\(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{x (f+g x+h x^2)} \, dx\) [86]

Optimal result

Integrand size = 34, antiderivative size = 800 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx =\text {Too large to display} \] Output:

n*ln(-b*x/a)*ln(b*x+a)/f-n*ln(-d*x/c)*ln(d*x+c)/f-g*arctanh((2*h*x+g)/(-4* 
f*h+g^2)^(1/2))*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))/f/(-4* 
f*h+g^2)^(1/2)-ln(x)*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))/f 
-1/2*(1+g/(-4*f*h+g^2)^(1/2))*n*ln(b*x+a)*ln(-b*(g-(-4*f*h+g^2)^(1/2)+2*h* 
x)/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))/f+1/2*(1+g/(-4*f*h+g^2)^(1/2))*n*ln(d 
*x+c)*ln(-d*(g-(-4*f*h+g^2)^(1/2)+2*h*x)/(2*c*h-d*(g-(-4*f*h+g^2)^(1/2)))) 
/f-1/2*(1-g/(-4*f*h+g^2)^(1/2))*n*ln(b*x+a)*ln(-b*(g+(-4*f*h+g^2)^(1/2)+2* 
h*x)/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))/f+1/2*(1-g/(-4*f*h+g^2)^(1/2))*n*ln 
(d*x+c)*ln(-d*(g+(-4*f*h+g^2)^(1/2)+2*h*x)/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2)) 
))/f+1/2*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))*ln(h*x^2+g*x+ 
f)/f-1/2*(1+g/(-4*f*h+g^2)^(1/2))*n*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g-(-4* 
f*h+g^2)^(1/2))))/f-1/2*(1-g/(-4*f*h+g^2)^(1/2))*n*polylog(2,2*h*(b*x+a)/( 
2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))/f+n*polylog(2,1+b*x/a)/f+1/2*(1+g/(-4*f*h 
+g^2)^(1/2))*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g-(-4*f*h+g^2)^(1/2))))/f+1 
/2*(1-g/(-4*f*h+g^2)^(1/2))*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g+(-4*f*h+g^ 
2)^(1/2))))/f-n*polylog(2,1+d*x/c)/f
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 625, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx=\frac {2 \log (x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\left (1+\frac {g}{\sqrt {g^2-4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )-\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )-2 n \left (\log (x) \left (\log \left (1+\frac {b x}{a}\right )-\log \left (1+\frac {d x}{c}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )\right )+\frac {\left (g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{2 c h+d \left (-g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}+\frac {\left (-g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right )}{\sqrt {g^2-4 f h}}}{2 f} \] Input:

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

Output:

(2*Log[x]*Log[e*((a + b*x)/(c + d*x))^n] - (1 + g/Sqrt[g^2 - 4*f*h])*Log[e 
*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] - (1 - g/Sqrt 
[g^2 - 4*f*h])*Log[e*((a + b*x)/(c + d*x))^n]*Log[g + Sqrt[g^2 - 4*f*h] + 
2*h*x] - 2*n*(Log[x]*(Log[1 + (b*x)/a] - Log[1 + (d*x)/c]) + PolyLog[2, -( 
(b*x)/a)] - PolyLog[2, -((d*x)/c)]) + ((g + Sqrt[g^2 - 4*f*h])*n*((Log[(2* 
h*(a + b*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - Log[(2*h*(c + d*x)) 
/(-(d*g) + 2*c*h + d*Sqrt[g^2 - 4*f*h])])*Log[g - Sqrt[g^2 - 4*f*h] + 2*h* 
x] + PolyLog[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*S 
qrt[g^2 - 4*f*h])] - PolyLog[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c* 
h + d*(-g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h] + ((-g + Sqrt[g^2 - 4 
*f*h])*n*((Log[(2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - Log[ 
(2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])*Log[g + Sqrt[g^2 - 4 
*f*h] + 2*h*x] + PolyLog[2, (b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(-2*a*h + 
b*(g + Sqrt[g^2 - 4*f*h]))] - PolyLog[2, (d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x 
))/(-2*c*h + d*(g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h])/(2*f)
 

Rubi [A] (verified)

Time = 2.50 (sec) , antiderivative size = 719, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2993, 1144, 25, 1142, 1083, 219, 1103, 2865, 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.

