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

Optimal. Leaf size=637 \[ a g m \text{PolyLog}\left (2,\frac{j x}{i}+1\right )-b g m \text{PolyLog}\left (2,\frac{j x}{i}+1\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+b f n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )+b g m n \text{PolyLog}\left (3,\frac{i (d+e x)}{d (i+j x)}\right )-b g m n \text{PolyLog}\left (3,\frac{j (d+e x)}{e (i+j x)}\right )+b g m n \log \left (\frac{d (i+j x)}{i (d+e x)}\right ) \text{PolyLog}\left (2,\frac{i (d+e x)}{d (i+j x)}\right )-b g m n \log \left (\frac{d (i+j x)}{i (d+e x)}\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{e (i+j x)}\right )+b g m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (\log (i+j x)-\log \left (\frac{d (i+j x)}{i (d+e x)}\right )\right )+b g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right ) \left (\log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )-b g m n \text{PolyLog}\left (3,\frac{e x}{d}+1\right )-b g m n \text{PolyLog}\left (3,\frac{j x}{i}+1\right )+f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g \log \left (-\frac{j x}{i}\right ) \log \left (h (i+j x)^m\right )-b g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )-b g m \log \left (-\frac{j x}{i}\right ) \log (i+j x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+\frac{1}{2} b g m n \left (\log \left (\frac{e i-d j}{e (i+j x)}\right )-\log \left (-\frac{x (e i-d j)}{d (i+j x)}\right )+\log \left (-\frac{e x}{d}\right )\right ) \log ^2\left (\frac{d (i+j x)}{i (d+e x)}\right )-\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{j x}{i}\right )\right ) \left (\log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )^2+b g m n \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log (i+j x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.432721, antiderivative size = 637, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {2438, 2394, 2315, 2437, 2435} \[ a g m \text{PolyLog}\left (2,\frac{j x}{i}+1\right )-b g m \text{PolyLog}\left (2,\frac{j x}{i}+1\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+b f n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )+b g m n \text{PolyLog}\left (3,\frac{i (d+e x)}{d (i+j x)}\right )-b g m n \text{PolyLog}\left (3,\frac{j (d+e x)}{e (i+j x)}\right )+b g m n \log \left (\frac{d (i+j x)}{i (d+e x)}\right ) \text{PolyLog}\left (2,\frac{i (d+e x)}{d (i+j x)}\right )-b g m n \log \left (\frac{d (i+j x)}{i (d+e x)}\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{e (i+j x)}\right )+b g m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (\log (i+j x)-\log \left (\frac{d (i+j x)}{i (d+e x)}\right )\right )+b g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right ) \left (\log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )-b g m n \text{PolyLog}\left (3,\frac{e x}{d}+1\right )-b g m n \text{PolyLog}\left (3,\frac{j x}{i}+1\right )+f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g \log \left (-\frac{j x}{i}\right ) \log \left (h (i+j x)^m\right )-b g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )-b g m \log \left (-\frac{j x}{i}\right ) \log (i+j x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+\frac{1}{2} b g m n \left (\log \left (\frac{e i-d j}{e (i+j x)}\right )-\log \left (-\frac{x (e i-d j)}{d (i+j x)}\right )+\log \left (-\frac{e x}{d}\right )\right ) \log ^2\left (\frac{d (i+j x)}{i (d+e x)}\right )-\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{j x}{i}\right )\right ) \left (\log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )^2+b g m n \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log (i+j x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2438

