∫ log 3 (e(a+bx c+dx)n) log(h(f+gx)m) ________________________________________

3.67 \(\int \frac{\log ^3(e (\frac{a+b x}{c+d x})^n) \log (h (f+g x)^m)}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=620 \[ -\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (4,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (3,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{6 m n^2 \text{PolyLog}\left (4,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{3 m n \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{6 m n^3 \text{PolyLog}\left (5,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{6 m n^3 \text{PolyLog}\left (5,\frac{d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{4 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)} \]

[Out]

(m*Log[e*((a + b*x)/(c + d*x))^n]^4*Log[(b*c - a*d)/(b*(c + d*x))])/(4*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c +

d*x))^n]^4*Log[h*(f + g*x)^m])/(4*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c + d*x))^n]^4*Log[1 - ((d*f - c*g)*(

a + b*x))/((b*f - a*g)*(c + d*x))])/(4*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, (d*(a +

b*x))/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, ((d*f - c*g)*(a + b*x))/((

b*f - a*g)*(c + d*x))])/(b*c - a*d) - (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, (d*(a + b*x))/(b*(c +

d*x))])/(b*c - a*d) + (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)

*(c + d*x))])/(b*c - a*d) + (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))])/(

b*c - a*d) - (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x)

)])/(b*c - a*d) - (6*m*n^3*PolyLog[5, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (6*m*n^3*PolyLog[5, ((d*f -

c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)

________________________________________________________________________________________

Rubi [A] time = 1.13263, antiderivative size = 649, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {2507, 2489, 2488, 2506, 2508, 6610, 2503} \[ -\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (4,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (3,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{6 m n^2 \text{PolyLog}\left (4,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac{3 m n \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{6 m n^3 \text{PolyLog}\left (5,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{6 m n^3 \text{PolyLog}\left (5,1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{4 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(Log[e*((a + b*x)/(c + d*x))^n]^3*Log[h*(f + g*x)^m])/((a + b*x)*(c + d*x)),x]

[Out]

(m*Log[e*((a + b*x)/(c + d*x))^n]^4*Log[(b*c - a*d)/(b*(c + d*x))])/(4*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c

+ d*x))^n]^4*Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])/(4*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c +

d*x))^n]^4*Log[h*(f + g*x)^m])/(4*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, 1 - (b*c -

a*d)/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, 1 - ((b*c - a*d)*(f + g*x))/

((b*f - a*g)*(c + d*x))])/(b*c - a*d) - (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, 1 - (b*c - a*d)/(b*

(c + d*x))])/(b*c - a*d) + (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, 1 - ((b*c - a*d)*(f + g*x))/((b*

f - a*g)*(c + d*x))])/(b*c - a*d) + (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, 1 - (b*c - a*d)/(b*(c +

d*x))])/(b*c - a*d) - (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, 1 - ((b*c - a*d)*(f + g*x))/((b*f -

a*g)*(c + d*x))])/(b*c - a*d) - (6*m*n^3*PolyLog[5, 1 - (b*c - a*d)/(b*(c + d*x))])/(b*c - a*d) + (6*m*n^3*Pol

yLog[5, 1 - ((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)

Rule 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_

.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j

*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/

(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]

/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,

-1]

Rule 2489

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)/((g_.) + (h_.)*(x_)),

x_Symbol] :> Dist[d/h, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[

Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r

, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IGtQ[s, 1]

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),

x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p

*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a

+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,

0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo

l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo

g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log

[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,

c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 2508

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_)*PolyLog[n_, v_],

x_Symbol] :> With[{g = Simplify[(v*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*PolyL

og[n + 1, v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] - Dist[h*p*r*s, Int[(PolyLog[n + 1, v]*Lo

g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,

c, d, e, f, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /

; !FalseQ[w]] /; FreeQ[n, x]

Rule 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi

th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(

a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(

p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x)

)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ

[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/

(u*(a + b*x))], x]

