\(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3} \, dx\) [61]

Optimal result

Integrand size = 40, antiderivative size = 141 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {B^2 i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {B i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.98 (sec) , antiderivative size = 765, normalized size of antiderivative = 5.43 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {i \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 B d (a+b x) \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 b^2 (b c-a d) g^3 (a+b x)^2} \] Input:

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

Output:

-1/4*(i*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 4*d*(-(b 
*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 4*B*d*(a + b 
*x)*(2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d*(a + b*x)*Lo 
g[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*d*(a + b*x)*(A + B*Log 
[(e*(a + b*x))/(c + d*x)])*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)] - Lo 
g[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B*( 
2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d*(-(b*c) + a*d)* 
(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*d^2*(a + b*x)^2*Log[a + 
 b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d^2*(a + b*x)^2*(A + B*Log[ 
(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x)*(b*c - a*d + d*(a 
 + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*((b*c - a*d)^2 + 2*d* 
(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x 
)^2*Log[c + d*x]) + 2*B*d^2*(a + b*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Lo 
g[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] 
) - 2*B*d^2*(a + b*x)^2*((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^2*(b*c - 
 a*d)*g^3*(a + b*x)^2)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 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.

\(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^3} \, dx\)

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2741

\(\displaystyle \frac {i \left (B \left (-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}\right )-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}\right )}{g^3 (b c-a d)}\)

Input:

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

Output:

(i*(-1/2*((c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x)^2 
+ B*(-1/4*(B*(c + d*x)^2)/(a + b*x)^2 - ((c + d*x)^2*(A + B*Log[(e*(a + b* 
x))/(c + d*x)]))/(2*(a + b*x)^2))))/((b*c - a*d)*g^3)
 

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 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(135)=270\).

Time = 1.47 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.34

method	result	size
parts	\(\frac {i \,A^{2} \left (-\frac {-d a +b c}{2 b^{2} \left (b x +a \right )^{2}}-\frac {d}{b^{2} \left (b x +a \right )}\right )}{g^{3}}-\frac {i \,B^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{g^{3} \left (d a -b c \right )}-\frac {2 i A B \,e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{g^{3} \left (d a -b c \right )}\)	\(330\)
norman	\(\frac {\frac {B^{2} c d i x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (d a -b c \right )}+\frac {c i B d \left (2 A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (d a -b c \right )}-\frac {2 A^{2} a i d +2 A^{2} b c i +2 a d i B A +2 b c i B A +a d i \,B^{2}+b c i \,B^{2}}{4 g \,b^{2}}-\frac {\left (2 A^{2} i d +2 d i B A +d i \,B^{2}\right ) x}{2 g b}+\frac {B^{2} i \,c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (d a -b c \right )}+\frac {B^{2} d^{2} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (d a -b c \right ) g}+\frac {\left (2 A +B \right ) c^{2} i B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (d a -b c \right )}+\frac {d^{2} i B \left (2 A +B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (d a -b c \right )}}{\left (b x +a \right )^{2} g^{2}}\)	\(336\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e \,A^{2}}{2 \left (d a -b c \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 i \,d^{2} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}+\frac {i \,d^{2} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}\right )}{d^{2}}\)	\(355\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e \,A^{2}}{2 \left (d a -b c \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 i \,d^{2} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}+\frac {i \,d^{2} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}\right )}{d^{2}}\)	\(355\)
risch	\(\frac {i \,A^{2} d a}{2 g^{3} b^{2} \left (b x +a \right )^{2}}-\frac {i \,A^{2} c}{2 g^{3} b \left (b x +a \right )^{2}}-\frac {i \,A^{2} d}{g^{3} b^{2} \left (b x +a \right )}+\frac {i \,B^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,B^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,B^{2} e^{2}}{4 g^{3} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {i A B \,e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {i A B \,e^{2}}{2 g^{3} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}\)	\(426\)
parallelrisch	\(-\frac {2 A^{2} a^{2} b^{2} d^{3} i +B^{2} a^{2} b^{2} d^{3} i -B^{2} b^{4} c^{2} d i +4 A^{2} x a \,b^{3} d^{3} i -4 A^{2} x \,b^{4} c \,d^{2} i +2 B^{2} x a \,b^{3} d^{3} i -2 B^{2} x \,b^{4} c \,d^{2} i -2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{2} d i -8 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{2} i -4 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{3} i -4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{4} c \,d^{2} i -4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{2} i +4 A B x a \,b^{3} d^{3} i -4 A B x \,b^{4} c \,d^{2} i -4 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{2} d i -2 A B \,b^{4} c^{2} d i -2 A^{2} c^{2} i \,b^{4} d +2 A B \,a^{2} b^{2} d^{3} i -2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{4} d^{3} i -2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{3} i -2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{4} c^{2} d i}{4 g^{3} \left (b x +a \right )^{2} b^{4} d \left (d a -b c \right )}\)	\(457\)
orering	\(\text {Expression too large to display}\)	\(1126\)