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

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 2437

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (390+j x)^m\right )\right )}{x} \, dx &=f \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx+g \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (390+j x)^m\right )}{x} \, dx\\ &=f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+(a g) \int \frac{\log \left (h (390+j x)^m\right )}{x} \, dx+(b g) \int \frac{\log \left (c (d+e x)^n\right ) \log \left (h (390+j x)^m\right )}{x} \, dx-(b e f n) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx\\ &=f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g \log \left (-\frac{j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text{Li}_2\left (1+\frac{e x}{d}\right )+(b g m) \int \frac{\log \left (c (d+e x)^n\right ) \log (390+j x)}{x} \, dx-(a g j m) \int \frac{\log \left (-\frac{j x}{390}\right )}{390+j x} \, dx-\left (b g \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )\right ) \int \frac{\log \left (c (d+e x)^n\right )}{x} \, dx\\ &=f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac{j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text{Li}_2\left (1+\frac{e x}{d}\right )+a g m \text{Li}_2\left (1+\frac{j x}{390}\right )+(b g m n) \int \frac{\log (d+e x) \log (390+j x)}{x} \, dx-\left (b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right ) \int \frac{\log (390+j x)}{x} \, dx+\left (b e g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx\\ &=-b g m \log (390) \log (x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log (390+j x)+\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )+\log \left (\frac{390 e-d j}{e (390+j x)}\right )-\log \left (-\frac{(390 e-d j) x}{d (390+j x)}\right )\right ) \log ^2\left (\frac{d (390+j x)}{390 (d+e x)}\right )-\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{j x}{390}\right )\right ) \left (\log (d+e x)+\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right )^2-b g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac{j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text{Li}_2\left (1+\frac{e x}{d}\right )+b g m n \left (\log (390+j x)-\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )-b g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )+a g m \text{Li}_2\left (1+\frac{j x}{390}\right )+b g m n \left (\log (d+e x)+\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right ) \text{Li}_2\left (1+\frac{j x}{390}\right )+b g m n \log \left (\frac{d (390+j x)}{390 (d+e x)}\right ) \text{Li}_2\left (\frac{390 (d+e x)}{d (390+j x)}\right )-b g m n \log \left (\frac{d (390+j x)}{390 (d+e x)}\right ) \text{Li}_2\left (\frac{j (d+e x)}{e (390+j x)}\right )-b g m n \text{Li}_3\left (1+\frac{e x}{d}\right )-b g m n \text{Li}_3\left (1+\frac{j x}{390}\right )+b g m n \text{Li}_3\left (\frac{390 (d+e x)}{d (390+j x)}\right )-b g m n \text{Li}_3\left (\frac{j (d+e x)}{e (390+j x)}\right )-\left (b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right ) \int \frac{\log \left (1+\frac{j x}{390}\right )}{x} \, dx\\ &=-b g m \log (390) \log (x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+f \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log (390+j x)+\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )+\log \left (\frac{390 e-d j}{e (390+j x)}\right )-\log \left (-\frac{(390 e-d j) x}{d (390+j x)}\right )\right ) \log ^2\left (\frac{d (390+j x)}{390 (d+e x)}\right )-\frac{1}{2} b g m n \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{j x}{390}\right )\right ) \left (\log (d+e x)+\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right )^2-b g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac{j x}{390}\right ) \log \left (h (390+j x)^m\right )+b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j x}{390}\right )+b f n \text{Li}_2\left (1+\frac{e x}{d}\right )+b g m n \left (\log (390+j x)-\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )-b g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )+a g m \text{Li}_2\left (1+\frac{j x}{390}\right )+b g m n \left (\log (d+e x)+\log \left (\frac{d (390+j x)}{390 (d+e x)}\right )\right ) \text{Li}_2\left (1+\frac{j x}{390}\right )+b g m n \log \left (\frac{d (390+j x)}{390 (d+e x)}\right ) \text{Li}_2\left (\frac{390 (d+e x)}{d (390+j x)}\right )-b g m n \log \left (\frac{d (390+j x)}{390 (d+e x)}\right ) \text{Li}_2\left (\frac{j (d+e x)}{e (390+j x)}\right )-b g m n \text{Li}_3\left (1+\frac{e x}{d}\right )-b g m n \text{Li}_3\left (1+\frac{j x}{390}\right )+b g m n \text{Li}_3\left (\frac{390 (d+e x)}{d (390+j x)}\right )-b g m n \text{Li}_3\left (\frac{j (d+e x)}{e (390+j x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.343942, size = 605, normalized size = 0.95 \[ a g m \left (\log (x) \left (\log (i+j x)-\log \left (\frac{j x}{i}+1\right )\right )-\text{PolyLog}\left (2,-\frac{j x}{i}\right )\right )+b g m \left (\log (x) \left (\log (i+j x)-\log \left (\frac{j x}{i}+1\right )\right )-\text{PolyLog}\left (2,-\frac{j x}{i}\right )\right ) \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )+b n \left (\log (x) \left (\log (d+e x)-\log \left (\frac{e x}{d}+1\right )\right )-\text{PolyLog}\left (2,-\frac{e x}{d}\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )-g m \log (i+j x)\right )+b g m n \left (\text{PolyLog}\left (3,\frac{d (i+j x)}{i (d+e x)}\right )-\text{PolyLog}\left (3,\frac{e (i+j x)}{j (d+e x)}\right )+\log \left (\frac{d (i+j x)}{i (d+e x)}\right ) \left (\text{PolyLog}\left (2,\frac{e (i+j x)}{j (d+e x)}\right )-\text{PolyLog}\left (2,\frac{d (i+j x)}{i (d+e x)}\right )\right )+\text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (\log (i+j x)-\log \left (\frac{d (i+j x)}{i (d+e x)}\right )\right )+\text{PolyLog}\left (2,\frac{j x}{i}+1\right ) \left (\log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )-\text{PolyLog}\left (3,\frac{e x}{d}+1\right )-\text{PolyLog}\left (3,\frac{j x}{i}+1\right )+\frac{1}{2} \left (\log \left (\frac{d j-e i}{j (d+e x)}\right )-\log \left (\frac{e i x-d j x}{d i+e i x}\right )+\log \left (-\frac{e x}{d}\right )\right ) \log ^2\left (\frac{d (i+j x)}{i (d+e x)}\right )+\log \left (\frac{j x}{i}+1\right ) \left (\log \left (-\frac{j x}{i}\right )-\log \left (-\frac{e x}{d}\right )\right ) \log \left (\frac{d (i+j x)}{i (d+e x)}\right )+\log \left (-\frac{e x}{d}\right ) \log (d+e x) \log (i+j x)+\frac{1}{2} \log \left (\frac{j x}{i}+1\right ) \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{j x}{i}\right )\right ) \left (\log \left (\frac{j x}{i}+1\right )-2 \log (d+e x)\right )\right )+\log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right ) \left (f+g \log \left (h (i+j x)^m\right )-g m \log (i+j x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.334, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a f \log \left (x\right ) + \int \frac{{\left (g \log \left (h\right ) + f\right )} b \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (g \log \left (h\right ) + f\right )} b \log \left (c\right ) + a g \log \left (h\right ) +{\left (b g \log \left ({\left (e x + d\right )}^{n}\right ) + b g \log \left (c\right ) + a g\right )} \log \left ({\left (j x + i\right )}^{m}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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