Rubi steps

\begin{align*} \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx &=\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-\frac{(g m) \int \frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{4 (b c-a d) n}\\ &=\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-\frac{(d m) \int \frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4 (b c-a d) n}+\frac{((d f-c g) m) \int \frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x) (f+g x)} \, dx}{4 (b c-a d) n}\\ &=\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-m \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+m \int \frac{\log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (-\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-(3 m n) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+(3 m n) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\left (6 m n^2\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx-\left (6 m n^2\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1+\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_4\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_4\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\left (6 m n^3\right ) \int \frac{\text{Li}_4\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+\left (6 m n^3\right ) \int \frac{\text{Li}_4\left (1+\frac{(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac{m \log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac{\log ^4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{3 m n \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_3\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_4\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{6 m n^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_4\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac{6 m n^3 \text{Li}_5\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{6 m n^3 \text{Li}_5\left (1-\frac{(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}\\ \end{align*}

Mathematica [B] time = 21.5443, size = 18164, normalized size = 29.3 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Log[e*((a + b*x)/(c + d*x))^n]^3*Log[h*(f + g*x)^m])/((a + b*x)*(c + d*x)),x]

[Out]

Result too large to show

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Maple [F] time = 7.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{ \left ( bx+a \right ) \left ( dx+c \right ) } \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*((b*x+a)/(d*x+c))^n)^3*ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x)

[Out]

int(ln(e*((b*x+a)/(d*x+c))^n)^3*ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x)

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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-1/4*(n^3*log(b*x + a)^4 + n^3*log(d*x + c)^4 - 4*n^2*log(b*x + a)^3*log(e) + 6*n*log(b*x + a)^2*log(e)^2 - 4*

(n^3*log(b*x + a) - n^2*log(e))*log(d*x + c)^3 - 4*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n)^3 + 4*(log(b

*x + a) - log(d*x + c))*log((d*x + c)^n)^3 - 4*log(b*x + a)*log(e)^3 + 6*(n^3*log(b*x + a)^2 - 2*n^2*log(b*x +

a)*log(e) + n*log(e)^2)*log(d*x + c)^2 + 6*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))

*log(d*x + c) - 2*log(b*x + a)*log(e))*log((b*x + a)^n)^2 + 6*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(

b*x + a) - log(e))*log(d*x + c) - 2*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n) - 2*log(b*x + a)*log(e))*lo

g((d*x + c)^n)^2 - 4*(n^3*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(e) + 3*n*log(b*x + a)*log(e)^2 - log(e)^3)

*log(d*x + c) - 4*(n^2*log(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) + 3*(n^2*log(b*x + a) -

n*log(e))*log(d*x + c)^2 + 3*log(b*x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n*log(b*x + a)*log(e) + log(e)

^2)*log(d*x + c))*log((b*x + a)^n) + 4*(n^2*log(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) +

3*(n^2*log(b*x + a) - n*log(e))*log(d*x + c)^2 + 3*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n)^2 + 3*log(b*

x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n*log(b*x + a)*log(e) + log(e)^2)*log(d*x + c) - 3*(n*log(b*x + a)

^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))*log(d*x + c) - 2*log(b*x + a)*log(e))*log((b*x + a)^n))*lo

g((d*x + c)^n))*log((g*x + f)^m)/(b*c - a*d) + integrate(1/4*(4*b*c*f*log(e)^3*log(h) - 4*a*d*f*log(e)^3*log(h

) + (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(b*x + a)^4 + (b*d*g*m*n^3*x^2 + a*c*g*

m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(d*x + c)^4 - 4*(b*d*g*m*n^2*x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c*

g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a)^3 + 4*(b*d*g*m*n^2*x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c

*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x - (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(

b*x + a))*log(d*x + c)^3 + 4*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a

*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*l

og((b*x + a)^n)^3 - 4*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g*m

+ (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((d*x

+ c)^n)^3 + 6*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*log(e)^2)*x)*log

