\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4} \, dx\) [8]

Optimal result

Integrand size = 33, antiderivative size = 183 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=-\frac {B n}{9 b g^4 (a+b x)^3}+\frac {B d n}{6 b (b c-a d) g^4 (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b g^4 (a+b x)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3 g^4} \] Output:

-1/9*B*n/b/g^4/(b*x+a)^3+1/6*B*d*n/b/(-a*d+b*c)/g^4/(b*x+a)^2-1/3*B*d^2*n/ 
b/(-a*d+b*c)^2/g^4/(b*x+a)-1/3*B*d^3*n*ln(b*x+a)/b/(-a*d+b*c)^3/g^4-1/3*(A 
+B*ln(e*((b*x+a)/(d*x+c))^n))/b/g^4/(b*x+a)^3+1/3*B*d^3*n*ln(d*x+c)/b/(-a* 
d+b*c)^3/g^4
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=-\frac {6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {B n \left ((b c-a d) \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )}{(b c-a d)^3}}{18 b g^4 (a+b x)^3} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4,x]
 

Output:

-1/18*(6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + (B*n*((b*c - a*d)*(11*a^ 
2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + 6*d^3 
*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]))/(b*c - a*d)^3 
)/(b*g^4*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2947, 27, 54, 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.

\(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^4} \, dx\)

\(\Big \downarrow \) 2947

\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{g^3 (a+b x)^4 (c+d x)}dx}{3 b g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 b g^4 (a+b x)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{(a+b x)^4 (c+d x)}dx}{3 b g^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 b g^4 (a+b x)^3}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {B n (b c-a d) \int \left (\frac {d^4}{(b c-a d)^4 (c+d x)}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b}{(b c-a d) (a+b x)^4}\right )dx}{3 b g^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 b g^4 (a+b x)^3}\)

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4,x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 2947

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + 
 B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d) 
/(g*(m + 1)))   Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; Free 
Q[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] 
&& NeQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(171)=342\).

Time = 10.96 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.40

method	result	size
parallelrisch	\(-\frac {-18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} d^{4} n -18 B x a \,b^{6} c \,d^{3} n^{2}-18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{5} c \,d^{3} n +18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} c^{2} d^{2} n -18 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{5} d^{4} n +11 B \,a^{3} b^{4} d^{4} n^{2}-2 B \,b^{7} c^{3} d \,n^{2}+6 A \,a^{3} b^{4} d^{4} n -6 A \,b^{7} c^{3} d n -6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} c^{3} d n -6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} d^{4} n +6 B \,x^{2} a \,b^{6} d^{4} n^{2}-6 B \,x^{2} b^{7} c \,d^{3} n^{2}+15 B x \,a^{2} b^{5} d^{4} n^{2}+3 B x \,b^{7} c^{2} d^{2} n^{2}-18 B \,a^{2} b^{5} c \,d^{3} n^{2}+9 B a \,b^{6} c^{2} d^{2} n^{2}-18 A \,a^{2} b^{5} c \,d^{3} n +18 A a \,b^{6} c^{2} d^{2} n}{18 g^{4} \left (b x +a \right )^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n \,b^{5} d}\)	\(440\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/18*(-18*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*d^4*n-18*B*x*a*b^6*c*d^3* 
n^2-18*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*c*d^3*n+18*B*ln(e*((b*x+a)/(d*x 
+c))^n)*a*b^6*c^2*d^2*n-18*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*d^4*n+11* 
B*a^3*b^4*d^4*n^2-2*B*b^7*c^3*d*n^2+6*A*a^3*b^4*d^4*n-6*A*b^7*c^3*d*n-6*B* 
ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^3*d*n-6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^ 
7*d^4*n+6*B*x^2*a*b^6*d^4*n^2-6*B*x^2*b^7*c*d^3*n^2+15*B*x*a^2*b^5*d^4*n^2 
+3*B*x*b^7*c^2*d^2*n^2-18*B*a^2*b^5*c*d^3*n^2+9*B*a*b^6*c^2*d^2*n^2-18*A*a 
^2*b^5*c*d^3*n+18*A*a*b^6*c^2*d^2*n)/g^4/(b*x+a)^3/(a^3*d^3-3*a^2*b*c*d^2+ 
3*a*b^2*c^2*d-b^3*c^3)/n/b^5/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (171) = 342\).

Time = 0.11 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.63 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=-\frac {6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right ) + 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{18 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="fri 
cas")
 

Output:

-1/18*(6*A*b^3*c^3 - 18*A*a*b^2*c^2*d + 18*A*a^2*b*c*d^2 - 6*A*a^3*d^3 + 6 
*(B*b^3*c*d^2 - B*a*b^2*d^3)*n*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5* 
B*a^2*b*d^3)*n*x + (2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 - 11* 
B*a^3*d^3)*n + 6*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2 - B*a^3*d^ 
3)*log(e) + 6*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n*x + 
 (B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*n)*log((b*x + a)/(d*x + c 
)))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3 
*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3 
*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + ( 
a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (171) = 342\).

Time = 0.05 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.36 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=-\frac {1}{18} \, B n {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {A}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="max 
ima")
 

Output:

-1/18*B*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c* 
d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^ 
5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3* 
c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) 
+ 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d 
^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 
- a^3*b*d^3)*g^4)) - 1/3*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^4*g^4 
*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A/(b^4*g^4*x^3 
 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (171) = 342\).

