\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(g+h x)^4} \, dx\) [301]

Optimal result

Integrand size = 31, antiderivative size = 284 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx=-\frac {B (b c-a d) n}{6 (b g-a h) (d g-c h) (g+h x)^2}-\frac {B (b c-a d) (2 b d g-b c h-a d h) n}{3 (b g-a h)^2 (d g-c h)^2 (g+h x)}+\frac {b^3 B n \log (a+b x)}{3 h (b g-a h)^3}-\frac {B d^3 n \log (c+d x)}{3 h (d g-c h)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}+\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log (g+h x)}{3 (b g-a h)^3 (d g-c h)^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.96 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx=-\frac {\frac {2 A}{(g+h x)^3}+\frac {2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}+B (b c-a d) n \left (\frac {h}{(b g-a h) (d g-c h) (g+h x)^2}-\frac {2 h (-2 b d g+b c h+a d h)}{(b g-a h)^2 (d g-c h)^2 (g+h x)}-\frac {2 b^3 \log (a+b x)}{(b c-a d) (b g-a h)^3}+\frac {2 d^3 \log (c+d x)}{(b c-a d) (d g-c h)^3}-\frac {2 h \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) \log (g+h x)}{(b g-a h)^3 (d g-c h)^3}\right )}{6 h} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2948, 93, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 93

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 93

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre 
eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
 

rule 2009

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

rule 2948

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3285\) vs. \(2(272)=544\).

Time = 99.20 (sec) , antiderivative size = 3286, normalized size of antiderivative = 11.57

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(3286\)
risch	\(\text {Expression too large to display}\)	\(9645\)

Input:

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

Output:

-1/6*(-2*B*ln(h*x+g)*x^3*b^4*c^3*d*h^8*n^2+6*B*x^2*ln(e*(b*x+a)^n/((d*x+c) 
^n))*a^3*b*d^4*g*h^7*n-18*B*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^2*d^4*g^ 
2*h^6*n+18*B*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^3*d^4*g^3*h^5*n-6*B*x^2*a 
^2*b^2*c*d^3*g*h^7*n^2+6*B*x^2*a*b^3*c^2*d^2*g*h^7*n^2+6*B*x*ln(e*(b*x+a)^ 
n/((d*x+c)^n))*a^3*b*d^4*g^2*h^6*n-18*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2* 
b^2*d^4*g^3*h^5*n-6*B*ln(h*x+g)*x^2*b^4*c^3*d*g*h^7*n^2+18*B*ln(h*x+g)*x^2 
*b^4*c^2*d^2*g^2*h^6*n^2-18*B*ln(h*x+g)*x^2*b^4*c*d^3*g^3*h^5*n^2-6*B*ln(b 
*x+a)*x*a^3*b*d^4*g^2*h^6*n^2+18*B*ln(b*x+a)*x*a^2*b^2*d^4*g^3*h^5*n^2-18* 
B*ln(b*x+a)*x*a*b^3*d^4*g^4*h^4*n^2-15*B*x*a^2*b^2*c*d^3*g^2*h^6*n^2-6*B*x 
*a*b^3*c^3*d*g*h^7*n^2+15*B*x*a*b^3*c^2*d^2*g^2*h^6*n^2-6*B*ln(e*(b*x+a)^n 
/((d*x+c)^n))*a^3*b*c^2*d^2*g*h^7*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b* 
c*d^3*g^2*h^6*n-6*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^2*c^3*d*g*h^7*n+18*B 
*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^2*c^2*d^2*g^2*h^6*n-18*B*ln(e*(b*x+a)^n 
/((d*x+c)^n))*a^2*b^2*c*d^3*g^3*h^5*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^ 
3*c^3*d*g^2*h^6*n-18*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^3*c^2*d^2*g^3*h^5*n 
-2*B*ln(b*x+a)*a^3*b*d^4*g^3*h^5*n^2+6*B*ln(b*x+a)*a^2*b^2*d^4*g^4*h^4*n^2 
-6*B*ln(b*x+a)*a*b^3*d^4*g^5*h^3*n^2+2*B*ln(b*x+a)*b^4*c^3*d*g^3*h^5*n^2-6 
*B*ln(b*x+a)*b^4*c^2*d^2*g^4*h^4*n^2+6*B*ln(b*x+a)*b^4*c*d^3*g^5*h^3*n^2+2 
*B*ln(h*x+g)*a^3*b*d^4*g^3*h^5*n^2-6*B*ln(h*x+g)*a^2*b^2*d^4*g^4*h^4*n^2+6 
*B*ln(h*x+g)*a*b^3*d^4*g^5*h^3*n^2-2*B*ln(h*x+g)*b^4*c^3*d*g^3*h^5*n^2+...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (272) = 544\).

Time = 0.11 (sec) , antiderivative size = 920, normalized size of antiderivative = 3.24 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1530 vs. \(2 (272) = 544\).