(b*x + a)^2 + 6*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 +

a*d*g*m*n^3)*x)*log(b*x + a)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*log(e)^2)*x - 2*(b*d*g*m*n^2*x^2*log(e) + a*

c*g*m*n^2*log(e) + (b*c*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a))*log(d*x + c)^2 + 6*(2*b*c*f*log(

e)*log(h) - 2*a*d*f*log(e)*log(h) + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (

b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*

log(h))*x - 2*(b*d*g*m*x^2*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*

d*g*m*x^2*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*

m*n + a*d*g*m*n)*x)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n)^2 + 6*(2*b*c*f*log(e)*log(h) - 2*a*d*f*log(e)

*log(h) + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (b*d*g*m*n*x^2 + a*c*g*m*n

+ (b*c*g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*log(h))*x - 2*(b*d*g*m*x^2

*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*d*g*m*x^2*log(e) + a*c*g*m

*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*

x + a))*log(d*x + c) + 2*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g

*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((

b*x + a)^n))*log((d*x + c)^n)^2 + 4*(b*c*g*log(e)^3*log(h) - a*d*g*log(e)^3*log(h))*x - 4*(b*d*g*m*x^2*log(e)^

3 + a*c*g*m*log(e)^3 + (b*c*g*m*log(e)^3 + a*d*g*m*log(e)^3)*x)*log(b*x + a) + 4*(b*d*g*m*x^2*log(e)^3 + a*c*g

*m*log(e)^3 - (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(b*x + a)^3 + 3*(b*d*g*m*n^2*

x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a)^2 + (b*c*g*m*log(e

)^3 + a*d*g*m*log(e)^3)*x - 3*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*l

og(e)^2)*x)*log(b*x + a))*log(d*x + c) + 4*(3*b*c*f*log(e)^2*log(h) - 3*a*d*f*log(e)^2*log(h) - (b*d*g*m*n^2*x

^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a)^3 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n

^2 + a*d*g*m*n^2)*x)*log(d*x + c)^3 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m

*n*log(e))*x)*log(b*x + a)^2 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(

e))*x - (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a))*log(d*x + c)^2 + 3*(b*c*

g*log(e)^2*log(h) - a*d*g*log(e)^2*log(h))*x - 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*c*g*m*log(e)^2

+ a*d*g*m*log(e)^2)*x)*log(b*x + a) + 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*d*g*m*n^2*x^2 + a*c*g*m*

n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x - 2*(b*d*g*m*n*x

^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a))*log(d*x + c))*log((b*x +

a)^n) - 4*(3*b*c*f*log(e)^2*log(h) - 3*a*d*f*log(e)^2*log(h) - (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2

+ a*d*g*m*n^2)*x)*log(b*x + a)^3 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(d*x + c

)^3 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a)^2 + 3

*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x - (b*d*g*m*n^2*x^2 + a*c*g

*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a))*log(d*x + c)^2 + 3*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g

*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*

g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((b*x + a)^n)^2 + 3*(b*c*g*log(e)^2*log(h) - a*d*g*log(e)^2*log(

h))*x - 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x)*log(b*x + a) + 3

*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*lo

g(b*x + a)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x - 2*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m

*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a))*log(d*x + c) + 3*(2*b*c*f*log(e)*log(h) - 2*a*d*f*log(e)*log(h)

+ (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*

g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*log(h))*x - 2*(b*d*g*m*x^2*log(e)

+ a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*d*g*m*x^2*log(e) + a*c*g*m*log(e)

+ (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a))

*log(d*x + c))*log((b*x + a)^n))*log((d*x + c)^n))/(a*b*c^2*f - a^2*c*d*f + (b^2*c*d*g - a*b*d^2*g)*x^3 - (a*b

*d^2*f + a^2*d^2*g - (c*d*f + c^2*g)*b^2)*x^2 + (b^2*c^2*f + a*b*c^2*g - (d^2*f + c*d*g)*a^2)*x), x)

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

[Out]

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

[Out]

Timed out

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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{3}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{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)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

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