3.367 \(\int x^2 \log (f x^m) (a+b \log (c (d+e x)^n))^2 \, dx\)

Optimal. Leaf size=705 \[ -\frac{2 b d^3 m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{9 e^3}+\frac{2 b^2 d^3 m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{3 e^3}+\frac{d^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}-\frac{d^3 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{9 e^3}-\frac{d^3 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{b d n x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2}{9} b n x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{5 b d m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{18 e}-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{4}{27} b m n x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{2 a b d^2 n x \log \left (f x^m\right )}{3 e^2}+\frac{2 a b d^2 m n x}{9 e^2}+\frac{b d^2 m n x (6 a-11 b n)}{9 e^2}-\frac{2 b^2 d^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{2 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{8 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{5 b^2 d^3 n^2 \log (d+e x) \log \left (f x^m\right )}{9 e^3}+\frac{11 b^2 d^2 n^2 x \log \left (f x^m\right )}{9 e^2}-\frac{71 b^2 d^2 m n^2 x}{54 e^2}+\frac{23 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}+\frac{5 b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{9 e^3}-\frac{5 b^2 d n^2 x^2 \log \left (f x^m\right )}{18 e}+\frac{19 b^2 d m n^2 x^2}{54 e}+\frac{2}{27} b^2 n^2 x^3 \log \left (f x^m\right )-\frac{2}{27} b^2 m n^2 x^3 \]

[Out]

