\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(f+g x)^5} \, dx\) [239]

Optimal result

Integrand size = 27, antiderivative size = 379 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=-\frac {B (b c-a d)}{12 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{8 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{4 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x)}{4 g (b f-a g)^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{4 g (f+g x)^4}-\frac {B d^4 \log (c+d x)}{4 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4} \] Output:

-1/12*B*(-a*d+b*c)/(-a*g+b*f)/(-c*g+d*f)/(g*x+f)^3-1/8*B*(-a*d+b*c)*(-a*d* 
g-b*c*g+2*b*d*f)/(-a*g+b*f)^2/(-c*g+d*f)^2/(g*x+f)^2-1/4*B*(-a*d+b*c)*(a^2 
*d^2*g^2-a*b*d*g*(-c*g+3*d*f)+b^2*(c^2*g^2-3*c*d*f*g+3*d^2*f^2))/(-a*g+b*f 
)^3/(-c*g+d*f)^3/(g*x+f)+1/4*b^4*B*ln(b*x+a)/g/(-a*g+b*f)^4-1/4*(A+B*ln(e* 
(b*x+a)/(d*x+c)))/g/(g*x+f)^4-1/4*B*d^4*ln(d*x+c)/g/(-c*g+d*f)^4-1/4*B*(-a 
*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c 
*d*f*g+2*d^2*f^2))*ln(g*x+f)/(-a*g+b*f)^4/(-c*g+d*f)^4
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4}+B (b c-a d) \left (-\frac {g}{3 (b f-a g) (d f-c g) (f+g x)^3}+\frac {g (-2 b d f+b c g+a d g)}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 \log (a+b x)}{(b c-a d) (b f-a g)^4}-\frac {d^4 \log (c+d x)}{(b c-a d) (d f-c g)^4}-\frac {g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^4 (d f-c g)^4}\right )}{4 g} \] Input:

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(f+g x)^5} \, dx\)

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 93

\(\displaystyle \frac {B (b c-a d) \int \left (\frac {b^5}{(b c-a d) (b f-a g)^4 (a+b x)}-\frac {d^5}{(b c-a d) (c g-d f)^4 (c+d x)}+\frac {g^2 (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 a d^2 f g b+a^2 d^2 g^2\right )}{(b f-a g)^4 (d f-c g)^4 (f+g x)}+\frac {g^2 \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)^2}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^3}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^4}\right )dx}{4 g}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 g (f+g x)^4}\)

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^5,x]
 

Output:

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

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. \(2849\) vs. \(2(365)=730\).

Time = 4.50 (sec) , antiderivative size = 2850, normalized size of antiderivative = 7.52

method	result	size
parts	\(\text {Expression too large to display}\)	\(2850\)
derivativedivides	\(\text {Expression too large to display}\)	\(3309\)
default	\(\text {Expression too large to display}\)	\(3309\)
risch	\(\text {Expression too large to display}\)	\(4167\)
parallelrisch	\(\text {Expression too large to display}\)	\(5539\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(g*x+f)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1757 vs. \(2 (365) = 730\).

Time = 0.16 (sec) , antiderivative size = 1757, normalized size of antiderivative = 4.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,x, algorithm="maxima")
 

Output:

