3.398 \(\int (a+b \log (c (d+e x)^n))^3 (f+g \log (h (i+j x)^m)) \, dx\)

Optimal. Leaf size=1147 \[ \text{result too large to display} \]

[Out]

6*a*b^2*f*n^2*x - 18*a*b^2*g*m*n^2*x - 6*b^3*f*n^3*x + 24*b^3*g*m*n^3*x + (6*b^3*f*n^2*(d + e*x)*Log[c*(d + e*
x)^n])/e - (18*b^3*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/
e + (6*b*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (d*f*(a + b*Log[c*(d + e*x)^n])^3)/e - (g*m*(d + e*
x)*(a + b*Log[c*(d + e*x)^n])^3)/e + (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)
])/j + (3*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e - (3*b*g*i*m*n*(a + b*Log[c
*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*
i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/j - (6*b^3*g*n^3*(i + j*x)*
Log[h*(i + j*x)^m])/j + (6*b^3*d*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/e + 6*b^2*g*n^2*x
*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (3*b*d*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e
- 3*b*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*
x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) + (6*b^3*g*i*m*n^3*PolyLog[2, -((j*(d + e
*x))/(e*i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/
e - (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (3*b*d*g*m*n*(a
+ b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e + (3*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n]
)^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^3*d*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e -
(6*b^3*d*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*
i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e - (6*b
^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j - (6*b^3*d*g*m*n^3*PolyLog
[4, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j

________________________________________________________________________________________

Rubi [A]  time = 3.12295, antiderivative size = 1147, normalized size of antiderivative = 1., number of steps used = 64, number of rules used = 22, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.71, Rules used = {2430, 2416, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589, 6742, 2411, 2346, 2302, 30, 2301, 43, 2394, 2393, 2391, 2375, 2317} \[ -6 f n^3 x b^3+24 g m n^3 x b^3+\frac{6 f n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e}-\frac{18 g m n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e}-\frac{6 g n^3 (i+j x) \log \left (h (i+j x)^m\right ) b^3}{j}+\frac{6 d g n^3 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) b^3}{e}+\frac{6 g i m n^3 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b^3}{j}+\frac{6 d g m n^3 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right ) b^3}{e}-\frac{6 d g m n^3 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^3}{e}+\frac{6 g i m n^3 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^3}{j}-\frac{6 d g m n^3 \text{PolyLog}\left (4,-\frac{j (d+e x)}{e i-d j}\right ) b^3}{e}+\frac{6 g i m n^3 \text{PolyLog}\left (4,-\frac{j (d+e x)}{e i-d j}\right ) b^3}{j}+6 a f n^2 x b^2-18 a g m n^2 x b^2+\frac{6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right ) b^2}{j}+6 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b^2+\frac{6 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{e}-\frac{6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{j}+\frac{6 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{e}-\frac{6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right ) b^2}{j}-\frac{3 f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e}+\frac{6 g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e}+\frac{3 d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (i+j x)}{e i-d j}\right ) b}{e}-\frac{3 g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (i+j x)}{e i-d j}\right ) b}{j}-\frac{3 d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right ) b}{e}-3 g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right ) b-\frac{3 d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b}{e}+\frac{3 g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) b}{j}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{g i m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (i+j x)}{e i-d j}\right )}{j}+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (i+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

6*a*b^2*f*n^2*x - 18*a*b^2*g*m*n^2*x - 6*b^3*f*n^3*x + 24*b^3*g*m*n^3*x + (6*b^3*f*n^2*(d + e*x)*Log[c*(d + e*
x)^n])/e - (18*b^3*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/
e + (6*b*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (d*f*(a + b*Log[c*(d + e*x)^n])^3)/e - (g*m*(d + e*
x)*(a + b*Log[c*(d + e*x)^n])^3)/e + (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)
])/j + (3*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e - (3*b*g*i*m*n*(a + b*Log[c
*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*
i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/j - (6*b^3*g*n^3*(i + j*x)*
Log[h*(i + j*x)^m])/j + (6*b^3*d*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/e + 6*b^2*g*n^2*x
*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (3*b*d*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e
- 3*b*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*
x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) + (6*b^3*g*i*m*n^3*PolyLog[2, -((j*(d + e
*x))/(e*i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/
e - (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (3*b*d*g*m*n*(a
+ b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e + (3*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n]
)^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^3*d*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e -
(6*b^3*d*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*
i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e - (6*b
^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j - (6*b^3*d*g*m*n^3*PolyLog
[4, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j

