\(\int \frac {(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^2} \, dx\) [195]

Optimal result

Integrand size = 45, antiderivative size = 500 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^2} \, dx=-\frac {2 A B (b c-a d) g^2 n (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B^2 (b c-a d) g^2 n^2 (a+b x)}{d^2 i^2 (c+d x)}-\frac {2 B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^2 (c+d x)}+\frac {b g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 i^2}+\frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 i^2 (c+d x)}+\frac {2 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i^2}+\frac {2 b B^2 (b c-a d) g^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {4 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}-\frac {4 b B^2 (b c-a d) g^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3522\) vs. \(2(500)=1000\).

Time = 7.26 (sec) , antiderivative size = 3522, normalized size of antiderivative = 7.04 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(b^2*g^2*x*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d 
*x)]))^2)/(d^2*i^2) + (-(A^2*b^2*c^2*g^2) + 2*a*A^2*b*c*d*g^2 - a^2*A^2*d^ 
2*g^2 - 2*A*b^2*B*c^2*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x 
)/(c + d*x)]) + 4*a*A*b*B*c*d*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[ 
(a + b*x)/(c + d*x)]) - 2*a^2*A*B*d^2*g^2*(Log[e*((a + b*x)/(c + d*x))^n] 
- n*Log[(a + b*x)/(c + d*x)]) - b^2*B^2*c^2*g^2*(Log[e*((a + b*x)/(c + d*x 
))^n] - n*Log[(a + b*x)/(c + d*x)])^2 + 2*a*b*B^2*c*d*g^2*(Log[e*((a + b*x 
)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2 - a^2*B^2*d^2*g^2*(Log[e*( 
(a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2)/(d^3*i^2*(c + d*x 
)) - (2*b*(b*c - a*d)*g^2*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[( 
a + b*x)/(c + d*x)]))^2*Log[c + d*x])/(d^3*i^2) + (2*a^2*B*g^2*n*(A + B*(L 
og[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*(((c/d + x)*( 
Log[c/d + x] + Log[c/d + x]^2))/((c + d*x)^2*Log[c/d + x]) + ((d*(a/b + x) 
*Log[a/b + x])/((-c + (a*d)/b)^2*(1 - (d*(a/b + x))/(-c + (a*d)/b))) + Log 
[1 - (d*(a/b + x))/(-c + (a*d)/b)]/(-c + (a*d)/b))/d - (-Log[a/b + x] + Lo 
g[c/d + x] + Log[a/(c + d*x) + (b*x)/(c + d*x)])/(d*(c + d*x))))/i^2 + (2* 
b^2*B*g^2*n*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + 
d*x)]))*(((a/b + x)*(-1 + Log[a/b + x]))/d^2 - ((c/d + x)*(-1 + Log[c/d + 
x]))/d^2 + (c*Log[c/d + x]^2)/d^3 + (c^2*(1 + Log[c/d + x]))/(d^3*(c + d*x 
)) + (c^2*(-(Log[a/b + x]/(d*(c + d*x))) + ((b*Log[a + b*x])/(b*c - a*d...
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

2*A*B*a^2*g^2*n*(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*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^2 + 2*A^2*a*b*g^2 
*(c/(d^3*i^2*x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - 2*A*B*a^2*g^2*log( 
e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^2*i^2*x + c*d*i^2) - A^2*a^2*g^2/(d^ 
2*i^2*x + c*d*i^2) + (B^2*b^2*d^2*g^2*x^2 + B^2*b^2*c*d*g^2*x - (b^2*c^2*g 
^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*B^2 - 2*((b^2*c*d*g^2 - a*b*d^2*g^2)*B^2 
*x + (b^2*c^2*g^2 - a*b*c*d*g^2)*B^2)*log(d*x + c))*log((d*x + c)^n)^2/(d^ 
4*i^2*x + c*d^3*i^2) - integrate(-(B^2*a^2*d^2*g^2*log(e)^2 + (B^2*b^2*d^2 
*g^2*log(e)^2 + 2*A*B*b^2*d^2*g^2*log(e))*x^2 + (B^2*b^2*d^2*g^2*x^2 + 2*B 
^2*a*b*d^2*g^2*x + B^2*a^2*d^2*g^2)*log((b*x + a)^n)^2 + 2*(B^2*a*b*d^2*g^ 
2*log(e)^2 + 2*A*B*a*b*d^2*g^2*log(e))*x + 2*(B^2*a^2*d^2*g^2*log(e) + (B^ 
2*b^2*d^2*g^2*log(e) + A*B*b^2*d^2*g^2)*x^2 + 2*(B^2*a*b*d^2*g^2*log(e) + 
A*B*a*b*d^2*g^2)*x)*log((b*x + a)^n) + 2*((b^2*c^2*g^2*n - 2*a*b*c*d*g^2*n 
 + (g^2*n - g^2*log(e))*a^2*d^2)*B^2 - (A*B*b^2*d^2*g^2 + (g^2*n + g^2*log 
(e))*B^2*b^2*d^2)*x^2 - (2*A*B*a*b*d^2*g^2 + (b^2*c*d*g^2*n + 2*a*b*d^2*g^ 
2*log(e))*B^2)*x + 2*((b^2*c*d*g^2*n - a*b*d^2*g^2*n)*B^2*x + (b^2*c^2*g^2 
*n - a*b*c*d*g^2*n)*B^2)*log(d*x + c) - (B^2*b^2*d^2*g^2*x^2 + 2*B^2*a*b*d 
^2*g^2*x + B^2*a^2*d^2*g^2)*log((b*x + a)^n))*log((d*x + c)^n))/(d^4*i^2*x 
^2 + 2*c*d^3*i^2*x + c^2*d^2*i^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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