1/24*(6*b^4*log(b*x + a)/(b^4*f^4*g - 4*a*b^3*f^3*g^2 + 6*a^2*b^2*f^2*g^3 
- 4*a^3*b*f*g^4 + a^4*g^5) - 6*d^4*log(d*x + c)/(d^4*f^4*g - 4*c*d^3*f^3*g 
^2 + 6*c^2*d^2*f^2*g^3 - 4*c^3*d*f*g^4 + c^4*g^5) + 6*(4*(b^4*c*d^3 - a*b^ 
3*d^4)*f^3 - 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*f^2*g + 4*(b^4*c^3*d - a^3*b*d^ 
4)*f*g^2 - (b^4*c^4 - a^4*d^4)*g^3)*log(g*x + f)/(b^4*d^4*f^8 + a^4*c^4*g^ 
8 - 4*(b^4*c*d^3 + a*b^3*d^4)*f^7*g + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3 
*a^2*b^2*d^4)*f^6*g^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + 
 a^3*b*d^4)*f^5*g^3 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16* 
a^3*b*c*d^3 + a^4*d^4)*f^4*g^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b* 
c^2*d^2 + a^4*c*d^3)*f^3*g^5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^ 
2*d^2)*f^2*g^6 - 4*(a^3*b*c^4 + a^4*c^3*d)*f*g^7) - (26*(b^3*c*d^2 - a*b^2 
*d^3)*f^4 - 31*(b^3*c^2*d - a^2*b*d^3)*f^3*g + (11*b^3*c^3 + 15*a*b^2*c^2* 
d - 15*a^2*b*c*d^2 - 11*a^3*d^3)*f^2*g^2 - 7*(a*b^2*c^3 - a^3*c*d^2)*f*g^3 
 + 2*(a^2*b*c^3 - a^3*c^2*d)*g^4 + 6*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2*g^2 - 
3*(b^3*c^2*d - a^2*b*d^3)*f*g^3 + (b^3*c^3 - a^3*d^3)*g^4)*x^2 + 3*(14*(b^ 
3*c*d^2 - a*b^2*d^3)*f^3*g - 15*(b^3*c^2*d - a^2*b*d^3)*f^2*g^2 + (5*b^3*c 
^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 5*a^3*d^3)*f*g^3 - (a*b^2*c^3 - a^3*c 
*d^2)*g^4)*x)/(b^3*d^3*f^9 + a^3*c^3*f^3*g^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*f 
^8*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^7*g^2 - (b^3*c^3 + 9*a* 
b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^6*g^3 + 3*(a*b^2*c^3 + 3*a^2*b*c...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20791 vs. \(2 (365) = 730\).

Time = 0.61 (sec) , antiderivative size = 20791, normalized size of antiderivative = 54.86 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,x, algorithm="giac")
 

Output:

1/24*(6*(4*B*b^5*c^2*d^3*e*f^3 - 8*B*a*b^4*c*d^4*e*f^3 + 4*B*a^2*b^3*d^5*e 
*f^3 - 6*B*b^5*c^3*d^2*e*f^2*g + 6*B*a*b^4*c^2*d^3*e*f^2*g + 6*B*a^2*b^3*c 
*d^4*e*f^2*g - 6*B*a^3*b^2*d^5*e*f^2*g + 4*B*b^5*c^4*d*e*f*g^2 - 4*B*a*b^4 
*c^3*d^2*e*f*g^2 - 4*B*a^3*b^2*c*d^4*e*f*g^2 + 4*B*a^4*b*d^5*e*f*g^2 - B*b 
^5*c^5*e*g^3 + B*a*b^4*c^4*d*e*g^3 + B*a^4*b*c*d^4*e*g^3 - B*a^5*d^5*e*g^3 
)*log(-b*e*f + a*e*g + (b*e*x + a*e)*d*f/(d*x + c) - (b*e*x + a*e)*c*g/(d* 
x + c))/(b^4*d^4*f^8 - 4*b^4*c*d^3*f^7*g - 4*a*b^3*d^4*f^7*g + 6*b^4*c^2*d 
^2*f^6*g^2 + 16*a*b^3*c*d^3*f^6*g^2 + 6*a^2*b^2*d^4*f^6*g^2 - 4*b^4*c^3*d* 
f^5*g^3 - 24*a*b^3*c^2*d^2*f^5*g^3 - 24*a^2*b^2*c*d^3*f^5*g^3 - 4*a^3*b*d^ 
4*f^5*g^3 + b^4*c^4*f^4*g^4 + 16*a*b^3*c^3*d*f^4*g^4 + 36*a^2*b^2*c^2*d^2* 
f^4*g^4 + 16*a^3*b*c*d^3*f^4*g^4 + a^4*d^4*f^4*g^4 - 4*a*b^3*c^4*f^3*g^5 - 
 24*a^2*b^2*c^3*d*f^3*g^5 - 24*a^3*b*c^2*d^2*f^3*g^5 - 4*a^4*c*d^3*f^3*g^5 
 + 6*a^2*b^2*c^4*f^2*g^6 + 16*a^3*b*c^3*d*f^2*g^6 + 6*a^4*c^2*d^2*f^2*g^6 
- 4*a^3*b*c^4*f*g^7 - 4*a^4*c^3*d*f*g^7 + a^4*c^4*g^8) + 6*(4*B*b^5*c^2*d^ 
3*e^5*f^3 - 8*B*a*b^4*c*d^4*e^5*f^3 + 4*B*a^2*b^3*d^5*e^5*f^3 - 6*B*b^5*c^ 
3*d^2*e^5*f^2*g + 6*B*a*b^4*c^2*d^3*e^5*f^2*g + 6*B*a^2*b^3*c*d^4*e^5*f^2* 
g - 6*B*a^3*b^2*d^5*e^5*f^2*g + 4*B*b^5*c^4*d*e^5*f*g^2 - 4*B*a*b^4*c^3*d^ 
2*e^5*f*g^2 - 4*B*a^3*b^2*c*d^4*e^5*f*g^2 + 4*B*a^4*b*d^5*e^5*f*g^2 - B*b^ 
5*c^5*e^5*g^3 + B*a*b^4*c^4*d*e^5*g^3 + B*a^4*b*c*d^4*e^5*g^3 - B*a^5*d^5* 
e^5*g^3 - 12*(b*e*x + a*e)*B*b^4*c^2*d^4*e^4*f^3/(d*x + c) + 24*(b*e*x ...
 