Time = 0.49 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.08 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=-\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} n - \frac {3 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, B b^{2} n - \frac {9 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 6 \, B b^{2} \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B b d \log \left (e\right )}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} - \frac {18 \, {\left (b x + a\right )} A b d}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="gia 
c")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 26.32 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.91 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=\frac {2\,A\,a\,c\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2\,n}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{3\,b\,g^4\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,n\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d\,n}{18\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2\,n}{18\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))/(a*g + b*g*x)^4,x)
 

Output:

(2*A*a*c*d)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (A*b*c^2)/(3*g^4*(a*d - b* 
c)^2*(a + b*x)^3) - (A*a^2*d^2)/(3*b*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*b 
*c^2*n)/(9*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*d^3*n*atan((a*d*1i + b*c*1i 
 + b*d*x*2i)/(a*d - b*c))*2i)/(3*b*g^4*(a*d - b*c)^3) - (B*log(e*((a + b*x 
)/(c + d*x))^n))/(3*b*g^4*(a + b*x)^3) - (B*b*d^2*n*x^2)/(3*g^4*(a*d - b*c 
)^2*(a + b*x)^3) + (7*B*a*c*d*n)/(18*g^4*(a*d - b*c)^2*(a + b*x)^3) - (11* 
B*a^2*d^2*n)/(18*b*g^4*(a*d - b*c)^2*(a + b*x)^3) - (5*B*a*d^2*n*x)/(6*g^4 
*(a*d - b*c)^2*(a + b*x)^3) + (B*b*c*d*n*x)/(6*g^4*(a*d - b*c)^2*(a + b*x) 
^3)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 631, normalized size of antiderivative = 3.45 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4} \, dx=\frac {18 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} d^{3} n x +18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} d^{3} n \,x^{2}+6 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} d^{3} n \,x^{3}-18 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} d^{3} n x -18 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} d^{3} n \,x^{2}-6 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} d^{3} n \,x^{3}+12 a^{2} b^{3} c \,d^{2} n x -3 a \,b^{4} c^{2} d n x -6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{4} b \,d^{3}+6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{4} c^{3}+18 a^{4} b c \,d^{2}-9 a^{4} b \,d^{3} n -18 a^{3} b^{2} c^{2} d +2 a \,b^{4} c^{3} n -6 a^{5} d^{3}+6 a^{2} b^{3} c^{3}+6 \,\mathrm {log}\left (b x +a \right ) a^{4} b \,d^{3} n -6 \,\mathrm {log}\left (d x +c \right ) a^{4} b \,d^{3} n +18 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{3} b^{2} c \,d^{2}-18 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{3} c^{2} d +16 a^{3} b^{2} c \,d^{2} n -9 a^{3} b^{2} d^{3} n x -9 a^{2} b^{3} c^{2} d n +2 a \,b^{4} d^{3} n \,x^{3}-2 b^{5} c \,d^{2} n \,x^{3}}{18 a b \,g^{4} \left (a^{3} b^{3} d^{3} x^{3}-3 a^{2} b^{4} c \,d^{2} x^{3}+3 a \,b^{5} c^{2} d \,x^{3}-b^{6} c^{3} x^{3}+3 a^{4} b^{2} d^{3} x^{2}-9 a^{3} b^{3} c \,d^{2} x^{2}+9 a^{2} b^{4} c^{2} d \,x^{2}-3 a \,b^{5} c^{3} x^{2}+3 a^{5} b \,d^{3} x -9 a^{4} b^{2} c \,d^{2} x +9 a^{3} b^{3} c^{2} d x -3 a^{2} b^{4} c^{3} x +a^{6} d^{3}-3 a^{5} b c \,d^{2}+3 a^{4} b^{2} c^{2} d -a^{3} b^{3} c^{3}\right )} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)
 

Output:

(6*log(a + b*x)*a**4*b*d**3*n + 18*log(a + b*x)*a**3*b**2*d**3*n*x + 18*lo 
g(a + b*x)*a**2*b**3*d**3*n*x**2 + 6*log(a + b*x)*a*b**4*d**3*n*x**3 - 6*l 
og(c + d*x)*a**4*b*d**3*n - 18*log(c + d*x)*a**3*b**2*d**3*n*x - 18*log(c 
+ d*x)*a**2*b**3*d**3*n*x**2 - 6*log(c + d*x)*a*b**4*d**3*n*x**3 - 6*log(( 
(a + b*x)**n*e)/(c + d*x)**n)*a**4*b*d**3 + 18*log(((a + b*x)**n*e)/(c + d 
*x)**n)*a**3*b**2*c*d**2 - 18*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3 
*c**2*d + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4*c**3 - 6*a**5*d**3 + 
 18*a**4*b*c*d**2 - 9*a**4*b*d**3*n - 18*a**3*b**2*c**2*d + 16*a**3*b**2*c 
*d**2*n - 9*a**3*b**2*d**3*n*x + 6*a**2*b**3*c**3 - 9*a**2*b**3*c**2*d*n + 
 12*a**2*b**3*c*d**2*n*x + 2*a*b**4*c**3*n - 3*a*b**4*c**2*d*n*x + 2*a*b** 
4*d**3*n*x**3 - 2*b**5*c*d**2*n*x**3)/(18*a*b*g**4*(a**6*d**3 - 3*a**5*b*c 
*d**2 + 3*a**5*b*d**3*x + 3*a**4*b**2*c**2*d - 9*a**4*b**2*c*d**2*x + 3*a* 
*4*b**2*d**3*x**2 - a**3*b**3*c**3 + 9*a**3*b**3*c**2*d*x - 9*a**3*b**3*c* 
d**2*x**2 + a**3*b**3*d**3*x**3 - 3*a**2*b**4*c**3*x + 9*a**2*b**4*c**2*d* 
x**2 - 3*a**2*b**4*c*d**2*x**3 - 3*a*b**5*c**3*x**2 + 3*a*b**5*c**2*d*x**3 
 - b**6*c**3*x**3))