\(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^3} \, dx\) [202]

Optimal result

Integrand size = 45, antiderivative size = 676 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {B^2 (b c-a d) g^3 n^2 (a+b x)^2}{4 d^2 i^3 (c+d x)^2}-\frac {4 A b B (b c-a d) g^3 n (a+b x)}{d^3 i^3 (c+d x)}+\frac {4 b B^2 (b c-a d) g^3 n^2 (a+b x)}{d^3 i^3 (c+d x)}-\frac {4 b B^2 (b c-a d) g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}-\frac {B (b c-a d) g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^2 i^3 (c+d x)^2}+\frac {b^2 g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^3}+\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^2 i^3 (c+d x)^2}+\frac {2 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^3 (c+d x)}+\frac {2 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^3}+\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^3}+\frac {2 b^2 B^2 (b c-a d) g^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac {6 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}-\frac {6 b^2 B^2 (b c-a d) g^3 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6878\) vs. \(2(676)=1352\).

Time = 7.79 (sec) , antiderivative size = 6878, normalized size of antiderivative = 10.17 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**3*i*(4*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c**3 + 3*c**2 
*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**5*c**4*d**6 + 8*int((log(((a 
+ b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d 
**3*x**3),x)*a**2*b**5*c**3*d**7*x + 4*int((log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**5 
*c**2*d**8*x**2 - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c**3 
 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b**6*c**5*d**5 - 16*int((l 
og(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x 
**2 + d**3*x**3),x)*a*b**6*c**4*d**6*x - 8*int((log(((a + b*x)**n*e)/(c + 
d*x)**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b** 
6*c**3*d**7*x**2 + 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c** 
3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**7*c**6*d**4 + 8*int((log 
(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x** 
2 + d**3*x**3),x)*b**7*c**5*d**5*x + 4*int((log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*x**3)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**7*c**4 
*d**6*x**2 + 12*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3 
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**3*b**4*c**4*d**6 + 24*int((lo 
g(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x* 
*2 + d**3*x**3),x)*a**3*b**4*c**3*d**7*x + 12*int((log(((a + b*x)**n*e)/(c 
 + d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)...