Time = 0.56 (sec) , antiderivative size = 1530, normalized size of antiderivative = 5.39 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*B*b^4*n*log(abs(b*x + a))/(b^4*g^3*h - 3*a*b^3*g^2*h^2 + 3*a^2*b^2*g*h 
^3 - a^3*b*h^4) - 1/3*B*d^4*n*log(abs(d*x + c))/(d^4*g^3*h - 3*c*d^3*g^2*h 
^2 + 3*c^2*d^2*g*h^3 - c^3*d*h^4) - 1/3*B*n*log(b*x + a)/(h^4*x^3 + 3*g*h^ 
3*x^2 + 3*g^2*h^2*x + g^3*h) + 1/3*B*n*log(d*x + c)/(h^4*x^3 + 3*g*h^3*x^2 
 + 3*g^2*h^2*x + g^3*h) + 1/3*(3*B*b^3*c*d^2*g^2*n - 3*B*a*b^2*d^3*g^2*n - 
 3*B*b^3*c^2*d*g*h*n + 3*B*a^2*b*d^3*g*h*n + B*b^3*c^3*h^2*n - B*a^3*d^3*h 
^2*n)*log(h*x + g)/(b^3*d^3*g^6 - 3*b^3*c*d^2*g^5*h - 3*a*b^2*d^3*g^5*h + 
3*b^3*c^2*d*g^4*h^2 + 9*a*b^2*c*d^2*g^4*h^2 + 3*a^2*b*d^3*g^4*h^2 - b^3*c^ 
3*g^3*h^3 - 9*a*b^2*c^2*d*g^3*h^3 - 9*a^2*b*c*d^2*g^3*h^3 - a^3*d^3*g^3*h^ 
3 + 3*a*b^2*c^3*g^2*h^4 + 9*a^2*b*c^2*d*g^2*h^4 + 3*a^3*c*d^2*g^2*h^4 - 3* 
a^2*b*c^3*g*h^5 - 3*a^3*c^2*d*g*h^5 + a^3*c^3*h^6) - 1/6*(4*B*b^2*c*d*g*h^ 
3*n*x^2 - 4*B*a*b*d^2*g*h^3*n*x^2 - 2*B*b^2*c^2*h^4*n*x^2 + 2*B*a^2*d^2*h^ 
4*n*x^2 + 9*B*b^2*c*d*g^2*h^2*n*x - 9*B*a*b*d^2*g^2*h^2*n*x - 5*B*b^2*c^2* 
g*h^3*n*x + 5*B*a^2*d^2*g*h^3*n*x + B*a*b*c^2*h^4*n*x - B*a^2*c*d*h^4*n*x 
+ 5*B*b^2*c*d*g^3*h*n - 5*B*a*b*d^2*g^3*h*n - 3*B*b^2*c^2*g^2*h^2*n + 3*B* 
a^2*d^2*g^2*h^2*n + B*a*b*c^2*g*h^3*n - B*a^2*c*d*g*h^3*n + 2*B*b^2*d^2*g^ 
4*log(e) - 4*B*b^2*c*d*g^3*h*log(e) - 4*B*a*b*d^2*g^3*h*log(e) + 2*B*b^2*c 
^2*g^2*h^2*log(e) + 8*B*a*b*c*d*g^2*h^2*log(e) + 2*B*a^2*d^2*g^2*h^2*log(e 
) - 4*B*a*b*c^2*g*h^3*log(e) - 4*B*a^2*c*d*g*h^3*log(e) + 2*B*a^2*c^2*h^4* 
log(e) + 2*A*b^2*d^2*g^4 - 4*A*b^2*c*d*g^3*h - 4*A*a*b*d^2*g^3*h + 2*A*...
 

Mupad [B] (verification not implemented)

