\(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^2} \, dx\) [194]

Optimal result

Integrand size = 45, antiderivative size = 770 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 A B (b c-a d)^2 g^3 n (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 n^2 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^2}+\frac {b B^2 (b c-a d)^2 g^3 n^2 \log (c+d x)}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 (b c-a d)^2 g^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \] Output:

2*A*B*(-a*d+b*c)^2*g^3*n*(b*x+a)/d^3/i^2/(d*x+c)-2*B^2*(-a*d+b*c)^2*g^3*n^ 
2*(b*x+a)/d^3/i^2/(d*x+c)+2*B^2*(-a*d+b*c)^2*g^3*n*(b*x+a)*ln(e*((b*x+a)/( 
d*x+c))^n)/d^3/i^2/(d*x+c)-b*B*(-a*d+b*c)*g^3*n*(b*x+a)*(A+B*ln(e*((b*x+a) 
/(d*x+c))^n))/d^3/i^2-3*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^2-(-a*d+b*c)^2*g^3*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^ 
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))^n))^2/ 
d^4/i^2-6*b*B*(-a*d+b*c)^2*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^2-3*b*(-a*d+b*c)^2*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^2+b*B^2*(-a*d+b*c)^2*g^3*n^2*ln(d*x+c) 
/d^4/i^2+b*B*(-a*d+b*c)^2*g^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*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*n^2*polylog(2,d*(b*x+a)/b 
/(d*x+c))/d^4/i^2-6*b*B*(-a*d+b*c)^2*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^2-b*B^2*(-a*d+b*c)^2*g^3*n^2*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*n^2*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. \(5850\) vs. \(2(770)=1540\).

Time = 7.73 (sec) , antiderivative size = 5850, normalized size of antiderivative = 7.60 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 640, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

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)^2,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)^2,x)
 

Fricas [F]

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

Output:

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

Giac [F(-1)]

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

Output:

Timed out
 

Mupad [F(-1)]

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

Reduce [F]

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

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

Output:

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