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

Optimal. Leaf size=800 \[ \frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \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 ) \log (a+b x)}{2 f}-\frac{\left (1-\frac{g}{\sqrt{g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{g \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 \sqrt{g^2-4 f h}}-\frac{\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}+\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\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}+\frac{\left (1-\frac{g}{\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}+\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 ) \log \left (h x^2+g x+f\right )}{2 f}-\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \text{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 ) n \text{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{n \text{PolyLog}\left (2,\frac{b x}{a}+1\right )}{f}+\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \text{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 ) n \text{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{n \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.980036, antiderivative size = 800, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2513, 2418, 2394, 2315, 2393, 2391, 705, 29, 634, 618, 206, 628} \[ \frac{n \log \left (-\frac{b x}{a}\right ) \log (a+b x)}{f}-\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \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 ) \log (a+b x)}{2 f}-\frac{\left (1-\frac{g}{\sqrt{g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 f}-\frac{n \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{f}-\frac{g \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 \sqrt{g^2-4 f h}}-\frac{\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}+\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\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}+\frac{\left (1-\frac{g}{\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}+\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 ) \log \left (h x^2+g x+f\right )}{2 f}-\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \text{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 ) n \text{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{n \text{PolyLog}\left (2,\frac{b x}{a}+1\right )}{f}+\frac{\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) n \text{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 ) n \text{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{n \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2513

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]]

Rule 2418

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 2394

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 2315

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

Rule 2393

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 2391

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 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[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] && 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]

Rule 29

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

Rule 634

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 618

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 206

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

Rule 628

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]

Rubi steps

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

Mathematica [A]  time = 0.940115, size = 625, normalized size = 0.78 \[ \frac{\frac{n \left (\sqrt{g^2-4 f h}+g\right ) \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{g^2-4 f h}-g-2 h x\right )}{2 a h+b \sqrt{g^2-4 f h}+b (-g)}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{g^2-4 f h}-g-2 h x\right )}{2 c h+d \left (\sqrt{g^2-4 f h}-g\right )}\right )+\log \left (-\sqrt{g^2-4 f h}+g+2 h x\right ) \left (\log \left (\frac{2 h (a+b x)}{2 a h+b \sqrt{g^2-4 f h}+b (-g)}\right )-\log \left (\frac{2 h (c+d x)}{2 c h+d \sqrt{g^2-4 f h}+d (-g)}\right )\right )\right )}{\sqrt{g^2-4 f h}}+\frac{n \left (\sqrt{g^2-4 f h}-g\right ) \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{g^2-4 f h}+g+2 h x\right )}{b \left (\sqrt{g^2-4 f h}+g\right )-2 a h}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{g^2-4 f h}+g+2 h x\right )}{d \left (\sqrt{g^2-4 f h}+g\right )-2 c h}\right )+\log \left (\sqrt{g^2-4 f h}+g+2 h x\right ) \left (\log \left (\frac{2 h (a+b x)}{2 a h-b \left (\sqrt{g^2-4 f h}+g\right )}\right )-\log \left (\frac{2 h (c+d x)}{2 c h-d \left (\sqrt{g^2-4 f h}+g\right )}\right )\right )\right )}{\sqrt{g^2-4 f h}}-2 n \left (\text{PolyLog}\left (2,-\frac{b x}{a}\right )-\text{PolyLog}\left (2,-\frac{d x}{c}\right )+\log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log \left (\frac{d x}{c}+1\right )\right )\right )-\left (\frac{g}{\sqrt{g^2-4 f h}}+1\right ) \log \left (-\sqrt{g^2-4 f h}+g+2 h x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\left (1-\frac{g}{\sqrt{g^2-4 f h}}\right ) \log \left (\sqrt{g^2-4 f h}+g+2 h x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 \log (x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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 - Sq
rt[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*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 + 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)

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Maple [F]  time = 1.414, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( h{x}^{2}+gx+f \right ) }\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{h x^{3} + g x^{2} + f x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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