\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(g+h x)^3} \, dx\) [300]

Optimal result

Integrand size = 31, antiderivative size = 191 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx=-\frac {B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac {b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac {B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.93 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx=-\frac {\frac {A}{(g+h x)^2}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+B (b c-a d) n \left (-\frac {b^2 \log (a+b x)}{(b c-a d) (b g-a h)^2}+\frac {\frac {d^2 \log (c+d x)}{b c-a d}+\frac {h \left (\frac {(b g-a h) (d g-c h)}{g+h x}+(-2 b d g+b c h+a d h) \log (g+h x)\right )}{(b g-a h)^2}}{(d g-c h)^2}\right )}{2 h} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 93

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(h*(g + h*x)^2) + (B*(b*c - 
a*d)*n*(-(h/((b*g - a*h)*(d*g - c*h)*(g + h*x))) + (b^2*Log[a + b*x])/((b* 
c - a*d)*(b*g - a*h)^2) - (d^2*Log[c + d*x])/((b*c - a*d)*(d*g - c*h)^2) + 
 (h*(2*b*d*g - b*c*h - a*d*h)*Log[g + h*x])/((b*g - a*h)^2*(d*g - c*h)^2)) 
)/(2*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. \(1384\) vs. \(2(181)=362\).

Time = 33.44 (sec) , antiderivative size = 1385, normalized size of antiderivative = 7.25

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1385\)
risch	\(\text {Expression too large to display}\)	\(4925\)

Input:

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (181) = 362\).

Time = 44.09 (sec) , antiderivative size = 1127, normalized size of antiderivative = 5.90 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^3,x, algorithm="frica 
s")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (181) = 362\).

Time = 0.05 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.00 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx=\frac {{\left (\frac {b^{2} e n \log \left (b x + a\right )}{b^{2} g^{2} h - 2 \, a b g h^{2} + a^{2} h^{3}} - \frac {d^{2} e n \log \left (d x + c\right )}{d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}} - \frac {{\left (2 \, a b d^{2} e g n - a^{2} d^{2} e h n - {\left (2 \, c d e g n - c^{2} e h n\right )} b^{2}\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac {b c e n - a d e n}{{\left (d g^{2} h - c g h^{2}\right )} a - {\left (d g^{3} - c g^{2} h\right )} b + {\left ({\left (d g h^{2} - c h^{3}\right )} a - {\left (d g^{2} h - c g h^{2}\right )} b\right )} x}\right )} B}{2 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} - \frac {A}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (181) = 362\).

