\(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(f+g x)^5} \, dx\) [271]

Optimal result

Integrand size = 29, antiderivative size = 381 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=-\frac {B (b c-a d)}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4} \] Output:

-1/6*B*(-a*d+b*c)/(-a*g+b*f)/(-c*g+d*f)/(g*x+f)^3-1/4*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)^2-1/2*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)+1/2*b^4*B*ln(b*x+a)/g/(-a*g+b*f)^4-1/4*(A+B*ln(e*( 
b*x+a)^2/(d*x+c)^2))/g/(g*x+f)^4-1/2*B*d^4*ln(d*x+c)/g/(-c*g+d*f)^4-1/2*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 = 0.98 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4}+2 B (b c-a d) \left (-\frac {g}{3 (b f-a g) (d f-c g) (f+g x)^3}+\frac {g (-2 b d f+b c g+a d g)}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 \log (a+b x)}{(b c-a d) (b f-a g)^4}-\frac {d^4 \log (c+d x)}{(b c-a d) (d f-c g)^4}-\frac {g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^4 (d f-c g)^4}\right )}{4 g} \] Input:

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^5,x]
 

Output:

(-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^4) + 2*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 = 0.86 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 93

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^5,x]
 

Output:

-1/4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(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)))/(2*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. \(2298\) vs. \(2(367)=734\).

Time = 5.90 (sec) , antiderivative size = 2299, normalized size of antiderivative = 6.03

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2299\)
default	\(\text {Expression too large to display}\)	\(2299\)
risch	\(\text {Expression too large to display}\)	\(4169\)
parallelrisch	\(\text {Expression too large to display}\)	\(5619\)

Input:

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/d*(-d^5*A*(-1/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g)-3/2*g/(c*g-d*f)^4 
/(c*g/(d*x+c)-f/(d*x+c)*d-g)^2-g^2/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g) 
^3-1/4*g^3/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g)^4)+((c*g-d*f)*b^4*g^2*B 
*d/(a^4*g^4-4*a^3*b*f*g^3+6*a^2*b^2*f^2*g^2-4*a*b^3*f^3*g+b^4*f^4)/(d*x+c) 
*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)+(c*g-d*f)*(c^2*g^2-2*c*d*f*g+d^2* 
f^2)*b^4*B*d/(a^4*g^4-4*a^3*b*f*g^3+6*a^2*b^2*f^2*g^2-4*a*b^3*f^3*g+b^4*f^ 
4)/(d*x+c)^3*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)-1/2*(B*a^3*d^5*g^6-3* 
B*a^2*b*d^5*f*g^5+3*B*a*b^2*d^5*f^2*g^4-B*b^3*c^3*d^2*g^6+3*B*b^3*c^2*d^3* 
f*g^5-3*B*b^3*c*d^4*f^2*g^4)/g/(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^3*f^4*g^2+3*a*b^2*c^3*f^2*g^4-9*a*b^2*c^2*d*f^3*g^3+ 
9*a*b^2*c*d^2*f^4*g^2-3*a*b^2*d^3*f^5*g-b^3*c^3*f^3*g^3+3*b^3*c^2*d*f^4*g^ 
2-3*b^3*c*d^2*f^5*g+b^3*d^3*f^6)/(d*x+c)+1/12*(11*B*a^3*d^5*g^6-2*B*a^2*b* 
c*d^4*g^6-31*B*a^2*b*d^5*f*g^5-3*B*a*b^2*c^2*d^3*g^6+10*B*a*b^2*c*d^4*f*g^ 
5+26*B*a*b^2*d^5*f^2*g^4-6*B*b^3*c^3*d^2*g^6+21*B*b^3*c^2*d^3*f*g^5-26*B*b 
^3*c*d^4*f^2*g^4)/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3)/g^4/(d*x+c 
)^4-1/6*(13*B*a^3*d^5*g^6-B*a^2*b*c*d^4*g^6-38*B*a^2*b*d^5*f*g^5-3*B*a*b^2 
*c^2*d^3*g^6+8*B*a*b^2*c*d^4*f*g^5+34*B*a*b^2*d^5*f^2*g^4-9*B*b^3*c^3*d^2* 
g^6+30*B*b^3*c^2*d^3*f*g^5-34*B*b^3*c*d^4*f^2*g^4)/g^3/(a^3*c*g^4-a^3*d*f* 
g^3-3*a^2*b*c*f*g^3+3*a^2*b*d*f^2*g^2+3*a*b^2*c*f^2*g^2-3*a*b^2*d*f^3*g...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))/(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. 1809 vs. \(2 (367) = 734\).

Time = 0.16 (sec) , antiderivative size = 1809, normalized size of antiderivative = 4.75 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="maxima" 
)
 

Output:

1/12*(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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.16 (sec) , antiderivative size = 2059, normalized size of antiderivative = 5.40 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 39.97 (sec) , antiderivative size = 2520, normalized size of antiderivative = 6.61 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(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))/(2*a^4*c^4*g^8 + 2*b^4*d^4*f^8 + 2*a^4*d^4*f^4* 
g^4 + 2*b^4*c^4*f^4*g^4 + 12*a^2*b^2*c^4*f^2*g^6 + 12*a^2*b^2*d^4*f^6*g^2 
+ 12*a^4*c^2*d^2*f^2*g^6 + 12*b^4*c^2*d^2*f^6*g^2 - 8*a^3*b*c^4*f*g^7 - 8* 
a*b^3*d^4*f^7*g - 8*a^4*c^3*d*f*g^7 - 8*b^4*c*d^3*f^7*g - 8*a*b^3*c^4*f^3* 
g^5 - 8*a^3*b*d^4*f^5*g^3 - 8*a^4*c*d^3*f^3*g^5 - 8*b^4*c^3*d*f^5*g^3 + 32 
*a*b^3*c*d^3*f^6*g^2 + 32*a*b^3*c^3*d*f^4*g^4 + 32*a^3*b*c*d^3*f^4*g^4 + 3 
2*a^3*b*c^3*d*f^2*g^6 - 48*a*b^3*c^2*d^2*f^5*g^3 - 48*a^2*b^2*c*d^3*f^5*g^ 
3 - 48*a^2*b^2*c^3*d*f^3*g^5 - 48*a^3*b*c^2*d^2*f^3*g^5 + 72*a^2*b^2*c^2*d 
^2*f^4*g^4) - ((3*A*a^3*c^3*g^6 + 3*A*b^3*d^3*f^6 - 3*A*a^3*d^3*f^3*g^3 - 
3*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 + 9*A*a* 
b^2*c^3*f^2*g^4 + 9*A*a^2*b*d^3*f^4*g^2 - 7*B*a*b^2*c^3*f^2*g^4 + 9*A*a^3* 
c*d^2*f^2*g^4 + 31*B*a^2*b*d^3*f^4*g^2 + 9*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 - 9*A*a^2*b*c^3*f*g^5 - 9*A*a*b^2*d^ 
3*f^5*g + 2*B*a^2*b*c^3*f*g^5 - 9*A*a^3*c^2*d*f*g^5 - 26*B*a*b^2*d^3*f^5*g 
 - 9*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 + 27*A 
*a*b^2*c*d^2*f^4*g^2 - 27*A*a*b^2*c^2*d*f^3*g^3 - 27*A*a^2*b*c*d^2*f^3*g^3 
 + 27*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^3 - b^3*c^3*f^3*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 7411, normalized size of antiderivative = 19.45 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(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 + ...