\(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(c i+d i x)^2} \, dx\) [143]

Optimal result

Integrand size = 43, antiderivative size = 359 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^2} \, dx=\frac {3 B (b c-a d)^2 g^3 n (a+b x)}{d^3 i^2 (c+d x)}-\frac {(b c-a d)^2 g^3 (6 A+5 B n) (a+b x)}{2 d^3 i^2 (c+d x)}-\frac {3 B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^2 (c+d x)}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d i^2 (c+d x)}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^2 i^2 (c+d x)}-\frac {b (b c-a d)^2 g^3 \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 d^4 i^2}-\frac {3 b B (b c-a d)^2 g^3 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \] Output:

3*B*(-a*d+b*c)^2*g^3*n*(b*x+a)/d^3/i^2/(d*x+c)-1/2*(-a*d+b*c)^2*g^3*(5*B*n 
+6*A)*(b*x+a)/d^3/i^2/(d*x+c)-3*B*(-a*d+b*c)^2*g^3*(b*x+a)*ln(e*((b*x+a)/( 
d*x+c))^n)/d^3/i^2/(d*x+c)+1/2*g^3*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n 
))/d/i^2/(d*x+c)-1/2*(-a*d+b*c)*g^3*(b*x+a)^2*(3*A+B*n+3*B*ln(e*((b*x+a)/( 
d*x+c))^n))/d^2/i^2/(d*x+c)-1/2*b*(-a*d+b*c)^2*g^3*(6*A+5*B*n+6*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*B*(-a*d+b*c)^2*g^3 
*n*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i^2
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.04 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^2} \, dx=\frac {g^3 \left (-2 A b^2 d (2 b c-3 a d) x-2 b B d (2 b c-3 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^3 d^2 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {2 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+2 b B (2 b c-3 a d) (b c-a d) n \log (c+d x)+6 b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-2 B (b c-a d)^2 n \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )+b B n \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-3 b B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^4 i^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2784, 2784, 2793, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{i^2}\)

\(\Big \downarrow \) 2793

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \left (-\frac {6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d}-\frac {b \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{i^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {b \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d^2}-\frac {(a+b x) (6 A+5 B n)}{d (c+d x)}-\frac {6 b B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2}-\frac {6 B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d (c+d x)}+\frac {6 B n (a+b x)}{d (c+d x)}}{d}}{2 d}\right )}{i^2}\)

Input:

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

Output:

((b*c - a*d)^2*g^3*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/( 
2*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)^2*(3*A + B* 
n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*(c + d*x)^2*(b - (d*(a + b*x)) 
/(c + d*x))) - ((6*B*n*(a + b*x))/(d*(c + d*x)) - ((6*A + 5*B*n)*(a + b*x) 
)/(d*(c + d*x)) - (6*B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d*(c + d 
*x)) - (b*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a 
 + b*x))/(b*(c + d*x))])/d^2 - (6*b*B*n*PolyLog[2, (d*(a + b*x))/(b*(c + d 
*x))])/d^2)/d)/(2*d)))/i^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], 
 (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, 
 f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer 
Q[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]

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

Input:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1892 vs. \(2 (350) = 700\).

Time = 0.37 (sec) , antiderivative size = 1892, normalized size of antiderivative = 5.27 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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*b^3*g^3 
 - 3*A*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*a^2*b*g^3*(c/(d^3*i^2*x + c*d^2*i^2) + log(d*x + c)/ 
(d^2*i^2)) - 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*a^3*g^3/(d^2*i^2*x + c*d*i^2) - 1/2*(6*a^3*b*d^3*g^3*log(e) 
- (7*g^3*n + 6*g^3*log(e))*b^4*c^3 + (17*g^3*n + 18*g^3*log(e))*a*b^3*c^2* 
d - 6*(2*g^3*n + 3*g^3*log(e))*a^2*b^2*c*d^2)*B*log(d*x + c)/(b*c*d^4*i^2 
- a*d^5*i^2) + 1/2*((b^4*c*d^3*g^3*log(e) - a*b^3*d^4*g^3*log(e))*B*x^3 - 
((g^3*n + 3*g^3*log(e))*b^4*c^2*d^2 - (2*g^3*n + 9*g^3*log(e))*a*b^3*c*d^3 
 + (g^3*n + 6*g^3*log(e))*a^2*b^2*d^4)*B*x^2 - ((g^3*n + 4*g^3*log(e))*b^4 
*c^3*d - 2*(g^3*n + 5*g^3*log(e))*a*b^3*c^2*d^2 + (g^3*n + 6*g^3*log(e))*a 
^2*b^2*c*d^3)*B*x - 6*((b^4*c^3*d*g^3*n - 3*a*b^3*c^2*d^2*g^3*n + 3*a^2*b^ 
2*c*d^3*g^3*n - a^3*b*d^4*g^3*n)*B*x + (b^4*c^4*g^3*n - 3*a*b^3*c^3*d*g^3* 
n + 3*a^2*b^2*c^2*d^2*g^3*n - a^3*b*c*d^3*g^3*n)*B)*log(b*x + a)*log(d*x + 
 c) + 3*((b^4*c^3*d*g^3*n - 3*a*b^3*c^2*d^2*g^3*n + 3*a^2*b^2*c*d^3*g^3*n 
- a^3*b*d^4*g^3*n)*B*x + (b^4*c^4*g^3*n - 3*a*b^3*c^3*d*g^3*n + 3*a^2*b^2* 
c^2*d^2*g^3*n - a^3*b*c*d^3*g^3*n)*B)*log(d*x + c)^2 - 2*((g^3*n - g^3*log 
(e))*b^4*c^4 - 4*(g^3*n - g^3*log(e))*a*b^3*c^3*d + 6*(g^3*n - g^3*log(...
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(g**3*( - 2*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*b**4*c**2*d**5 - 2*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*b**4*c*d**6*x + 2*int((log(((a 
+ b*x)**n*e)/(c + d*x)**n)*x**3)/(c**2 + 2*c*d*x + d**2*x**2),x)*b**5*c**3 
*d**4 + 2*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**3)/(c**2 + 2*c*d*x + 
d**2*x**2),x)*b**5*c**2*d**5*x - 6*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**2*d**5 - 6*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**6*x + 6*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**4 + 6*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**5*x - 6*int((l 
og(((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**5 - 6*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**6*x + 6*int((log(((a + b*x)**n*e)/(c + 
d*x)**n)*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**2*b**3*c**3*d**4 + 6*int((l 
og(((a + b*x)**n*e)/(c + d*x)**n)*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**2* 
b**3*c**2*d**5*x + 2*log(a + b*x)*a**4*b*c*d**4*n + 2*log(a + b*x)*a**4*b* 
d**5*n*x - 6*log(c + d*x)*a**4*b*c**2*d**3 - 2*log(c + d*x)*a**4*b*c*d**4* 
n - 6*log(c + d*x)*a**4*b*c*d**4*x - 2*log(c + d*x)*a**4*b*d**5*n*x + 18*l 
og(c + d*x)*a**3*b**2*c**3*d**2 + 18*log(c + d*x)*a**3*b**2*c**2*d**3*x...