\(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(f+g x)^4} \, dx\) [270]

Optimal result

Integrand size = 29, antiderivative size = 277 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx=-\frac {B (b c-a d)}{3 (b f-a g) (d f-c g) (f+g x)^2}-\frac {2 B (b c-a d) (2 b d f-b c g-a d g)}{3 (b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {2 b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 g (f+g x)^3}-\frac {2 B d^3 \log (c+d x)}{3 g (d f-c g)^3}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^3 (d f-c g)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3}+2 B (b c-a d) \left (-\frac {g}{2 (b f-a g) (d f-c g) (f+g x)^2}+\frac {g (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}+\frac {d^3 \log (c+d x)}{(b c-a d) (-d f+c g)^3}+\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}\right )}{3 g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 93

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(g*(f + g*x)^3) + (2*B*(b*c 
- a*d)*(-1/2*g/((b*f - a*g)*(d*f - c*g)*(f + g*x)^2) - (g*(2*b*d*f - b*c*g 
 - a*d*g))/((b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (b^3*Log[a + b*x])/(( 
b*c - a*d)*(b*f - a*g)^3) - (d^3*Log[c + d*x])/((b*c - a*d)*(d*f - c*g)^3) 
 + (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))*Log[f + g*x])/((b*f - a*g)^3*(d*f - c*g)^3)))/(3*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. \(1292\) vs. \(2(265)=530\).

Time = 2.75 (sec) , antiderivative size = 1293, normalized size of antiderivative = 4.67

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1293\)
default	\(\text {Expression too large to display}\)	\(1293\)
risch	\(\text {Expression too large to display}\)	\(2293\)
parallelrisch	\(\text {Expression too large to display}\)	\(2946\)

Input:

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/d*(d^4*A*(-1/(c*g-d*f)^3/(c*g/(d*x+c)-f/(d*x+c)*d-g)-g/(c*g-d*f)^3/(c*g 
/(d*x+c)-f/(d*x+c)*d-g)^2-1/3*g^2/(c*g-d*f)^3/(c*g/(d*x+c)-f/(d*x+c)*d-g)^ 
3)+((c*g-d*f)*b^3*g*B*d/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3)/(d*x 
+c)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)-1/3*(2*B*a^2*d^4*g^4-4*B*a*b*d 
^4*f*g^3-2*B*b^2*c^2*d^2*g^4+4*B*b^2*c*d^3*f*g^3)/g/(a^2*c^2*g^4-2*a^2*c*d 
*f*g^3+a^2*d^2*f^2*g^2-2*a*b*c^2*f*g^3+4*a*b*c*d*f^2*g^2-2*a*b*d^2*f^3*g+b 
^2*c^2*f^2*g^2-2*b^2*c*d*f^3*g+b^2*d^2*f^4)/(d*x+c)-1/3*(3*B*a^2*d^4*g^4-B 
*a*b*c*d^3*g^4-5*B*a*b*d^4*f*g^3-2*B*b^2*c^2*d^2*g^4+5*B*b^2*c*d^3*f*g^3)/ 
(a^2*g^2-2*a*b*f*g+b^2*f^2)/g^3/(d*x+c)^3+1/3*(5*B*a^2*d^4*g^4-B*a*b*c*d^3 
*g^4-9*B*a*b*d^4*f*g^3-4*B*b^2*c^2*d^2*g^4+9*B*b^2*c*d^3*f*g^3)/(c*g-d*f)/ 
g^2/(a^2*g^2-2*a*b*f*g+b^2*f^2)/(d*x+c)^2-1/3*B*d*(a^3*d^3*g^2-3*a^2*b*d^3 
*f*g+3*a*b^2*d^3*f^2-b^3*c^3*g^2+3*b^3*c^2*d*f*g-3*b^3*c*d^2*f^2)/(a^3*g^3 
-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3)/(d*x+c)^3*ln(e*(a*d/(d*x+c)-b*c/(d*x 
+c)+b)^2/d^2)-1/3*b^3*g^2*B*d/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3 
)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)-(c^2*g^2-2*c*d*f*g+d^2*f^2)*b^3* 
B*d/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3)/(d*x+c)^2*ln(e*(a*d/(d*x 
+c)-b*c/(d*x+c)+b)^2/d^2))/(c*g/(d*x+c)-f/(d*x+c)*d-g)^3+2/3*B*d*(a^3*d^3* 
g^2-3*a^2*b*d^3*f*g+3*a*b^2*d^3*f^2-b^3*c^3*g^2+3*b^3*c^2*d*f*g-3*b^3*c*d^ 
2*f^2)/(a^3*c^3*g^6-3*a^3*c^2*d*f*g^5+3*a^3*c*d^2*f^2*g^4-a^3*d^3*f^3*g^3- 
3*a^2*b*c^3*f*g^5+9*a^2*b*c^2*d*f^2*g^4-9*a^2*b*c*d^2*f^3*g^3+3*a^2*b*d...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx=\text {Timed out} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))/(g*x+f)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (265) = 530\).

