\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) [46]

Optimal result

Integrand size = 40, antiderivative size = 457 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {B d^4 (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {2 b B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2} \] Output:

-B*d^4*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-6*b^2*B*d^2*(d*x+c)/(-a*d+b*c) 
^5/g^4/i^2/(b*x+a)+b^3*B*d*(d*x+c)^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-1/9*b^ 
4*B*(d*x+c)^3/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3+2*b*B*d^3*ln((b*x+a)/(d*x+c)) 
^2/(-a*d+b*c)^5/g^4/i^2+d^4*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c) 
^5/g^4/i^2/(d*x+c)-6*b^2*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c 
)^5/g^4/i^2/(b*x+a)+2*b^3*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b* 
c)^5/g^4/i^2/(b*x+a)^2-1/3*b^4*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d 
+b*c)^5/g^4/i^2/(b*x+a)^3-4*b*d^3*ln((b*x+a)/(d*x+c))*(A+B*ln(e*(b*x+a)/(d 
*x+c)))/(-a*d+b*c)^5/g^4/i^2
 

Mathematica [C] (verified)

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

Time = 1.23 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {\frac {b B (b c-a d)^3}{(a+b x)^3}-\frac {6 b B d (b c-a d)^2}{(a+b x)^2}+\frac {27 b^2 B c d^2}{a+b x}-\frac {27 a b B d^3}{a+b x}+\frac {12 b B d^2 (b c-a d)}{a+b x}-\frac {9 b B c d^3}{c+d x}+\frac {9 a B d^4}{c+d x}+30 b B d^3 \log (a+b x)+\frac {3 b (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}-\frac {9 b d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {27 b d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {9 d^3 (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+36 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-30 b B d^3 \log (c+d x)-36 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-18 b B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+18 b B d^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{9 (b c-a d)^5 g^4 i^2} \] Input:

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

Output:

-1/9*((b*B*(b*c - a*d)^3)/(a + b*x)^3 - (6*b*B*d*(b*c - a*d)^2)/(a + b*x)^ 
2 + (27*b^2*B*c*d^2)/(a + b*x) - (27*a*b*B*d^3)/(a + b*x) + (12*b*B*d^2*(b 
*c - a*d))/(a + b*x) - (9*b*B*c*d^3)/(c + d*x) + (9*a*B*d^4)/(c + d*x) + 3 
0*b*B*d^3*Log[a + b*x] + (3*b*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + 
d*x)]))/(a + b*x)^3 - (9*b*d*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d 
*x)]))/(a + b*x)^2 + (27*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d 
*x)]))/(a + b*x) - (9*d^3*(-(b*c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x 
)]))/(c + d*x) + 36*b*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
) - 30*b*B*d^3*Log[c + d*x] - 36*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
)*Log[c + d*x] - 18*b*B*d^3*(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)]) + 18*b*B*d 
^3*((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)^5*g^4*i^2)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 2772, 2009}

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.

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

\(\Big \downarrow \) 2962

\(\displaystyle \frac {\int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 i^2 (b c-a d)^5}\)

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B \int \left (d^4-\frac {4 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) d^3}{a+b x}-\frac {6 b^2 (c+d x)^2 d^2}{(a+b x)^2}+\frac {2 b^3 (c+d x)^3 d}{(a+b x)^3}-\frac {b^4 (c+d x)^4}{3 (a+b x)^4}\right )d\frac {a+b x}{c+d x}-\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-B \left (\frac {b^4 (c+d x)^3}{9 (a+b x)^3}-\frac {b^3 d (c+d x)^2}{(a+b x)^2}+\frac {6 b^2 d^2 (c+d x)}{a+b x}+\frac {d^4 (a+b x)}{c+d x}-2 b d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{g^4 i^2 (b c-a d)^5}\)

Input:

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

Output:

(-(B*((d^4*(a + b*x))/(c + d*x) + (6*b^2*d^2*(c + d*x))/(a + b*x) - (b^3*d 
*(c + d*x)^2)/(a + b*x)^2 + (b^4*(c + d*x)^3)/(9*(a + b*x)^3) - 2*b*d^3*Lo 
g[(a + b*x)/(c + d*x)]^2)) + (d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + 
d*x)]))/(c + d*x) - (6*b^2*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x 
)]))/(a + b*x) + (2*b^3*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) 
)/(a + b*x)^2 - (b^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3* 
(a + b*x)^3) - 4*b*d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/( 
c + d*x)]))/((b*c - a*d)^5*g^4*i^2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