Rule 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f
+ g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 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 2375

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right ) \, dx &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g j m) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{398+j x} \, dx-(3 b e n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (398+j x)^m\right )\right )}{d+e x} \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g j m) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j}-\frac{398 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j (398+j x)}\right ) \, dx-(3 b e n) \int \left (\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x}+\frac{g x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x}\right ) \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx+(398 g m) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{398+j x} \, dx-(3 b e f n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx-(3 b e g n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac{(g m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}-(3 b f n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )-(3 b e g n) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac{(1194 b e g m n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}\\ &=-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac{(3 b f n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac{(3 b d f n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )}{e}-(3 b g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right ) \, dx+(3 b d g n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x} \, dx+\frac{(3 b g m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac{(1194 b g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{e \left (\frac{398 e-d j}{e}+\frac{j x}{e}\right )}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{(3 d f) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{(3 b d g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (h \left (\frac{398 e-d j}{e}+\frac{j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+(3 b g j m n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{398+j x} \, dx+\frac{\left (6 b^2 f n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\left (6 b^2 e g n^2\right ) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{d+e x} \, dx-\frac{\left (6 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (2388 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{(d g j m) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\frac{398 e-d j}{e}+\frac{j x}{e}} \, dx,x,d+e x\right )}{e^2}+(3 b g j m n) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{398 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j (398+j x)}\right ) \, dx+\frac{\left (6 b^3 f n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\left (6 b^2 e g n^2\right ) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{e}-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac{\left (6 b^3 g m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (2388 b^3 g m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+(3 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx-(1194 b g m n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{398+j x} \, dx+\frac{(3 b d g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\left (6 b^2 g n^2\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right ) \, dx-\left (6 b^2 d g n^2\right ) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{(3 b g m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^2 d g n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac{398 e-d j}{e}+\frac{j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (6 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (2388 b^2 e g m n^2\right ) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}-\left (6 b^2 g j m n^2\right ) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{398+j x} \, dx-\left (6 b^3 e g n^3\right ) \int \frac{x \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac{3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{(3 b d g j m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\frac{398 e-d j}{e}+\frac{j x}{e}} \, dx,x,d+e x\right )}{e^2}-\frac{\left (6 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (2388 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{398 e-d j}{e}+\frac{j x}{e}\right )}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}-\left (6 b^2 g j m n^2\right ) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{j}-\frac{398 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (398+j x)}\right ) \, dx-\left (6 b^3 e g n^3\right ) \int \left (\frac{\log \left (h (398+j x)^m\right )}{e}-\frac{d \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac{\left (6 b^3 d g m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 f n^2 x-12 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac{3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{6 b^3 d g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\left (6 b^2 g m n^2\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\left (2388 b^2 g m n^2\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{398+j x} \, dx-\frac{\left (6 b^3 g m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}-\left (6 b^3 g n^3\right ) \int \log \left (h (398+j x)^m\right ) \, dx+\left (6 b^3 d g n^3\right ) \int \frac{\log \left (h (398+j x)^m\right )}{d+e x} \, dx+\frac{\left (2388 b^3 g m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+12 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{12 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{6 b^3 d g n^3 \log \left (-\frac{j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac{3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{2388 b^3 g m n^3 \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{6 b^3 d g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\left (6 b^3 g m n^2\right ) \int \log \left (c (d+e x)^n\right ) \, dx-\frac{\left (6 b^3 g n^3\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,398+j x\right )}{j}-\frac{\left (6 b^3 d g m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac{\left (2388 b^3 e g m n^3\right ) \int \frac{\log \left (\frac{e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}-\frac{\left (6 b^3 d g j m n^3\right ) \int \frac{\log \left (\frac{j (d+e x)}{-398 e+d j}\right )}{398+j x} \, dx}{e}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+18 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{12 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-\frac{6 b^3 g n^3 (398+j x) \log \left (h (398+j x)^m\right )}{j}+\frac{6 b^3 d g n^3 \log \left (-\frac{j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac{3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{6 b^3 d g m n^3 \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{6 b^3 d g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{\left (6 b^3 g m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^3 d g m n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-398 e+d j}\right )}{x} \, dx,x,398+j x\right )}{e}-\frac{\left (2388 b^3 g m n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+24 b^3 g m n^3 x+\frac{6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{18 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}+\frac{398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (398+j x)}{398 e-d j}\right )}{j}-\frac{6 b^3 g n^3 (398+j x) \log \left (h (398+j x)^m\right )}{j}+\frac{6 b^3 d g n^3 \log \left (-\frac{j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac{3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^3 d g m n^3 \text{Li}_2\left (\frac{e (398+j x)}{398 e-d j}\right )}{e}-\frac{6 b^3 d g m n^3 \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}+\frac{6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}-\frac{2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}-\frac{6 b^3 d g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{e}+\frac{2388 b^3 g m n^3 \text{Li}_4\left (-\frac{j (d+e x)}{398 e-d j}\right )}{j}\\ \end{align*}