Time = 0.09 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.25 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x, algorithm="maxima" 
)
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1359 vs. \(2 (265) = 530\).

Time = 0.58 (sec) , antiderivative size = 1359, normalized size of antiderivative = 4.91 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x, algorithm="giac")
 

Output:

2/3*B*b^4*log(abs(b*x + a))/(b^4*f^3*g - 3*a*b^3*f^2*g^2 + 3*a^2*b^2*f*g^3 
 - a^3*b*g^4) - 2/3*B*d^4*log(abs(d*x + c))/(d^4*f^3*g - 3*c*d^3*f^2*g^2 + 
 3*c^2*d^2*f*g^3 - c^3*d*g^4) + 2/3*(3*B*b^3*c*d^2*f^2 - 3*B*a*b^2*d^3*f^2 
 - 3*B*b^3*c^2*d*f*g + 3*B*a^2*b*d^3*f*g + B*b^3*c^3*g^2 - B*a^3*d^3*g^2)* 
log(g*x + f)/(b^3*d^3*f^6 - 3*b^3*c*d^2*f^5*g - 3*a*b^2*d^3*f^5*g + 3*b^3* 
c^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 + 3*a^2*b*d^3*f^4*g^2 - b^3*c^3*f^3* 
g^3 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 - a^3*d^3*f^3*g^3 + 3* 
a*b^2*c^3*f^2*g^4 + 9*a^2*b*c^2*d*f^2*g^4 + 3*a^3*c*d^2*f^2*g^4 - 3*a^2*b* 
c^3*f*g^5 - 3*a^3*c^2*d*f*g^5 + a^3*c^3*g^6) - 1/3*B*log((b^2*e*x^2 + 2*a* 
b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g 
^2*x + f^3*g) - 1/3*(4*B*b^2*c*d*f*g^3*x^2 - 4*B*a*b*d^2*f*g^3*x^2 - 2*B*b 
^2*c^2*g^4*x^2 + 2*B*a^2*d^2*g^4*x^2 + 9*B*b^2*c*d*f^2*g^2*x - 9*B*a*b*d^2 
*f^2*g^2*x - 5*B*b^2*c^2*f*g^3*x + 5*B*a^2*d^2*f*g^3*x + B*a*b*c^2*g^4*x - 
 B*a^2*c*d*g^4*x + A*b^2*d^2*f^4 - 2*A*b^2*c*d*f^3*g + 5*B*b^2*c*d*f^3*g - 
 2*A*a*b*d^2*f^3*g - 5*B*a*b*d^2*f^3*g + A*b^2*c^2*f^2*g^2 - 3*B*b^2*c^2*f 
^2*g^2 + 4*A*a*b*c*d*f^2*g^2 + A*a^2*d^2*f^2*g^2 + 3*B*a^2*d^2*f^2*g^2 - 2 
*A*a*b*c^2*f*g^3 + B*a*b*c^2*f*g^3 - 2*A*a^2*c*d*f*g^3 - B*a^2*c*d*f*g^3 + 
 A*a^2*c^2*g^4)/(b^2*d^2*f^4*g^4*x^3 - 2*b^2*c*d*f^3*g^5*x^3 - 2*a*b*d^2*f 
^3*g^5*x^3 + b^2*c^2*f^2*g^6*x^3 + 4*a*b*c*d*f^2*g^6*x^3 + a^2*d^2*f^2*g^6 
*x^3 - 2*a*b*c^2*f*g^7*x^3 - 2*a^2*c*d*f*g^7*x^3 + a^2*c^2*g^8*x^3 + 3*...
 

Mupad [B] (verification not implemented)

Time = 32.56 (sec) , antiderivative size = 1147, normalized size of antiderivative = 4.14 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(f + g*x)^4,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 7464, normalized size of antiderivative = 26.95 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x)
 

Output:

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