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 [A] (verified)

Time = 3.40 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.40

method	result	size
parts	\(\frac {A \left (-\frac {d^{3}}{\left (d a -b c \right )^{4} \left (d x +c \right )}-\frac {4 d^{3} b \ln \left (d x +c \right )}{\left (d a -b c \right )^{5}}-\frac {b}{3 \left (d a -b c \right )^{2} \left (b x +a \right )^{3}}+\frac {4 d^{3} b \ln \left (b x +a \right )}{\left (d a -b c \right )^{5}}-\frac {3 b \,d^{2}}{\left (d a -b c \right )^{4} \left (b x +a \right )}-\frac {b d}{\left (d a -b c \right )^{3} \left (b x +a \right )^{2}}\right )}{g^{4} i^{2}}-\frac {B \left (\frac {d^{4} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{4}}-\frac {2 b \,d^{3} e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\left (d a -b c \right )^{4}}+\frac {6 b^{2} d^{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 )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4}}-\frac {4 b^{3} d \,e^{3} \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 )^{4}}+\frac {b^{4} e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4}}\right )}{g^{4} i^{2} \left (d a -b c \right ) e}\)	\(640\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e^{2} A \,b^{4}}{3 i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {2 d^{3} e A \,b^{3}}{i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {6 d^{4} A \,b^{2}}{i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {4 d^{5} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{6} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{2} e^{2} B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i^{2} \left (d a -b c \right )^{6} g^{4}}-\frac {4 d^{3} e B \,b^{3} \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 )}{i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {6 d^{4} B \,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 )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{6} g^{4}}-\frac {2 d^{5} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{6} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{6} g^{4}}\right )}{d^{2}}\)	\(803\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e^{2} A \,b^{4}}{3 i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {2 d^{3} e A \,b^{3}}{i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {6 d^{4} A \,b^{2}}{i^{2} \left (d a -b c \right )^{6} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {4 d^{5} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{6} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{2} e^{2} B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i^{2} \left (d a -b c \right )^{6} g^{4}}-\frac {4 d^{3} e B \,b^{3} \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 )}{i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {6 d^{4} B \,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 )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{6} g^{4}}-\frac {2 d^{5} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{6} g^{4}}+\frac {d^{6} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{6} g^{4}}\right )}{d^{2}}\)	\(803\)
risch	\(-\frac {A \,d^{3}}{g^{4} i^{2} \left (d a -b c \right )^{4} \left (d x +c \right )}-\frac {4 A \,d^{3} b \ln \left (d x +c \right )}{g^{4} i^{2} \left (d a -b c \right )^{5}}-\frac {A b}{3 g^{4} i^{2} \left (d a -b c \right )^{2} \left (b x +a \right )^{3}}+\frac {4 A \,d^{3} b \ln \left (b x +a \right )}{g^{4} i^{2} \left (d a -b c \right )^{5}}-\frac {3 A b \,d^{2}}{g^{4} i^{2} \left (d a -b c \right )^{4} \left (b x +a \right )}-\frac {A b d}{g^{4} i^{2} \left (d a -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b}{g^{4} i^{2} \left (d a -b c \right )^{5}}-\frac {B \,d^{4} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (d x +c \right )}+\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b c}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (d x +c \right )}+\frac {B \,d^{4} a}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (d x +c \right )}-\frac {B \,d^{3} b c}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (d x +c \right )}+\frac {B b \,d^{3}}{g^{4} i^{2} \left (d a -b c \right )^{5}}+\frac {2 B b \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{4} i^{2} \left (d a -b c \right )^{5}}+\frac {6 B e \,b^{2} d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {6 B e \,b^{2} d^{2}}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {2 B \,e^{2} b^{3} d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}-\frac {B \,e^{2} b^{3} d}{g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {B \,e^{3} b^{4} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}+\frac {B \,e^{3} b^{4}}{9 g^{4} i^{2} \left (d a -b c \right )^{5} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}\)	\(918\)
norman	\(\text {Expression too large to display}\)	\(1722\)
parallelrisch	\(\text {Expression too large to display}\)	\(1730\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x,method=_RETU 
RNVERBOSE)
 

Output:

A/g^4/i^2*(-d^3/(a*d-b*c)^4/(d*x+c)-4*d^3/(a*d-b*c)^5*b*ln(d*x+c)-1/3*b/(a 
*d-b*c)^2/(b*x+a)^3+4*d^3/(a*d-b*c)^5*b*ln(b*x+a)-3*b/(a*d-b*c)^4*d^2/(b*x 
+a)-b/(a*d-b*c)^3*d/(b*x+a)^2)-B/g^4/i^2/(a*d-b*c)/e*(d^4/(a*d-b*c)^4*((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))-(a*d-b*c)*e/d/( 
d*x+c)-b*e/d)-2/(a*d-b*c)^4*b*d^3*e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+6/(a 
*d-b*c)^4*b^2*d^2*e^2*(-1/(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))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))-4/(a*d-b*c)^4*b^3*d*e^3*(-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)+1/(a*d-b*c)^4*b^4*e^4*(-1/3/(b*e/d+(a*d-b*c) 
*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))^3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (453) = 906\).

Time = 0.10 (sec) , antiderivative size = 1019, normalized size of antiderivative = 2.23 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algo 
rithm="fricas")
 

Output:

-1/9*((3*A + B)*b^4*c^4 - 9*(2*A + B)*a*b^3*c^3*d + 54*(A + B)*a^2*b^2*c^2 
*d^2 - 5*(6*A + 11*B)*a^3*b*c*d^3 - 9*(A - B)*a^4*d^4 + 6*((6*A + 5*B)*b^4 
*c*d^3 - (6*A + 5*B)*a*b^3*d^4)*x^3 + 3*((6*A + 11*B)*b^4*c^2*d^2 + 8*(3*A 
 + B)*a*b^3*c*d^3 - (30*A + 19*B)*a^2*b^2*d^4)*x^2 + 18*(B*b^4*d^4*x^4 + B 
*a^3*b*c*d^3 + (B*b^4*c*d^3 + 3*B*a*b^3*d^4)*x^3 + 3*(B*a*b^3*c*d^3 + B*a^ 
2*b^2*d^4)*x^2 + (3*B*a^2*b^2*c*d^3 + B*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d 
*x + c))^2 - ((6*A + 5*B)*b^4*c^3*d - 27*(2*A + 3*B)*a*b^3*c^2*d^2 - 3*(6* 
A - 19*B)*a^2*b^2*c*d^3 + (66*A + 19*B)*a^3*b*d^4)*x + 3*(2*(6*A + 5*B)*b^ 
4*d^4*x^4 + B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*B*a^2*b^2*c^2*d^2 + 12*A*a^3* 
b*c*d^3 - 3*B*a^4*d^4 + 2*((6*A + 11*B)*b^4*c*d^3 + 9*(2*A + B)*a*b^3*d^4) 
*x^3 + 6*(B*b^4*c^2*d^2 + 3*(2*A + 3*B)*a*b^3*c*d^3 + 6*A*a^2*b^2*d^4)*x^2 
 - 2*(B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2 - 18*(A + B)*a^2*b^2*c*d^3 - 6*(A - 
B)*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^8*c^5*d - 5*a*b^7*c^4*d 
^2 + 10*a^2*b^6*c^3*d^3 - 10*a^3*b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d 
^6)*g^4*i^2*x^4 + (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 + 20*a^3*b^ 
5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*g^4*i^2 
*x^3 + 3*(a*b^7*c^6 - 4*a^2*b^6*c^5*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2* 
d^4 + 4*a^6*b^2*c*d^5 - a^7*b*d^6)*g^4*i^2*x^2 + (3*a^2*b^6*c^6 - 14*a^3*b 
^5*c^5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*a^6*b^2*c^2*d^4 + 2 
*a^7*b*c*d^5 - a^8*d^6)*g^4*i^2*x + (a^3*b^5*c^6 - 5*a^4*b^4*c^5*d + 10...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4/(d*i*x+c*i)**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2560 vs. \(2 (453) = 906\).

Time = 0.27 (sec) , antiderivative size = 2560, normalized size of antiderivative = 5.60 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algo 
rithm="maxima")
 

Output:

-1/3*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3 
*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11 
*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b 
^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c 
^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^ 
3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 
 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4* 
c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^ 
4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2 
*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4 
*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^ 
4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3* 
d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(b*e*x/(d 
*x + c) + a*e/(d*x + c)) - 1/3*A*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2* 
d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3* 
c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6 
*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - 
 a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 
 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c 
^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 +...
 