Mathematica [B]  time = 1.05337, size = 3163, normalized size = 2.76 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^2*b*d*f*j*n + 3*a^2*b*d*g*j*m*n - 6*a*b^2*d*g*j*m*n^2 + 6*b^3*d*g*j*m*n^3 + a^3*e*f*j*x - a^3*e*g*j*m*x
- 3*a^2*b*e*f*j*n*x + 6*a^2*b*e*g*j*m*n*x + 6*a*b^2*e*f*j*n^2*x - 18*a*b^2*e*g*j*m*n^2*x - 6*b^3*e*f*j*n^3*x +
24*b^3*e*g*j*m*n^3*x + 3*a^2*b*d*f*j*n*Log[d + e*x] - 3*a^2*b*d*g*j*m*n*Log[d + e*x] + 6*a*b^2*d*g*j*m*n^2*Lo
g[d + e*x] + 6*b^3*d*f*j*n^3*Log[d + e*x] - 12*b^3*d*g*j*m*n^3*Log[d + e*x] - 3*a*b^2*d*f*j*n^2*Log[d + e*x]^2
+ 3*a*b^2*d*g*j*m*n^2*Log[d + e*x]^2 - 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2 + b^3*d*f*j*n^3*Log[d + e*x]^3 - b^3*
d*g*j*m*n^3*Log[d + e*x]^3 - 6*a*b^2*d*f*j*n*Log[c*(d + e*x)^n] + 6*a*b^2*d*g*j*m*n*Log[c*(d + e*x)^n] - 6*b^3
*d*g*j*m*n^2*Log[c*(d + e*x)^n] + 3*a^2*b*e*f*j*x*Log[c*(d + e*x)^n] - 3*a^2*b*e*g*j*m*x*Log[c*(d + e*x)^n] -
6*a*b^2*e*f*j*n*x*Log[c*(d + e*x)^n] + 12*a*b^2*e*g*j*m*n*x*Log[c*(d + e*x)^n] + 6*b^3*e*f*j*n^2*x*Log[c*(d +
e*x)^n] - 18*b^3*e*g*j*m*n^2*x*Log[c*(d + e*x)^n] + 6*a*b^2*d*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n] - 6*a*b^2*
d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] + 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n] - 3*b^3*d*f*j*n^
2*Log[d + e*x]^2*Log[c*(d + e*x)^n] + 3*b^3*d*g*j*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n] - 3*b^3*d*f*j*n*Log[
c*(d + e*x)^n]^2 + 3*b^3*d*g*j*m*n*Log[c*(d + e*x)^n]^2 + 3*a*b^2*e*f*j*x*Log[c*(d + e*x)^n]^2 - 3*a*b^2*e*g*j
*m*x*Log[c*(d + e*x)^n]^2 - 3*b^3*e*f*j*n*x*Log[c*(d + e*x)^n]^2 + 6*b^3*e*g*j*m*n*x*Log[c*(d + e*x)^n]^2 + 3*
b^3*d*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2 - 3*b^3*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2 + b^3*e*f*j*
x*Log[c*(d + e*x)^n]^3 - b^3*e*g*j*m*x*Log[c*(d + e*x)^n]^3 + a^3*e*g*i*m*Log[i + j*x] - 3*a^2*b*e*g*i*m*n*Log
[i + j*x] + 3*a^2*b*d*g*j*m*n*Log[i + j*x] + 6*a*b^2*e*g*i*m*n^2*Log[i + j*x] - 6*b^3*e*g*i*m*n^3*Log[i + j*x]
- 3*a^2*b*e*g*i*m*n*Log[d + e*x]*Log[i + j*x] + 6*a*b^2*e*g*i*m*n^2*Log[d + e*x]*Log[i + j*x] - 6*a*b^2*d*g*j
*m*n^2*Log[d + e*x]*Log[i + j*x] - 6*b^3*e*g*i*m*n^3*Log[d + e*x]*Log[i + j*x] + 3*a*b^2*e*g*i*m*n^2*Log[d + e
*x]^2*Log[i + j*x] - 3*b^3*e*g*i*m*n^3*Log[d + e*x]^2*Log[i + j*x] + 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2*Log[i +
j*x] - b^3*e*g*i*m*n^3*Log[d + e*x]^3*Log[i + j*x] + 3*a^2*b*e*g*i*m*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*a*b^2
*e*g*i*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*a*b^2*d*g*j*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*b^3*e*g*i*m
*n^2*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*a*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*b^3*
e*g*i*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n]*L
og[i + j*x] + 3*b^3*e*g*i*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[i + j*x] + 3*a*b^2*e*g*i*m*Log[c*(d + e*
x)^n]^2*Log[i + j*x] - 3*b^3*e*g*i*m*n*Log[c*(d + e*x)^n]^2*Log[i + j*x] + 3*b^3*d*g*j*m*n*Log[c*(d + e*x)^n]^