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 1144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

\(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \left (\frac {\log (x)}{f}-\frac {\frac {1}{2} \int \frac {g+2 h x}{h x^2+g x+f}dx-\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{\sqrt {g^2-4 f h}}}{f}\right )\right )+n \int \frac {\log (a+b x)}{x \left (h x^2+g x+f\right )}dx-n \int \frac {\log (c+d x)}{x \left (h x^2+g x+f\right )}dx\)

\(\Big \downarrow \) 1103

\(\displaystyle n \int \frac {\log (a+b x)}{x \left (h x^2+g x+f\right )}dx-n \int \frac {\log (c+d x)}{x \left (h x^2+g x+f\right )}dx-\left (\left (\frac {\log (x)}{f}-\frac {\frac {1}{2} \log \left (f+g x+h x^2\right )-\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{\sqrt {g^2-4 f h}}}{f}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\)

\(\Big \downarrow \) 2865

\(\displaystyle n \int \left (\frac {\log (a+b x)}{f x}+\frac {(-g-h x) \log (a+b x)}{f \left (h x^2+g x+f\right )}\right )dx-n \int \left (\frac {\log (c+d x)}{f x}+\frac {(-g-h x) \log (c+d x)}{f \left (h x^2+g x+f\right )}\right )dx-\left (\left (\frac {\log (x)}{f}-\frac {\frac {1}{2} \log \left (f+g x+h x^2\right )-\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{\sqrt {g^2-4 f h}}}{f}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (\left (\frac {\log (x)}{f}-\frac {\frac {1}{2} \log \left (f+g x+h x^2\right )-\frac {g \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{\sqrt {g^2-4 f h}}}{f}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )+n \left (-\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (a+b x) \log \left (-\frac {b \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (a+b x) \log \left (-\frac {b \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 a h-b \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 f}+\frac {\operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right )}{f}+\frac {\log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f}\right )-n \left (-\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \operatorname {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log (c+d x) \log \left (-\frac {d \left (-\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f}-\frac {\left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log (c+d x) \log \left (-\frac {d \left (\sqrt {g^2-4 f h}+g+2 h x\right )}{2 c h-d \left (\sqrt {g^2-4 f h}+g\right )}\right )}{2 f}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{f}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f}\right )\)

Input:

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

Output:

-((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*(Log[ 
x]/f - (-((g*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]])/Sqrt[g^2 - 4*f*h]) + 
Log[f + g*x + h*x^2]/2)/f)) + n*((Log[-((b*x)/a)]*Log[a + b*x])/f - ((1 + 
g/Sqrt[g^2 - 4*f*h])*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x) 
)/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*f) - ((1 - g/Sqrt[g^2 - 4*f*h] 
)*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + S 
qrt[g^2 - 4*f*h])))])/(2*f) - ((1 + g/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*( 
a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*f) - ((1 - g/Sqrt[g^2 - 
 4*f*h])*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/ 
(2*f) + PolyLog[2, 1 + (b*x)/a]/f) - n*((Log[-((d*x)/c)]*Log[c + d*x])/f - 
 ((1 + g/Sqrt[g^2 - 4*f*h])*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 
 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*f) - ((1 - g/Sqrt[g^2 - 
 4*f*h])*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d 
*(g + Sqrt[g^2 - 4*f*h])))])/(2*f) - ((1 + g/Sqrt[g^2 - 4*f*h])*PolyLog[2, 
 (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*f) - ((1 - g/Sqr 
t[g^2 - 4*f*h])*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f* 
h]))])/(2*f) + PolyLog[2, 1 + (d*x)/c]/f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1144

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] 
 :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S 
imp[1/(c*d^2 - b*d*e + a*e^2)   Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), 
x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2009

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

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

rule 2993

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r   Int[RFx*Log[a + b*x], x], x] + (Si 
mp[q*r   Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c 
+ d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])   Int[RFx, x], x]) /; FreeQ[ 
{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a 
*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ 
m, n]]
 

Maple [F]

\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{x \left (h \,x^{2}+g x +f \right )}d x\]

Input:

int(ln(e*((b*x+a)/(d*x+c))^n)/x/(h*x^2+g*x+f),x)
 

Output:

int(ln(e*((b*x+a)/(d*x+c))^n)/x/(h*x^2+g*x+f),x)
 

Fricas [F]

\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx=\int { \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (h x^{2} + g x + f\right )} x} \,d x } \] Input:

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

Output:

integral(log(e*((b*x + a)/(d*x + c))^n)/(h*x^3 + g*x^2 + f*x), x)
 

Sympy [F(-1)]

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

integrate(ln(e*((b*x+a)/(d*x+c))**n)/x/(h*x**2+g*x+f),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*f*h-g^2>0)', see `assume?` for 
 more deta
 

Giac [F]

\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx=\int { \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (h x^{2} + g x + f\right )} x} \,d x } \] Input:

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

Output:

integrate(log(e*((b*x + a)/(d*x + c))^n)/((h*x^2 + g*x + f)*x), x)
 

Mupad [F(-1)]

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

int(log(e*((a + b*x)/(c + d*x))^n)/(x*(f + g*x + h*x^2)),x)
 

Output:

int(log(e*((a + b*x)/(c + d*x))^n)/(x*(f + g*x + h*x^2)), x)
 

Reduce [F]

\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x \left (f+g x+h x^2\right )} \, dx=\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{h \,x^{3}+g \,x^{2}+f x}d x \] Input:

int(log(e*((b*x+a)/(d*x+c))^n)/x/(h*x^2+g*x+f),x)
 

Output:

int(log(((a + b*x)**n*e)/(c + d*x)**n)/(f*x + g*x**2 + h*x**3),x)