3} e \left (b x +a \right )^{2}}+\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) B \left (\frac {2 e \,b^{2}}{\left (d x +c \right )^{2}}-\frac {8 e \left (b x +a \right ) b d}{\left (d x +c \right )^{3}}+\frac {6 e \left (b x +a \right )^{2} d^{2}}{\left (d x +c \right )^{4}}\right ) \left (d x +c \right )^{2}}{\left (b g x +a g \right )^{2} e \left (b x +a \right )^{2}}-\frac {4 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )^{2} b}{\left (b g x +a g \right )^{2} e \left (b x +a \right )^{3}}+\frac {4 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right ) d}{\left (b g x +a g \right )^{2} e \left (b x +a \right )^{2}}+\frac {6 {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2} b^{2} g^{2}}{\left (b g x +a g \right )^{4}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}\)	\(789\)

Input:

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

Output:

((A^2+4*A*B+8*B^2)/g/a*x+B^2*c/g/(a*d-b*c)*ln(e*(b*x+a)^2/(d*x+c)^2)^2+B^2 
*d/g/(a*d-b*c)*x*ln(e*(b*x+a)^2/(d*x+c)^2)^2+2*c*B*(A+2*B)/g/(a*d-b*c)*ln( 
e*(b*x+a)^2/(d*x+c)^2)+2*B*d*(A+2*B)/g/(a*d-b*c)*x*ln(e*(b*x+a)^2/(d*x+c)^ 
2))/g/(b*x+a)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.54 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} b c - {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left ({\left (A B + 2 \, B^{2}\right )} b d x + {\left (A B + 2 \, B^{2}\right )} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \] Input:

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

Output:

-((A^2 + 4*A*B + 8*B^2)*b*c - (A^2 + 4*A*B + 8*B^2)*a*d + (B^2*b*d*x + B^2 
*b*c)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))^2 + 2 
*((A*B + 2*B^2)*b*d*x + (A*B + 2*B^2)*b*c)*log((b^2*e*x^2 + 2*a*b*e*x + a^ 
2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((b^3*c - a*b^2*d)*g^2*x + (a*b^2*c - a^2 
*b*d)*g^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (112) = 224\).

Time = 1.22 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=- \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d - \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} + \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} - \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d + \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} - \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} + \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B - 4 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} - 4 A B - 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \] Input:

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

Output:

-4*B*d*(A + 2*B)*log(x + (4*A*B*a*d**2 + 4*A*B*b*c*d + 8*B**2*a*d**2 + 8*B 
**2*b*c*d - 4*B*a**2*d**3*(A + 2*B)/(a*d - b*c) + 8*B*a*b*c*d**2*(A + 2*B) 
/(a*d - b*c) - 4*B*b**2*c**2*d*(A + 2*B)/(a*d - b*c))/(8*A*B*b*d**2 + 16*B 
**2*b*d**2))/(b*g**2*(a*d - b*c)) + 4*B*d*(A + 2*B)*log(x + (4*A*B*a*d**2 
+ 4*A*B*b*c*d + 8*B**2*a*d**2 + 8*B**2*b*c*d + 4*B*a**2*d**3*(A + 2*B)/(a* 
d - b*c) - 8*B*a*b*c*d**2*(A + 2*B)/(a*d - b*c) + 4*B*b**2*c**2*d*(A + 2*B 
)/(a*d - b*c))/(8*A*B*b*d**2 + 16*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + (-2 
*A*B - 4*B**2)*log(e*(a + b*x)**2/(c + d*x)**2)/(a*b*g**2 + b**2*g**2*x) + 
 (B**2*c + B**2*d*x)*log(e*(a + b*x)**2/(c + d*x)**2)**2/(a**2*d*g**2 - a* 
b*c*g**2 + a*b*d*g**2*x - b**2*c*g**2*x) + (-A**2 - 4*A*B - 8*B**2)/(a*b*g 
**2 + b**2*g**2*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (130) = 260\).