Input:

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x,method=_RETU 
RNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (135) = 270\).

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.05 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {2 \, {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b d^{2}\right )} i x + 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} i + 2 \, {\left ({\left (2 \, A B + B^{2}\right )} b^{2} d^{2} i x^{2} + 2 \, {\left (2 \, A B + B^{2}\right )} b^{2} c d i x + {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algo 
rithm="fricas")
 

Output:

-1/4*(2*((2*A^2 + 2*A*B + B^2)*b^2*c*d - (2*A^2 + 2*A*B + B^2)*a*b*d^2)*i* 
x + 2*(B^2*b^2*d^2*i*x^2 + 2*B^2*b^2*c*d*i*x + B^2*b^2*c^2*i)*log((b*e*x + 
 a*e)/(d*x + c))^2 + ((2*A^2 + 2*A*B + B^2)*b^2*c^2 - (2*A^2 + 2*A*B + B^2 
)*a^2*d^2)*i + 2*((2*A*B + B^2)*b^2*d^2*i*x^2 + 2*(2*A*B + B^2)*b^2*c*d*i* 
x + (2*A*B + B^2)*b^2*c^2*i)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c - a*b^4 
*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (122) = 244\).

Time = 5.15 (sec) , antiderivative size = 714, normalized size of antiderivative = 5.06 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=- \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i - \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} + \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} - \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i + \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} - \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} + \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {\left (B^{2} c^{2} i + 2 B^{2} c d i x + B^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d g^{3} - 2 a^{2} b c g^{3} + 4 a^{2} b d g^{3} x - 4 a b^{2} c g^{3} x + 2 a b^{2} d g^{3} x^{2} - 2 b^{3} c g^{3} x^{2}} + \frac {- 2 A^{2} a d i - 2 A^{2} b c i - 2 A B a d i - 2 A B b c i - B^{2} a d i - B^{2} b c i + x \left (- 4 A^{2} b d i - 4 A B b d i - 2 B^{2} b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- 2 A B a d i - 2 A B b c i - 4 A B b d i x - B^{2} a d i - B^{2} b c i - 2 B^{2} b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \] Input:

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**3,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1987 vs. \(2 (135) = 270\).

Time = 0.17 (sec) , antiderivative size = 1987, normalized size of antiderivative = 14.09 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algo 
rithm="maxima")
 

Output:

-1/2*(2*b*x + a)*B^2*d*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^4*g^3*x 
^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) + 1/4*(2*((2*b*d*x - b*c + 3*a*d)/((b^4* 
c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b* 
d)*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2 
*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3))*log(b*e*x/(d* 
x + c) + a*e/(d*x + c)) - (b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2 + 2*(b^2*d^2*x^ 
2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + 
 a^2*d^2)*log(d*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a* 
b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d 
^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))/( 
a^2*b^3*c^2*g^3 - 2*a^3*b^2*c*d*g^3 + a^4*b*d^2*g^3 + (b^5*c^2*g^3 - 2*a*b 
^4*c*d*g^3 + a^2*b^3*d^2*g^3)*x^2 + 2*(a*b^4*c^2*g^3 - 2*a^2*b^3*c*d*g^3 + 
 a^3*b^2*d^2*g^3)*x))*B^2*c*i - 1/4*(2*((3*a*b*c - a^2*d + 2*(2*b^2*c - a* 
b*d)*x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2* 
b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a 
*b^3*c*d + a^2*b^2*d^2)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 
- 2*a*b^3*c*d + a^2*b^2*d^2)*g^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 
(7*a*b^2*c^2 - 8*a^2*b*c*d + a^3*d^2 - 2*(2*a^2*b*c*d - a^3*d^2 + (2*b^3*c 
*d - a*b^2*d^2)*x^2 + 2*(2*a*b^2*c*d - a^2*b*d^2)*x)*log(b*x + a)^2 - 2*(2 
*a^2*b*c*d - a^3*d^2 + (2*b^3*c*d - a*b^2*d^2)*x^2 + 2*(2*a*b^2*c*d - a...
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.46 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B^{2} e^{3} i \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )}^{2} g^{3}} + \frac {2 \, {\left (2 \, A B e^{3} i + B^{2} e^{3} i\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3}} + \frac {{\left (2 \, A^{2} e^{3} i + 2 \, A B e^{3} i + B^{2} e^{3} i\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algo 
rithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 28.20 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.33 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B+b\,d\,i\,B^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i}{2}+\frac {B^2\,b\,c\,i}{2}+A\,B\,a\,d\,i+A\,B\,b\,c\,i}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b^2\,g^3}+\frac {B^2\,a\,d\,i}{2\,b^3\,g^3}+\frac {B^2\,d\,i\,x}{b^2\,g^3}}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {B^2\,i}{b^2\,g^3}+\frac {2\,A\,B\,i}{b^2\,g^3}\right )+\frac {A\,B\,a\,i}{b^3\,g^3}+\frac {B\,i\,\left (A\,b\,c-B\,a\,d+B\,b\,c\right )}{b^3\,d\,g^3}+\frac {B^2\,d^2\,i\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b^2\,g^3\,\left (a\,d-b\,c\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,b^3\,g^3+2\,a\,d\,b^2\,g^3}{2\,b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+B\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 684, normalized size of antiderivative = 4.85 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {i \left (2 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} c^{2}+4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} c^{2}-2 a^{4} c d +2 a^{3} b \,c^{2}+2 a^{3} b \,d^{2} x^{2}+2 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} d^{2} x^{2}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} c d -2 a \,b^{3} c d \,x^{2}+2 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c d +2 \,\mathrm {log}\left (b x +a \right ) b^{4} c d \,x^{2}-4 \,\mathrm {log}\left (d x +c \right ) a^{3} b c d -2 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c d -2 \,\mathrm {log}\left (d x +c \right ) b^{4} c d \,x^{2}-4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b c d +4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} d^{2} x^{2}+2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} d^{2} x^{2}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} c d \,x^{2}-2 a^{2} b^{2} c d \,x^{2}+8 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c d x +4 \,\mathrm {log}\left (b x +a \right ) a^{3} b c d +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c d \,x^{2}+4 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c d x -8 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c d x -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c d \,x^{2}-4 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c d x +4 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} c d x -4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c d \,x^{2}-b^{4} c d \,x^{2}+2 a^{2} b^{2} c^{2}+a \,b^{3} c^{2}+2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c^{2}-2 a^{3} b c d -a^{2} b^{2} c d +2 a^{2} b^{2} d^{2} x^{2}+a \,b^{3} d^{2} x^{2}\right )}{4 a b \,g^{3} \left (a \,b^{2} d \,x^{2}-b^{3} c \,x^{2}+2 a^{2} b d x -2 a \,b^{2} c x +a^{3} d -a^{2} b c \right )} \] Input:

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

Output:

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