\(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{x^2 (f+g x+h x^2)} \, dx\) [87]

Optimal result

Integrand size = 34, antiderivative size = 995 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f+g x+h x^2\right )} \, dx=\frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {\left (g^2-2 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2 \sqrt {g^2-4 f h}}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}-\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}-\frac {g n \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}+\frac {g n \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2593, 2465, 2442, 36, 29, 31, 2441, 2352, 2440, 2438, 723, 814, 648, 632, 212, 642} \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f+g x+h x^2\right )} \, dx=\frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}+\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log (x)}{f^2}-\frac {b n \log (a+b x)}{a f}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (c+d x)}{f x}+\frac {\left (g^2-2 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2 \sqrt {g^2-4 f h}}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}-\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}-\frac {g n \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right )}{f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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^2}+\frac {g n \operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{f^2} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> 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] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2593

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[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_.) /; Integ
ersQ[m, n]]

Rubi steps \begin{align*} \text {integral}& = n \int \frac {\log (a+b x)}{x^2 \left (f+g x+h x^2\right )} \, dx-n \int \frac {\log (c+d x)}{x^2 \left (f+g x+h x^2\right )} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {1}{x^2 \left (f+g x+h x^2\right )} \, dx \\ & = \frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+n \int \left (\frac {\log (a+b x)}{f x^2}-\frac {g \log (a+b x)}{f^2 x}+\frac {\left (g^2-f h+g h x\right ) \log (a+b x)}{f^2 \left (f+g x+h x^2\right )}\right ) \, dx-n \int \left (\frac {\log (c+d x)}{f x^2}-\frac {g \log (c+d x)}{f^2 x}+\frac {\left (g^2-f h+g h x\right ) \log (c+d x)}{f^2 \left (f+g x+h x^2\right )}\right ) \, dx-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {-g-h x}{x \left (f+g x+h x^2\right )} \, dx}{f} \\ & = \frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {n \int \frac {\left (g^2-f h+g h x\right ) \log (a+b x)}{f+g x+h x^2} \, dx}{f^2}-\frac {n \int \frac {\left (g^2-f h+g h x\right ) \log (c+d x)}{f+g x+h x^2} \, dx}{f^2}+\frac {n \int \frac {\log (a+b x)}{x^2} \, dx}{f}-\frac {n \int \frac {\log (c+d x)}{x^2} \, dx}{f}-\frac {(g n) \int \frac {\log (a+b x)}{x} \, dx}{f^2}+\frac {(g n) \int \frac {\log (c+d x)}{x} \, dx}{f^2}-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \left (-\frac {g}{f x}+\frac {g^2-f h+g h x}{f \left (f+g x+h x^2\right )}\right ) \, dx}{f} \\ & = -\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}+\frac {n \int \left (\frac {\left (g h+\frac {h \left (g^2-2 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g h-\frac {h \left (g^2-2 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{f^2}-\frac {n \int \left (\frac {\left (g h+\frac {h \left (g^2-2 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g h-\frac {h \left (g^2-2 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{f^2}+\frac {(b n) \int \frac {1}{x (a+b x)} \, dx}{f}-\frac {(d n) \int \frac {1}{x (c+d x)} \, dx}{f}+\frac {(b g n) \int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx}{f^2}-\frac {(d g n) \int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx}{f^2}-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {g^2-f h+g h x}{f+g x+h x^2} \, dx}{f^2} \\ & = -\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}-\frac {g n \text {Li}_2\left (1+\frac {b x}{a}\right )}{f^2}+\frac {g n \text {Li}_2\left (1+\frac {d x}{c}\right )}{f^2}+\frac {(b n) \int \frac {1}{x} \, dx}{a f}-\frac {\left (b^2 n\right ) \int \frac {1}{a+b x} \, dx}{a f}-\frac {(d n) \int \frac {1}{x} \, dx}{c f}+\frac {\left (d^2 n\right ) \int \frac {1}{c+d x} \, dx}{c f}+\frac {\left (h \left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{f^2}-\frac {\left (h \left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{f^2}+\frac {\left (h \left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{f^2}-\frac {\left (h \left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{f^2}-\frac {\left (g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {g+2 h x}{f+g x+h x^2} \, dx}{2 f^2}-\frac {\left (\left (g^2-2 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {1}{f+g x+h x^2} \, dx}{2 f^2} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}-\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 f^2}-\frac {g n \text {Li}_2\left (1+\frac {b x}{a}\right )}{f^2}+\frac {g n \text {Li}_2\left (1+\frac {d x}{c}\right )}{f^2}-\frac {\left (b \left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\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 )}{a+b x} \, dx}{2 f^2}+\frac {\left (d \left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\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 )}{c+d x} \, dx}{2 f^2}-\frac {\left (b \left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\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 )}{a+b x} \, dx}{2 f^2}+\frac {\left (d \left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\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 )}{c+d x} \, dx}{2 f^2}+\frac {\left (\left (g^2-2 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \text {Subst}\left (\int \frac {1}{g^2-4 f h-x^2} \, dx,x,g+2 h x\right )}{f^2} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2 \sqrt {g^2-4 f h}}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}-\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 f^2}-\frac {g n \text {Li}_2\left (1+\frac {b x}{a}\right )}{f^2}+\frac {g n \text {Li}_2\left (1+\frac {d x}{c}\right )}{f^2}-\frac {\left (\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 f^2}+\frac {\left (\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 f^2}-\frac {\left (\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 f^2}+\frac {\left (\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 f^2} \\ & = \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}-\frac {g n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f^2}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {g n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f^2}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+\frac {\left (g^2-2 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2 \sqrt {g^2-4 f h}}+\frac {g \log (x) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\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 )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\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 )}{2 f^2}-\frac {g \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 f^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}-\frac {g n \text {Li}_2\left (1+\frac {b x}{a}\right )}{f^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 f^2}+\frac {g n \text {Li}_2\left (1+\frac {d x}{c}\right )}{f^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2 \left (f+g x+h x^2\right )} \, dx=\frac {-\frac {2 f \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x}-2 g \log (x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\frac {2 f n ((b c-a d) \log (x)-b c \log (a+b x)+a d \log (c+d x))}{a c}+\left (g+\frac {g^2-2 f h}{\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 (g+\frac {-g^2+2 f h}{\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 g 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^2-2 f h+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^2-2 f h-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^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(log(e*((b*x + a)/(d*x + c))^n)/(h*x^4 + g*x^3 + f*x^2), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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