Mupad [B] (verification not implemented)

Time = 38.69 (sec) , antiderivative size = 2518, normalized size of antiderivative = 6.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))/(f + g*x)^5,x)
 

Output:

(log(f + g*x)*(g*(6*B*a^2*b^2*d^4*f^2 - 6*B*b^4*c^2*d^2*f^2) - g^2*(4*B*a^ 
3*b*d^4*f - 4*B*b^4*c^3*d*f) + g^3*(B*a^4*d^4 - B*b^4*c^4) - 4*B*a*b^3*d^4 
*f^3 + 4*B*b^4*c*d^3*f^3))/(4*a^4*c^4*g^8 + 4*b^4*d^4*f^8 + 4*a^4*d^4*f^4* 
g^4 + 4*b^4*c^4*f^4*g^4 + 24*a^2*b^2*c^4*f^2*g^6 + 24*a^2*b^2*d^4*f^6*g^2 
+ 24*a^4*c^2*d^2*f^2*g^6 + 24*b^4*c^2*d^2*f^6*g^2 - 16*a^3*b*c^4*f*g^7 - 1 
6*a*b^3*d^4*f^7*g - 16*a^4*c^3*d*f*g^7 - 16*b^4*c*d^3*f^7*g - 16*a*b^3*c^4 
*f^3*g^5 - 16*a^3*b*d^4*f^5*g^3 - 16*a^4*c*d^3*f^3*g^5 - 16*b^4*c^3*d*f^5* 
g^3 + 64*a*b^3*c*d^3*f^6*g^2 + 64*a*b^3*c^3*d*f^4*g^4 + 64*a^3*b*c*d^3*f^4 
*g^4 + 64*a^3*b*c^3*d*f^2*g^6 - 96*a*b^3*c^2*d^2*f^5*g^3 - 96*a^2*b^2*c*d^ 
3*f^5*g^3 - 96*a^2*b^2*c^3*d*f^3*g^5 - 96*a^3*b*c^2*d^2*f^3*g^5 + 144*a^2* 
b^2*c^2*d^2*f^4*g^4) - ((6*A*a^3*c^3*g^6 + 6*A*b^3*d^3*f^6 - 6*A*a^3*d^3*f 
^3*g^3 - 6*A*b^3*c^3*f^3*g^3 - 11*B*a^3*d^3*f^3*g^3 + 11*B*b^3*c^3*f^3*g^3 
 + 18*A*a*b^2*c^3*f^2*g^4 + 18*A*a^2*b*d^3*f^4*g^2 - 7*B*a*b^2*c^3*f^2*g^4 
 + 18*A*a^3*c*d^2*f^2*g^4 + 31*B*a^2*b*d^3*f^4*g^2 + 18*A*b^3*c^2*d*f^4*g^ 
2 + 7*B*a^3*c*d^2*f^2*g^4 - 31*B*b^3*c^2*d*f^4*g^2 - 18*A*a^2*b*c^3*f*g^5 
- 18*A*a*b^2*d^3*f^5*g + 2*B*a^2*b*c^3*f*g^5 - 18*A*a^3*c^2*d*f*g^5 - 26*B 
*a*b^2*d^3*f^5*g - 18*A*b^3*c*d^2*f^5*g - 2*B*a^3*c^2*d*f*g^5 + 26*B*b^3*c 
*d^2*f^5*g + 54*A*a*b^2*c*d^2*f^4*g^2 - 54*A*a*b^2*c^2*d*f^3*g^3 - 54*A*a^ 
2*b*c*d^2*f^3*g^3 + 54*A*a^2*b*c^2*d*f^2*g^4 + 15*B*a*b^2*c^2*d*f^3*g^3 - 
15*B*a^2*b*c*d^2*f^3*g^3)/(6*(a^3*c^3*g^6 + b^3*d^3*f^6 - a^3*d^3*f^3*g...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 6836, normalized size of antiderivative = 18.04 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,x)
 