Giac [A] (verification not implemented)

Time = 64.85 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} e^{4} - \frac {3 \, {\left (b e x + a e\right )} B b d e^{3}}{d x + c} + \frac {3 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, A b^{2} e^{4} + 2 \, B b^{2} e^{4} - \frac {18 \, {\left (b e x + a e\right )} A b d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B b d e^{3}}{d x + c} + \frac {18 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algo 
rithm="giac")
 

Output:

-1/18*(6*(B*b^2*e^4 - 3*(b*e*x + a*e)*B*b*d*e^3/(d*x + c) + 3*(b*e*x + a*e 
)^2*B*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b 
^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4*i^2/(d*x + c)^3 
 + (b*e*x + a*e)^3*a^2*d^2*g^4*i^2/(d*x + c)^3) + (6*A*b^2*e^4 + 2*B*b^2*e 
^4 - 18*(b*e*x + a*e)*A*b*d*e^3/(d*x + c) - 9*(b*e*x + a*e)*B*b*d*e^3/(d*x 
 + c) + 18*(b*e*x + a*e)^2*A*d^2*e^2/(d*x + c)^2 + 18*(b*e*x + a*e)^2*B*d^ 
2*e^2/(d*x + c)^2)/((b*e*x + a*e)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*e*x 
 + a*e)^3*a*b*c*d*g^4*i^2/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4*i^2/(d 
*x + c)^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)))^2
 

Mupad [B] (verification not implemented)

Time = 34.90 (sec) , antiderivative size = 1679, normalized size of antiderivative = 3.67 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^4*(c*i + d*i*x)^2) 
,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 2879, normalized size of antiderivative = 6.30 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x)
 

Output:

( - 108*log(a + b*x)*a**5*b*c*d**4 - 108*log(a + b*x)*a**5*b*d**5*x - 36*l 
og(a + b*x)*a**4*b**2*c**2*d**3 - 360*log(a + b*x)*a**4*b**2*c*d**4*x - 54 
*log(a + b*x)*a**4*b**2*c*d**4 - 324*log(a + b*x)*a**4*b**2*d**5*x**2 - 54 
*log(a + b*x)*a**4*b**2*d**5*x - 108*log(a + b*x)*a**3*b**3*c**2*d**3*x - 
66*log(a + b*x)*a**3*b**3*c**2*d**3 - 432*log(a + b*x)*a**3*b**3*c*d**4*x* 
*2 - 228*log(a + b*x)*a**3*b**3*c*d**4*x - 324*log(a + b*x)*a**3*b**3*d**5 
*x**3 - 162*log(a + b*x)*a**3*b**3*d**5*x**2 - 108*log(a + b*x)*a**2*b**4* 
c**2*d**3*x**2 - 198*log(a + b*x)*a**2*b**4*c**2*d**3*x - 216*log(a + b*x) 
*a**2*b**4*c*d**4*x**3 - 360*log(a + b*x)*a**2*b**4*c*d**4*x**2 - 108*log( 
a + b*x)*a**2*b**4*d**5*x**4 - 162*log(a + b*x)*a**2*b**4*d**5*x**3 - 36*l 
og(a + b*x)*a*b**5*c**2*d**3*x**3 - 198*log(a + b*x)*a*b**5*c**2*d**3*x**2 
 - 36*log(a + b*x)*a*b**5*c*d**4*x**4 - 252*log(a + b*x)*a*b**5*c*d**4*x** 
3 - 54*log(a + b*x)*a*b**5*d**5*x**4 - 66*log(a + b*x)*b**6*c**2*d**3*x**3 
 - 66*log(a + b*x)*b**6*c*d**4*x**4 + 108*log(c + d*x)*a**5*b*c*d**4 + 108 
*log(c + d*x)*a**5*b*d**5*x + 36*log(c + d*x)*a**4*b**2*c**2*d**3 + 360*lo 
g(c + d*x)*a**4*b**2*c*d**4*x + 54*log(c + d*x)*a**4*b**2*c*d**4 + 324*log 
(c + d*x)*a**4*b**2*d**5*x**2 + 54*log(c + d*x)*a**4*b**2*d**5*x + 108*log 
(c + d*x)*a**3*b**3*c**2*d**3*x + 66*log(c + d*x)*a**3*b**3*c**2*d**3 + 43 
2*log(c + d*x)*a**3*b**3*c*d**4*x**2 + 228*log(c + d*x)*a**3*b**3*c*d**4*x 
 + 324*log(c + d*x)*a**3*b**3*d**5*x**3 + 162*log(c + d*x)*a**3*b**3*d*...