Time = 0.29 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.78 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx=\frac {B b^{3} n \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (b^{3} g^{2} h - 2 \, a b^{2} g h^{2} + a^{2} b h^{3}\right )}} - \frac {B d^{3} n \log \left ({\left | d x + c \right |}\right )}{2 \, {\left (d^{3} g^{2} h - 2 \, c d^{2} g h^{2} + c^{2} d h^{3}\right )}} - \frac {B n \log \left (b x + a\right )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} + \frac {B n \log \left (d x + c\right )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} + \frac {{\left (2 \, B b^{2} c d g n - 2 \, B a b d^{2} g n - B b^{2} c^{2} h n + B a^{2} d^{2} h n\right )} \log \left (h x + g\right )}{2 \, {\left (b^{2} d^{2} g^{4} - 2 \, b^{2} c d g^{3} h - 2 \, a b d^{2} g^{3} h + b^{2} c^{2} g^{2} h^{2} + 4 \, a b c d g^{2} h^{2} + a^{2} d^{2} g^{2} h^{2} - 2 \, a b c^{2} g h^{3} - 2 \, a^{2} c d g h^{3} + a^{2} c^{2} h^{4}\right )}} - \frac {B b c h^{2} n x - B a d h^{2} n x + B b c g h n - B a d g h n + B b d g^{2} \log \left (e\right ) - B b c g h \log \left (e\right ) - B a d g h \log \left (e\right ) + B a c h^{2} \log \left (e\right ) + A b d g^{2} - A b c g h - A a d g h + A a c h^{2}}{2 \, {\left (b d g^{2} h^{3} x^{2} - b c g h^{4} x^{2} - a d g h^{4} x^{2} + a c h^{5} x^{2} + 2 \, b d g^{3} h^{2} x - 2 \, b c g^{2} h^{3} x - 2 \, a d g^{2} h^{3} x + 2 \, a c g h^{4} x + b d g^{4} h - b c g^{3} h^{2} - a d g^{3} h^{2} + a c g^{2} h^{3}\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 27.85 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.26 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx=\frac {\ln \left (g+h\,x\right )\,\left (h\,\left (B\,a^2\,d^2\,n-B\,b^2\,c^2\,n\right )-2\,B\,a\,b\,d^2\,g\,n+2\,B\,b^2\,c\,d\,g\,n\right )}{2\,a^2\,c^2\,h^4-4\,a^2\,c\,d\,g\,h^3+2\,a^2\,d^2\,g^2\,h^2-4\,a\,b\,c^2\,g\,h^3+8\,a\,b\,c\,d\,g^2\,h^2-4\,a\,b\,d^2\,g^3\,h+2\,b^2\,c^2\,g^2\,h^2-4\,b^2\,c\,d\,g^3\,h+2\,b^2\,d^2\,g^4}-\frac {\frac {A\,a\,c\,h^2+A\,b\,d\,g^2-A\,a\,d\,g\,h-A\,b\,c\,g\,h-B\,a\,d\,g\,h\,n+B\,b\,c\,g\,h\,n}{a\,c\,h^2+b\,d\,g^2-a\,d\,g\,h-b\,c\,g\,h}-\frac {x\,\left (B\,a\,d\,h^2\,n-B\,b\,c\,h^2\,n\right )}{a\,c\,h^2+b\,d\,g^2-a\,d\,g\,h-b\,c\,g\,h}}{2\,g^2\,h+4\,g\,h^2\,x+2\,h^3\,x^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{2\,h\,\left (g^2+2\,g\,h\,x+h^2\,x^2\right )}+\frac {B\,b^2\,n\,\ln \left (a+b\,x\right )}{2\,a^2\,h^3-4\,a\,b\,g\,h^2+2\,b^2\,g^2\,h}-\frac {B\,d^2\,n\,\ln \left (c+d\,x\right )}{2\,c^2\,h^3-4\,c\,d\,g\,h^2+2\,d^2\,g^2\,h} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 2381, normalized size of antiderivative = 12.47 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*log(a + b*x)*a**2*b*c**2*g**2*h**4*n - 4*log(a + b*x)*a**2*b*c**2*g* 
h**5*n*x - 2*log(a + b*x)*a**2*b*c**2*h**6*n*x**2 + 4*log(a + b*x)*a**2*b* 
c*d*g**3*h**3*n + 8*log(a + b*x)*a**2*b*c*d*g**2*h**4*n*x + 4*log(a + b*x) 
*a**2*b*c*d*g*h**5*n*x**2 - 2*log(a + b*x)*a**2*b*d**2*g**4*h**2*n - 4*log 
(a + b*x)*a**2*b*d**2*g**3*h**3*n*x - 2*log(a + b*x)*a**2*b*d**2*g**2*h**4 
*n*x**2 + 4*log(a + b*x)*a*b**2*c**2*g**3*h**3*n + 8*log(a + b*x)*a*b**2*c 
**2*g**2*h**4*n*x + 4*log(a + b*x)*a*b**2*c**2*g*h**5*n*x**2 - 8*log(a + b 
*x)*a*b**2*c*d*g**4*h**2*n - 16*log(a + b*x)*a*b**2*c*d*g**3*h**3*n*x - 8* 
log(a + b*x)*a*b**2*c*d*g**2*h**4*n*x**2 + 4*log(a + b*x)*a*b**2*d**2*g**5 
*h*n + 8*log(a + b*x)*a*b**2*d**2*g**4*h**2*n*x + 4*log(a + b*x)*a*b**2*d* 
*2*g**3*h**3*n*x**2 + 2*log(c + d*x)*a**2*b*c**2*g**2*h**4*n + 4*log(c + d 
*x)*a**2*b*c**2*g*h**5*n*x + 2*log(c + d*x)*a**2*b*c**2*h**6*n*x**2 - 4*lo 
g(c + d*x)*a**2*b*c*d*g**3*h**3*n - 8*log(c + d*x)*a**2*b*c*d*g**2*h**4*n* 
x - 4*log(c + d*x)*a**2*b*c*d*g*h**5*n*x**2 - 4*log(c + d*x)*a*b**2*c**2*g 
**3*h**3*n - 8*log(c + d*x)*a*b**2*c**2*g**2*h**4*n*x - 4*log(c + d*x)*a*b 
**2*c**2*g*h**5*n*x**2 + 8*log(c + d*x)*a*b**2*c*d*g**4*h**2*n + 16*log(c 
+ d*x)*a*b**2*c*d*g**3*h**3*n*x + 8*log(c + d*x)*a*b**2*c*d*g**2*h**4*n*x* 
*2 + 2*log(c + d*x)*b**3*c**2*g**4*h**2*n + 4*log(c + d*x)*b**3*c**2*g**3* 
h**3*n*x + 2*log(c + d*x)*b**3*c**2*g**2*h**4*n*x**2 - 4*log(c + d*x)*b**3 
*c*d*g**5*h*n - 8*log(c + d*x)*b**3*c*d*g**4*h**2*n*x - 4*log(c + d*x)*...