\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2} \, dx\) [69]

Optimal result

Integrand size = 42, antiderivative size = 442 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {2 B^2 (b c-a d) i^2 (c+d x)}{b^2 g^2 (a+b x)}-\frac {2 B (b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {2 B d (b c-a d) i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {d^2 i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac {(b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {2 B^2 d (b c-a d) i^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac {4 B d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {4 B^2 d (b c-a d) i^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2649\) vs. \(2(442)=884\).

Time = 4.61 (sec) , antiderivative size = 2649, normalized size of antiderivative = 5.99 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 366, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Fricas [F]

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

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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