\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2} \, dx\) [79]

Optimal result

Integrand size = 42, antiderivative size = 692 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {2 B^2 (b c-a d)^2 i^3 (c+d x)}{b^3 g^2 (a+b x)}-\frac {B d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {2 B (b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {4 B d (b c-a d)^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {2 d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}-\frac {(b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}+\frac {B^2 d (b c-a d)^2 i^3 \log (c+d x)}{b^4 g^2}+\frac {B d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}-\frac {3 d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {4 B^2 d (b c-a d)^2 i^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g^2}-\frac {B^2 d (b c-a d)^2 i^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {6 B d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {6 B^2 d (b c-a d)^2 i^3 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5266\) vs. \(2(692)=1384\).

Time = 7.58 (sec) , antiderivative size = 5266, normalized size of antiderivative = 7.61 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.83, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(i*( - 2*int((log((a*e + b*e*x)/(c + d*x))**2*x**3)/(a**2 + 2*a*b*x + b**2 
*x**2),x)*a**3*b**6*d**4 + 2*int((log((a*e + b*e*x)/(c + d*x))**2*x**3)/(a 
**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**7*c*d**3 - 2*int((log((a*e + b*e*x)/ 
(c + d*x))**2*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**7*d**4*x + 2*i 
nt((log((a*e + b*e*x)/(c + d*x))**2*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)* 
a*b**8*c*d**3*x - 6*int((log((a*e + b*e*x)/(c + d*x))**2*x**2)/(a**2 + 2*a 
*b*x + b**2*x**2),x)*a**3*b**6*c*d**3 + 6*int((log((a*e + b*e*x)/(c + d*x) 
)**2*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**7*c**2*d**2 - 6*int((lo 
g((a*e + b*e*x)/(c + d*x))**2*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b 
**7*c*d**3*x + 6*int((log((a*e + b*e*x)/(c + d*x))**2*x**2)/(a**2 + 2*a*b* 
x + b**2*x**2),x)*a*b**8*c**2*d**2*x - 6*int((log((a*e + b*e*x)/(c + d*x)) 
**2*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**3*b**6*c**2*d**2 + 6*int((log((a 
*e + b*e*x)/(c + d*x))**2*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**7*c** 
3*d - 6*int((log((a*e + b*e*x)/(c + d*x))**2*x)/(a**2 + 2*a*b*x + b**2*x** 
2),x)*a**2*b**7*c**2*d**2*x + 6*int((log((a*e + b*e*x)/(c + d*x))**2*x)/(a 
**2 + 2*a*b*x + b**2*x**2),x)*a*b**8*c**3*d*x - 4*int((log((a*e + b*e*x)/( 
c + d*x))*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**4*b**5*d**4 + 4*int((lo 
g((a*e + b*e*x)/(c + d*x))*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**3*b**6 
*c*d**3 - 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**2 + 2*a*b*x + b**2 
*x**2),x)*a**3*b**6*d**4*x + 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/...