2*Log[i + j*x] - 3*b^3*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2*Log[i + j*x] + b^3*e*g*i*m*Log[c*(d + e*x)^
n]^3*Log[i + j*x] + 3*a^2*b*e*g*i*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a^2*b*d*g*j*m*n*Log[d +
e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 6*a*b^2*e*g*i*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*a*b^
2*d*g*j*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*b^3*e*g*i*m*n^3*Log[d + e*x]*Log[(e*(i + j*x))/(
e*i - d*j)] - 6*b^3*d*g*j*m*n^3*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a*b^2*e*g*i*m*n^2*Log[d + e*x]
^2*Log[(e*(i + j*x))/(e*i - d*j)] + 3*a*b^2*d*g*j*m*n^2*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] + 3*b^3*
e*g*i*m*n^3*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2*Log[(e*(i + j*x))
/(e*i - d*j)] + b^3*e*g*i*m*n^3*Log[d + e*x]^3*Log[(e*(i + j*x))/(e*i - d*j)] - b^3*d*g*j*m*n^3*Log[d + e*x]^3
*Log[(e*(i + j*x))/(e*i - d*j)] + 6*a*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d
*j)] - 6*a*b^2*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] - 6*b^3*e*g*i*m*n^2*Lo
g[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^
n]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*e*g*i*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i
- d*j)] + 3*b^3*d*g*j*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 3*b^3*e*g*i*m*
n*Log[d + e*x]*Log[c*(d + e*x)^n]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e
*x)^n]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a^2*b*d*g*j*n*Log[h*(i + j*x)^m] + a^3*e*g*j*x*Log[h*(i + j*x)^m]
- 3*a^2*b*e*g*j*n*x*Log[h*(i + j*x)^m] + 6*a*b^2*e*g*j*n^2*x*Log[h*(i + j*x)^m] - 6*b^3*e*g*j*n^3*x*Log[h*(i +
j*x)^m] + 3*a^2*b*d*g*j*n*Log[d + e*x]*Log[h*(i + j*x)^m] + 6*b^3*d*g*j*n^3*Log[d + e*x]*Log[h*(i + j*x)^m] -
3*a*b^2*d*g*j*n^2*Log[d + e*x]^2*Log[h*(i + j*x)^m] + b^3*d*g*j*n^3*Log[d + e*x]^3*Log[h*(i + j*x)^m] - 6*a*b
^2*d*g*j*n*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + 3*a^2*b*e*g*j*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] - 6*a
*b^2*e*g*j*n*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + 6*b^3*e*g*j*n^2*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m]
+ 6*a*b^2*d*g*j*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] - 3*b^3*d*g*j*n^2*Log[d + e*x]^2*Log[c*(
d + e*x)^n]*Log[h*(i + j*x)^m] - 3*b^3*d*g*j*n*Log[c*(d + e*x)^n]^2*Log[h*(i + j*x)^m] + 3*a*b^2*e*g*j*x*Log[c
*(d + e*x)^n]^2*Log[h*(i + j*x)^m] - 3*b^3*e*g*j*n*x*Log[c*(d + e*x)^n]^2*Log[h*(i + j*x)^m] + 3*b^3*d*g*j*n*L
og[d + e*x]*Log[c*(d + e*x)^n]^2*Log[h*(i + j*x)^m] + b^3*e*g*j*x*Log[c*(d + e*x)^n]^3*Log[h*(i + j*x)^m] + 3*
b*g*(e*i - d*j)*m*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b*(a - b*n)*Log[c*(d + e*x)^n] + b^2*Log[c*(d + e*x)^n]^2)*
PolyLog[2, (j*(d + e*x))/(-(e*i) + d*j)] - 6*b^2*g*(e*i - d*j)*m*n^2*(a - b*n + b*Log[c*(d + e*x)^n])*PolyLog[
3, (j*(d + e*x))/(-(e*i) + d*j)] + 6*b^3*e*g*i*m*n^3*PolyLog[4, (j*(d + e*x))/(-(e*i) + d*j)] - 6*b^3*d*g*j*m*
n^3*PolyLog[4, (j*(d + e*x))/(-(e*i) + d*j)])/(e*j)

