\(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^2} \, dx\) [92]

Optimal result

Integrand size = 42, antiderivative size = 722 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {b B^2 (b c-a d)^2 g^3 \log (c+d x)}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \] Output:

2*A*B*(-a*d+b*c)^2*g^3*(b*x+a)/d^3/i^2/(d*x+c)-2*B^2*(-a*d+b*c)^2*g^3*(b*x 
+a)/d^3/i^2/(d*x+c)+2*B^2*(-a*d+b*c)^2*g^3*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/d 
^3/i^2/(d*x+c)-b*B*(-a*d+b*c)*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3/ 
i^2-6*b*B*(-a*d+b*c)^2*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x 
+c)))/d^4/i^2-3*b*(-a*d+b*c)*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^3 
/i^2-(-a*d+b*c)^2*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^3/i^2/(d*x+c 
)+1/2*b^3*g^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i^2-3*b*(-a*d+b* 
c)^2*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i^2+b* 
B^2*(-a*d+b*c)^2*g^3*ln(d*x+c)/d^4/i^2+b*B*(-a*d+b*c)^2*g^3*(A+B*ln(e*(b*x 
+a)/(d*x+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/d^4/i^2-6*b*B^2*(-a*d+b*c)^2*g^3*p 
olylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i^2-6*b*B*(-a*d+b*c)^2*g^3*(A+B*ln(e*(b* 
x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i^2-b*B^2*(-a*d+b*c)^2*g 
^3*polylog(2,b*(d*x+c)/d/(b*x+a))/d^4/i^2+6*b*B^2*(-a*d+b*c)^2*g^3*polylog 
(3,d*(b*x+a)/b/(d*x+c))/d^4/i^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5199\) vs. \(2(722)=1444\).

Time = 8.08 (sec) , antiderivative size = 5199, normalized size of antiderivative = 7.20 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2795, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {6 b B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}-\frac {3 b \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^4}-\frac {6 b B \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}+\frac {b B \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}-\frac {3 b (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {b B (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 (c+d x)}+\frac {2 A B (a+b x)}{d^3 (c+d x)}-\frac {6 b B^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}-\frac {b B^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4}+\frac {6 b B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}-\frac {b B^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 (c+d x)}-\frac {2 B^2 (a+b x)}{d^3 (c+d x)}\right )}{i^2}\)

Input:

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

Output:

((b*c - a*d)^2*g^3*((2*A*B*(a + b*x))/(d^3*(c + d*x)) - (2*B^2*(a + b*x))/ 
(d^3*(c + d*x)) + (2*B^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*(c + 
 d*x)) - (b*B*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*(c + d* 
x)*(b - (d*(a + b*x))/(c + d*x))) - ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c 
 + d*x)])^2)/(d^3*(c + d*x)) + (b^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 
)/(2*d^4*(b - (d*(a + b*x))/(c + d*x))^2) - (3*b*(a + b*x)*(A + B*Log[(e*( 
a + b*x))/(c + d*x)])^2)/(d^3*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - ( 
b*B^2*Log[b - (d*(a + b*x))/(c + d*x)])/d^4 - (6*b*B*(A + B*Log[(e*(a + b* 
x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^4 - (3*b*(A + B*Lo 
g[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^4 + 
(b*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b* 
x))])/d^4 - (6*b*B^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^4 - (6*b*B 
*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x 
))])/d^4 - (b*B^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/d^4 + (6*b*B^2* 
PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/d^4))/i^2
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

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 [F]

\[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**3*( - 2*int((log((a*e + b*e*x)/(c + d*x))**2*x**3)/(c**2 + 2*c*d*x + d 
**2*x**2),x)*a*b**5*c**2*d**5 - 2*int((log((a*e + b*e*x)/(c + d*x))**2*x** 
3)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b**5*c*d**6*x + 2*int((log((a*e + b*e 
*x)/(c + d*x))**2*x**3)/(c**2 + 2*c*d*x + d**2*x**2),x)*b**6*c**3*d**4 + 2 
*int((log((a*e + b*e*x)/(c + d*x))**2*x**3)/(c**2 + 2*c*d*x + d**2*x**2),x 
)*b**6*c**2*d**5*x - 6*int((log((a*e + b*e*x)/(c + d*x))**2*x**2)/(c**2 + 
2*c*d*x + d**2*x**2),x)*a**2*b**4*c**2*d**5 - 6*int((log((a*e + b*e*x)/(c 
+ d*x))**2*x**2)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**2*b**4*c*d**6*x + 6*in 
t((log((a*e + b*e*x)/(c + d*x))**2*x**2)/(c**2 + 2*c*d*x + d**2*x**2),x)*a 
*b**5*c**3*d**4 + 6*int((log((a*e + b*e*x)/(c + d*x))**2*x**2)/(c**2 + 2*c 
*d*x + d**2*x**2),x)*a*b**5*c**2*d**5*x - 6*int((log((a*e + b*e*x)/(c + d* 
x))**2*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**3*b**3*c**2*d**5 - 6*int((log 
((a*e + b*e*x)/(c + d*x))**2*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**3*b**3* 
c*d**6*x + 6*int((log((a*e + b*e*x)/(c + d*x))**2*x)/(c**2 + 2*c*d*x + d** 
2*x**2),x)*a**2*b**4*c**3*d**4 + 6*int((log((a*e + b*e*x)/(c + d*x))**2*x) 
/(c**2 + 2*c*d*x + d**2*x**2),x)*a**2*b**4*c**2*d**5*x - 4*int((log((a*e + 
 b*e*x)/(c + d*x))*x**3)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**2*b**4*c**2*d* 
*5 - 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(c**2 + 2*c*d*x + d**2*x**2 
),x)*a**2*b**4*c*d**6*x + 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(c**2 
+ 2*c*d*x + d**2*x**2),x)*a*b**5*c**3*d**4 + 4*int((log((a*e + b*e*x)/(...