Output:

(12*log(a + b*x)*b**5*c**4*f**5*g**4 + 48*log(a + b*x)*b**5*c**4*f**4*g**5 
*x + 72*log(a + b*x)*b**5*c**4*f**3*g**6*x**2 + 48*log(a + b*x)*b**5*c**4* 
f**2*g**7*x**3 + 12*log(a + b*x)*b**5*c**4*f*g**8*x**4 - 48*log(a + b*x)*b 
**5*c**3*d*f**6*g**3 - 192*log(a + b*x)*b**5*c**3*d*f**5*g**4*x - 288*log( 
a + b*x)*b**5*c**3*d*f**4*g**5*x**2 - 192*log(a + b*x)*b**5*c**3*d*f**3*g* 
*6*x**3 - 48*log(a + b*x)*b**5*c**3*d*f**2*g**7*x**4 + 72*log(a + b*x)*b** 
5*c**2*d**2*f**7*g**2 + 288*log(a + b*x)*b**5*c**2*d**2*f**6*g**3*x + 432* 
log(a + b*x)*b**5*c**2*d**2*f**5*g**4*x**2 + 288*log(a + b*x)*b**5*c**2*d* 
*2*f**4*g**5*x**3 + 72*log(a + b*x)*b**5*c**2*d**2*f**3*g**6*x**4 - 48*log 
(a + b*x)*b**5*c*d**3*f**8*g - 192*log(a + b*x)*b**5*c*d**3*f**7*g**2*x - 
288*log(a + b*x)*b**5*c*d**3*f**6*g**3*x**2 - 192*log(a + b*x)*b**5*c*d**3 
*f**5*g**4*x**3 - 48*log(a + b*x)*b**5*c*d**3*f**4*g**5*x**4 + 12*log(a + 
b*x)*b**5*d**4*f**9 + 48*log(a + b*x)*b**5*d**4*f**8*g*x + 72*log(a + b*x) 
*b**5*d**4*f**7*g**2*x**2 + 48*log(a + b*x)*b**5*d**4*f**6*g**3*x**3 + 12* 
log(a + b*x)*b**5*d**4*f**5*g**4*x**4 - 12*log(c + d*x)*a**4*b*d**4*f**5*g 
**4 - 48*log(c + d*x)*a**4*b*d**4*f**4*g**5*x - 72*log(c + d*x)*a**4*b*d** 
4*f**3*g**6*x**2 - 48*log(c + d*x)*a**4*b*d**4*f**2*g**7*x**3 - 12*log(c + 
 d*x)*a**4*b*d**4*f*g**8*x**4 + 48*log(c + d*x)*a**3*b**2*d**4*f**6*g**3 + 
 192*log(c + d*x)*a**3*b**2*d**4*f**5*g**4*x + 288*log(c + d*x)*a**3*b**2* 
d**4*f**4*g**5*x**2 + 192*log(c + d*x)*a**3*b**2*d**4*f**3*g**6*x**3 + ...