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Maple [F]  time = 4.523, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3} \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m)),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((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

b^3*f*x*log((e*x + d)^n*c)^3 - 3*a^2*b*e*f*n*(x/e - d*log(e*x + d)/e^2) - a^3*g*j*m*(x/j - i*log(j*x + i)/j^2)
+ 3*a*b^2*f*x*log((e*x + d)^n*c)^2 + 3*a^2*b*f*x*log((e*x + d)^n*c) + a^3*g*x*log((j*x + i)^m*h) - 3*(2*e*n*(
x/e - d*log(e*x + d)/e^2)*log((e*x + d)^n*c) + (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*a*b^2*f -
(3*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)^n*c)^2 - e*n*((d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x +
6*d*log(e*x + d))*n^2/e^2 - 3*(d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n*log((e*x + d)^n*c)/e^2))*b^3*f
+ a^3*f*x + ((b^3*e*g*i*m*log(j*x + i) - (j*m - j*log(h))*b^3*e*g*x)*log((e*x + d)^n)^3 + (b^3*d*g*j*n^3*log(e
*x + d)^3 + b^3*e*g*j*x*log((e*x + d)^n)^3 - 3*(a*b^2*d*g*j*n^2 - (d*g*j*n^3 - d*g*j*n^2*log(c))*b^3)*log(e*x
+ d)^2 + 3*(b^3*d*g*j*n*log(e*x + d) + (a*b^2*e*g*j - (e*g*j*n - e*g*j*log(c))*b^3)*x)*log((e*x + d)^n)^2 - (3
*(e*g*j*n - e*g*j*log(c))*a^2*b - 3*(2*e*g*j*n^2 - 2*e*g*j*n*log(c) + e*g*j*log(c)^2)*a*b^2 + (6*e*g*j*n^3 - 6
*e*g*j*n^2*log(c) + 3*e*g*j*n*log(c)^2 - e*g*j*log(c)^3)*b^3)*x + 3*(a^2*b*d*g*j*n - 2*(d*g*j*n^2 - d*g*j*n*lo
g(c))*a*b^2 + (2*d*g*j*n^3 - 2*d*g*j*n^2*log(c) + d*g*j*n*log(c)^2)*b^3)*log(e*x + d) - 3*(b^3*d*g*j*n^2*log(e
*x + d)^2 - (a^2*b*e*g*j - 2*(e*g*j*n - e*g*j*log(c))*a*b^2 + (2*e*g*j*n^2 - 2*e*g*j*n*log(c) + e*g*j*log(c)^2
)*b^3)*x - 2*(a*b^2*d*g*j*n - (d*g*j*n^2 - d*g*j*n*log(c))*b^3)*log(e*x + d))*log((e*x + d)^n))*log((j*x + i)^
m))/(e*j) - integrate(-(b^3*d*e*g*i*j*log(c)^3*log(h) + 3*a*b^2*d*e*g*i*j*log(c)^2*log(h) + 3*a^2*b*d*e*g*i*j*
log(c)*log(h) - (b^3*d*e*g*j^2*m*n^3*x + b^3*d^2*g*j^2*m*n^3)*log(e*x + d)^3 + (3*(e^2*g*j^2*m*n - (j^2*m - j^
2*log(h))*e^2*g*log(c))*a^2*b - 3*(2*e^2*g*j^2*m*n^2 - 2*e^2*g*j^2*m*n*log(c) + (j^2*m - j^2*log(h))*e^2*g*log
(c)^2)*a*b^2 + (6*e^2*g*j^2*m*n^3 - 6*e^2*g*j^2*m*n^2*log(c) + 3*e^2*g*j^2*m*n*log(c)^2 - (j^2*m - j^2*log(h))