Time = 0.08 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.42 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-4 \, {\left ({\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {2}{b^{2} g^{2} x + a b g^{2}} + \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \] Input:

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

Output:

-4*((1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*lo 
g(d*x + c)/((b^2*c - a*b*d)*g^2))*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) 
+ 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 
 ((b*d*x + a*d)*log(b*x + a)^2 + (b*d*x + a*d)*log(d*x + c)^2 - 2*b*c + 2* 
a*d - 2*(b*d*x + a*d)*log(b*x + a) + 2*(b*d*x + a*d - (b*d*x + a*d)*log(b* 
x + a))*log(d*x + c))/(a*b^2*c*g^2 - a^2*b*d*g^2 + (b^3*c*g^2 - a*b^2*d*g^ 
2)*x))*B^2 - 2*A*B*(log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d 
^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(b^2*g^2*x + a* 
b*g^2) + 2/(b^2*g^2*x + a*b*g^2) + 2*d*log(b*x + a)/((b^2*c - a*b*d)*g^2) 
- 2*d*log(d*x + c)/((b^2*c - a*b*d)*g^2)) - B^2*log(b^2*e*x^2/(d^2*x^2 + 2 
*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c 
*d*x + c^2))^2/(b^2*g^2*x + a*b*g^2) - A^2/(b^2*g^2*x + a*b*g^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (130) = 260\).

Time = 0.32 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.92 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (\frac {B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac {B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac {4 \, {\left (A B d + 2 \, B^{2} d\right )} \log \left (\frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac {2 \, {\left (A B + 2 \, B^{2}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac {A^{2} + 4 \, A B + 8 \, B^{2}}{{\left (b g x + a g\right )} b g} \] Input:

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

Output:

-(B^2*d/(b^2*c*g^2 - a*b*d*g^2) + B^2/((b*g*x + a*g)*b*g))*log(b^2*e/(b^2* 
c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g 
*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))^2 
+ 4*(A*B*d + 2*B^2*d)*log(b*c*g/(b*g*x + a*g) - a*d*g/(b*g*x + a*g) + d)/( 
b^2*c*g^2 - a*b*d*g^2) - 2*(A*B + 2*B^2)*log(b^2*e/(b^2*c^2*g^2/(b*g*x + a 
*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b* 
c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))/((b*g*x + a*g)*b*g) 
- (A^2 + 4*A*B + 8*B^2)/((b*g*x + a*g)*b*g)
 

Mupad [B] (verification not implemented)

Time = 26.95 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2+4\,A\,B+8\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {4\,B^2}{b^2\,d\,g^2}+\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+2\,B\right )\,8{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \] Input:

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

Output:

- log((e*(a + b*x)^2)/(c + d*x)^2)^2*(B^2/(b^2*g^2*(x + a/b)) - (B^2*d)/(b 
*g^2*(a*d - b*c))) - (A^2 + 8*B^2 + 4*A*B)/(b^2*g^2*x + a*b*g^2) - (log((e 
*(a + b*x)^2)/(c + d*x)^2)*((4*B^2)/(b^2*d*g^2) + (2*A*B)/(b^2*d*g^2)))/(x 
/d + a/(b*d)) - (B*d*atan(((2*b*d*x + (b^2*c*g^2 + a*b*d*g^2)/(b*g^2))*1i) 
/(a*d - b*c))*(A + 2*B)*8i)/(b*g^2*(a*d - b*c))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.63 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {4 \,\mathrm {log}\left (b x +a \right ) a^{2} b c +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c x +8 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c +8 \,\mathrm {log}\left (b x +a \right ) b^{3} c x -4 \,\mathrm {log}\left (d x +c \right ) a^{2} b c -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c x -8 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c -8 \,\mathrm {log}\left (d x +c \right ) b^{3} c x +\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right )^{2} a \,b^{2} c +\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right )^{2} a \,b^{2} d x +2 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{2} b d x -2 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a \,b^{2} c x +4 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a \,b^{2} d x -4 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) b^{3} c x +a^{3} d x -a^{2} b c x +4 a^{2} b d x -4 a \,b^{2} c x +8 a \,b^{2} d x -8 b^{3} c x}{a \,g^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:

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

Output:

(4*log(a + b*x)*a**2*b*c + 4*log(a + b*x)*a*b**2*c*x + 8*log(a + b*x)*a*b* 
*2*c + 8*log(a + b*x)*b**3*c*x - 4*log(c + d*x)*a**2*b*c - 4*log(c + d*x)* 
a*b**2*c*x - 8*log(c + d*x)*a*b**2*c - 8*log(c + d*x)*b**3*c*x + log((a**2 
*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a*b**2*c + 
log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a* 
b**2*d*x + 2*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**2*b*d*x - 2*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**2*c*x + 4*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**2*d*x - 4*log((a**2*e + 2*a*b*e*x + b**2* 
e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b**3*c*x + a**3*d*x - a**2*b*c*x + 4 
*a**2*b*d*x - 4*a*b**2*c*x + 8*a*b**2*d*x - 8*b**3*c*x)/(a*g**2*(a**2*d - 
a*b*c + a*b*d*x - b**2*c*x))