Time = 30.99 (sec) , antiderivative size = 1183, normalized size of antiderivative = 4.17 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(B*d^3*n*log(c + d*x))/(3*c^3*h^4 - 3*d^3*g^3*h + 9*c*d^2*g^2*h^2 - 9*c^2* 
d*g*h^3) - (log(g + h*x)*(h^2*(B*a^3*d^3*n - B*b^3*c^3*n) - h*(3*B*a^2*b*d 
^3*g*n - 3*B*b^3*c^2*d*g*n) + 3*B*a*b^2*d^3*g^2*n - 3*B*b^3*c*d^2*g^2*n))/ 
(3*a^3*c^3*h^6 + 3*b^3*d^3*g^6 - 3*a^3*d^3*g^3*h^3 - 3*b^3*c^3*g^3*h^3 - 9 
*a^2*b*c^3*g*h^5 - 9*a*b^2*d^3*g^5*h - 9*a^3*c^2*d*g*h^5 - 9*b^3*c*d^2*g^5 
*h + 9*a*b^2*c^3*g^2*h^4 + 9*a^2*b*d^3*g^4*h^2 + 9*a^3*c*d^2*g^2*h^4 + 9*b 
^3*c^2*d*g^4*h^2 + 27*a*b^2*c*d^2*g^4*h^2 - 27*a*b^2*c^2*d*g^3*h^3 - 27*a^ 
2*b*c*d^2*g^3*h^3 + 27*a^2*b*c^2*d*g^2*h^4) - (B*log((e*(a + b*x)^n)/(c + 
d*x)^n))/(3*h*(g^3 + h^3*x^3 + 3*g^2*h*x + 3*g*h^2*x^2)) - (B*b^3*n*log(a 
+ b*x))/(3*a^3*h^4 - 3*b^3*g^3*h + 9*a*b^2*g^2*h^2 - 9*a^2*b*g*h^3) - ((2* 
A*a^2*c^2*h^4 + 2*A*b^2*d^2*g^4 + 2*A*a^2*d^2*g^2*h^2 + 2*A*b^2*c^2*g^2*h^ 
2 + 3*B*a^2*d^2*g^2*h^2*n - 3*B*b^2*c^2*g^2*h^2*n - 4*A*a*b*c^2*g*h^3 - 4* 
A*a*b*d^2*g^3*h - 4*A*a^2*c*d*g*h^3 - 4*A*b^2*c*d*g^3*h + 8*A*a*b*c*d*g^2* 
h^2 + B*a*b*c^2*g*h^3*n - 5*B*a*b*d^2*g^3*h*n - B*a^2*c*d*g*h^3*n + 5*B*b^ 
2*c*d*g^3*h*n)/(2*(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g 
^2*h^2 - 2*a*b*c^2*g*h^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*g 
^3*h + 4*a*b*c*d*g^2*h^2)) + (x*(B*a*b*c^2*h^4*n - B*a^2*c*d*h^4*n + 5*B*a 
^2*d^2*g*h^3*n - 5*B*b^2*c^2*g*h^3*n - 9*B*a*b*d^2*g^2*h^2*n + 9*B*b^2*c*d 
*g^2*h^2*n))/(2*(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g^2 
*h^2 - 2*a*b*c^2*g*h^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*...
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 6610, normalized size of antiderivative = 23.27 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*log(a + b*x)*a**3*b*c**3*g**3*h**6*n - 18*log(a + b*x)*a**3*b*c**3*g 
**2*h**7*n*x - 18*log(a + b*x)*a**3*b*c**3*g*h**8*n*x**2 - 6*log(a + b*x)* 
a**3*b*c**3*h**9*n*x**3 + 18*log(a + b*x)*a**3*b*c**2*d*g**4*h**5*n + 54*l 
og(a + b*x)*a**3*b*c**2*d*g**3*h**6*n*x + 54*log(a + b*x)*a**3*b*c**2*d*g* 
*2*h**7*n*x**2 + 18*log(a + b*x)*a**3*b*c**2*d*g*h**8*n*x**3 - 18*log(a + 
b*x)*a**3*b*c*d**2*g**5*h**4*n - 54*log(a + b*x)*a**3*b*c*d**2*g**4*h**5*n 
*x - 54*log(a + b*x)*a**3*b*c*d**2*g**3*h**6*n*x**2 - 18*log(a + b*x)*a**3 
*b*c*d**2*g**2*h**7*n*x**3 + 6*log(a + b*x)*a**3*b*d**3*g**6*h**3*n + 18*l 
og(a + b*x)*a**3*b*d**3*g**5*h**4*n*x + 18*log(a + b*x)*a**3*b*d**3*g**4*h 
**5*n*x**2 + 6*log(a + b*x)*a**3*b*d**3*g**3*h**6*n*x**3 + 18*log(a + b*x) 
*a**2*b**2*c**3*g**4*h**5*n + 54*log(a + b*x)*a**2*b**2*c**3*g**3*h**6*n*x 
 + 54*log(a + b*x)*a**2*b**2*c**3*g**2*h**7*n*x**2 + 18*log(a + b*x)*a**2* 
b**2*c**3*g*h**8*n*x**3 - 54*log(a + b*x)*a**2*b**2*c**2*d*g**5*h**4*n - 1 
62*log(a + b*x)*a**2*b**2*c**2*d*g**4*h**5*n*x - 162*log(a + b*x)*a**2*b** 
2*c**2*d*g**3*h**6*n*x**2 - 54*log(a + b*x)*a**2*b**2*c**2*d*g**2*h**7*n*x 
**3 + 54*log(a + b*x)*a**2*b**2*c*d**2*g**6*h**3*n + 162*log(a + b*x)*a**2 
*b**2*c*d**2*g**5*h**4*n*x + 162*log(a + b*x)*a**2*b**2*c*d**2*g**4*h**5*n 
*x**2 + 54*log(a + b*x)*a**2*b**2*c*d**2*g**3*h**6*n*x**3 - 18*log(a + b*x 
)*a**2*b**2*d**3*g**7*h**2*n - 54*log(a + b*x)*a**2*b**2*d**3*g**6*h**3*n* 
x - 54*log(a + b*x)*a**2*b**2*d**3*g**5*h**4*n*x**2 - 18*log(a + b*x)*a...