*e^2*g*log(c)^3)*b^3)*x^2 + 3*(a*b^2*d^2*g*j^2*m*n^2 - (d^2*g*j^2*m*n^3 - d^2*g*j^2*m*n^2*log(c))*b^3 + (a*b^2
*d*e*g*j^2*m*n^2 - (d*e*g*j^2*m*n^3 - d*e*g*j^2*m*n^2*log(c))*b^3)*x)*log(e*x + d)^2 + 3*(b^3*d*e*g*i*j*log(c)
*log(h) + a*b^2*d*e*g*i*j*log(h) - ((j^2*m - j^2*log(h))*a*b^2*e^2*g + ((j^2*m - j^2*log(h))*e^2*g*log(c) - (2
*j^2*m*n - j^2*n*log(h))*e^2*g)*b^3)*x^2 + ((e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*a*b^2 + (d*e*g*j^2
*m*n + (i*j*m*n - i*j*n*log(h))*e^2*g + (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*b^3)*x - (b^3*
d*e*g*j^2*m*n*x + b^3*d^2*g*j^2*m*n)*log(e*x + d) - (b^3*e^2*g*i*j*m*n*x + b^3*e^2*g*i^2*m*n)*log(j*x + i))*lo
g((e*x + d)^n)^2 + (3*(d*e*g*j^2*m*n + (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*a^2*b - 3*(2*d*
e*g*j^2*m*n^2 - 2*d*e*g*j^2*m*n*log(c) - (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c)^2)*a*b^2 + (6*
d*e*g*j^2*m*n^3 - 6*d*e*g*j^2*m*n^2*log(c) + 3*d*e*g*j^2*m*n*log(c)^2 + (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h
))*d*e*g)*log(c)^3)*b^3)*x - 3*(a^2*b*d^2*g*j^2*m*n - 2*(d^2*g*j^2*m*n^2 - d^2*g*j^2*m*n*log(c))*a*b^2 + (2*d^
2*g*j^2*m*n^3 - 2*d^2*g*j^2*m*n^2*log(c) + d^2*g*j^2*m*n*log(c)^2)*b^3 + (a^2*b*d*e*g*j^2*m*n - 2*(d*e*g*j^2*m
*n^2 - d*e*g*j^2*m*n*log(c))*a*b^2 + (2*d*e*g*j^2*m*n^3 - 2*d*e*g*j^2*m*n^2*log(c) + d*e*g*j^2*m*n*log(c)^2)*b
^3)*x)*log(e*x + d) + 3*(b^3*d*e*g*i*j*log(c)^2*log(h) + 2*a*b^2*d*e*g*i*j*log(c)*log(h) + a^2*b*d*e*g*i*j*log
(h) - ((j^2*m - j^2*log(h))*a^2*b*e^2*g - 2*(e^2*g*j^2*m*n - (j^2*m - j^2*log(h))*e^2*g*log(c))*a*b^2 + (2*e^2
*g*j^2*m*n^2 - 2*e^2*g*j^2*m*n*log(c) + (j^2*m - j^2*log(h))*e^2*g*log(c)^2)*b^3)*x^2 + (b^3*d*e*g*j^2*m*n^2*x
+ b^3*d^2*g*j^2*m*n^2)*log(e*x + d)^2 + ((e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*a^2*b + 2*(d*e*g*j^2
*m*n + (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*a*b^2 - (2*d*e*g*j^2*m*n^2 - 2*d*e*g*j^2*m*n*lo
g(c) - (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c)^2)*b^3)*x - 2*(a*b^2*d^2*g*j^2*m*n - (d^2*g*j^2*
m*n^2 - d^2*g*j^2*m*n*log(c))*b^3 + (a*b^2*d*e*g*j^2*m*n - (d*e*g*j^2*m*n^2 - d*e*g*j^2*m*n*log(c))*b^3)*x)*lo
g(e*x + d))*log((e*x + d)^n))/(e^2*j^2*x^2 + d*e*i*j + (e^2*i*j + d*e*j^2)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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