(2*a*b*d^2*m*n*x)/(9*e^2) - (71*b^2*d^2*m*n^2*x)/(54*e^2) + (b*d^2*m*n*(6*a - 11*b*n)*x)/(9*e^2) + (19*b^2*d*m
*n^2*x^2)/(54*e) - (2*b^2*m*n^2*x^3)/27 - (2*a*b*d^2*n*x*Log[f*x^m])/(3*e^2) + (11*b^2*d^2*n^2*x*Log[f*x^m])/(
9*e^2) - (5*b^2*d*n^2*x^2*Log[f*x^m])/(18*e) + (2*b^2*n^2*x^3*Log[f*x^m])/27 + (23*b^2*d^3*m*n^2*Log[d + e*x])
/(54*e^3) + (5*b^2*d^3*m*n^2*Log[-((e*x)/d)]*Log[d + e*x])/(9*e^3) - (5*b^2*d^3*n^2*Log[f*x^m]*Log[d + e*x])/(
9*e^3) + (8*b^2*d^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(9*e^3) + (2*b^2*d^3*m*n*Log[-((e*x)/d)]*Log[c*(d + e*x)
^n])/(3*e^3) - (2*b^2*d^2*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/(3*e^3) - (5*b*d*m*n*x^2*(a + b*Log[c*(d
+ e*x)^n]))/(18*e) + (4*b*m*n*x^3*(a + b*Log[c*(d + e*x)^n]))/27 + (b*d*n*x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x
)^n]))/(3*e) - (2*b*n*x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/9 - (d^3*m*(a + b*Log[c*(d + e*x)^n])^2)/(9*e
^3) - (m*x^3*(a + b*Log[c*(d + e*x)^n])^2)/9 - (d^3*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) +
(d^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/3 + (11*
b^2*d^3*m*n^2*PolyLog[2, 1 + (e*x)/d])/(9*e^3) - (2*b*d^3*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/
d])/(3*e^3) + (2*b^2*d^3*m*n^2*PolyLog[3, 1 + (e*x)/d])/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 2.18908, antiderivative size = 902, normalized size of antiderivative = 1.28, number of steps used = 50, number of rules used = 22, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846, Rules used = {2398, 2411, 43, 2334, 12, 14, 2301, 2428, 2396, 2433, 2374, 6589, 6741, 6742, 2394, 2315, 2389, 2295, 2395, 2434, 2375, 2317} \[ \frac{b^2 m n^2 \log ^2(d+e x) d^3}{9 e^3}+\frac{b^2 m n^2 \log (x) \log ^2(d+e x) d^3}{3 e^3}-\frac{b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x) d^3}{3 e^3}+\frac{b^2 m \log (x) \log ^2\left (c (d+e x)^n\right ) d^3}{3 e^3}-\frac{b^2 m \log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^n\right ) d^3}{3 e^3}+\frac{23 b^2 m n^2 \log (x) d^3}{54 e^3}+\frac{11 a b m n \log (x) d^3}{9 e^3}+\frac{13 b^2 m n^2 \log (d+e x) d^3}{54 e^3}-\frac{2 a b m n \log \left (-\frac{e x}{d}\right ) \log (d+e x) d^3}{3 e^3}+\frac{11 b^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right ) d^3}{9 e^3}-\frac{2 b^2 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right ) d^3}{3 e^3}+\frac{11 b^2 m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) d^3}{9 e^3}-\frac{2 a b m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) d^3}{3 e^3}-\frac{2 b^2 m n \log \left (c (d+e x)^n\right ) \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) d^3}{3 e^3}+\frac{2 b^2 m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right ) d^3}{3 e^3}-\frac{151 b^2 m n^2 x d^2}{54 e^2}+\frac{2 a b m n x d^2}{3 e^2}+\frac{2 b^2 n^2 x \log \left (f x^m\right ) d^2}{e^2}+\frac{2 b^2 m n (d+e x) \log \left (c (d+e x)^n\right ) d^2}{3 e^3}+\frac{7 b^2 m n^2 x^2 d}{27 e}-\frac{a b m n x^2 d}{6 e}+\frac{b^2 m n^2 (d+e x)^2 d}{6 e^3}-\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right ) d}{2 e^3}-\frac{b^2 m n x^2 \log \left (c (d+e x)^n\right ) d}{6 e}-\frac{4}{81} b^2 m n^2 x^3+\frac{2}{27} a b m n x^3-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{1}{27} b m n \left (-\frac{6 \log (d+e x) d^3}{e^3}+\frac{18 (d+e x) d^2}{e^3}-\frac{9 (d+e x)^2 d}{e^3}+\frac{2 (d+e x)^3}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (-\frac{6 \log (d+e x) d^3}{e^3}+\frac{18 (d+e x) d^2}{e^3}-\frac{9 (d+e x)^2 d}{e^3}+\frac{2 (d+e x)^3}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*d^2*m*n*x)/(3*e^2) - (151*b^2*d^2*m*n^2*x)/(54*e^2) - (a*b*d*m*n*x^2)/(6*e) + (7*b^2*d*m*n^2*x^2)/(27*e
) + (2*a*b*m*n*x^3)/27 - (4*b^2*m*n^2*x^3)/81 + (b^2*d*m*n^2*(d + e*x)^2)/(6*e^3) - (2*b^2*m*n^2*(d + e*x)^3)/
(81*e^3) + (11*a*b*d^3*m*n*Log[x])/(9*e^3) + (23*b^2*d^3*m*n^2*Log[x])/(54*e^3) + (2*b^2*d^2*n^2*x*Log[f*x^m])
/e^2 - (b^2*d*n^2*(d + e*x)^2*Log[f*x^m])/(2*e^3) + (2*b^2*n^2*(d + e*x)^3*Log[f*x^m])/(27*e^3) + (13*b^2*d^3*
m*n^2*Log[d + e*x])/(54*e^3) - (2*a*b*d^3*m*n*Log[-((e*x)/d)]*Log[d + e*x])/(3*e^3) + (b^2*d^3*m*n^2*Log[d + e
*x]^2)/(9*e^3) + (b^2*d^3*m*n^2*Log[x]*Log[d + e*x]^2)/(3*e^3) - (b^2*d^3*n^2*Log[f*x^m]*Log[d + e*x]^2)/(3*e^
3) - (b^2*d*m*n*x^2*Log[c*(d + e*x)^n])/(6*e) + (2*b^2*m*n*x^3*Log[c*(d + e*x)^n])/27 + (2*b^2*d^2*m*n*(d + e*
x)*Log[c*(d + e*x)^n])/(3*e^3) + (11*b^2*d^3*m*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/(9*e^3) - (2*b^2*d^3*m*n*
Log[x]*Log[d + e*x]*Log[c*(d + e*x)^n])/(3*e^3) + (b^2*d^3*m*Log[x]*Log[c*(d + e*x)^n]^2)/(3*e^3) - (b^2*d^3*m
*Log[-((e*x)/d)]*Log[c*(d + e*x)^n]^2)/(3*e^3) + (b*m*n*((18*d^2*(d + e*x))/e^3 - (9*d*(d + e*x)^2)/e^3 + (2*(
d + e*x)^3)/e^3 - (6*d^3*Log[d + e*x])/e^3)*(a + b*Log[c*(d + e*x)^n]))/27 - (b*n*Log[f*x^m]*((18*d^2*(d + e*x
))/e^3 - (9*d*(d + e*x)^2)/e^3 + (2*(d + e*x)^3)/e^3 - (6*d^3*Log[d + e*x])/e^3)*(a + b*Log[c*(d + e*x)^n]))/9
- (m*x^3*(a + b*Log[c*(d + e*x)^n])^2)/9 + (x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/3 - (2*a*b*d^3*m*n*P
olyLog[2, 1 + (e*x)/d])/(3*e^3) + (11*b^2*d^3*m*n^2*PolyLog[2, 1 + (e*x)/d])/(9*e^3) - (2*b^2*d^3*m*n*Log[c*(d
+ e*x)^n]*PolyLog[2, 1 + (e*x)/d])/(3*e^3) + (2*b^2*d^3*m*n^2*PolyLog[3, 1 + (e*x)/d])/(3*e^3)

Rule 2398

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

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 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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2428

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((g_.)*(x_))^(q_.), x_Symb
ol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - Dist[m, Int[Dist[
1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 1] && IGtQ[q, 0]

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 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 6741

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

Rule 6742

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

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 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 2295

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

Rule 2395

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

Rule 2434

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

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 x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-m \int \left (\frac{2 b^2 d^2 n^2}{e^2}-\frac{b^2 d n^2 (d+e x)^2}{2 e^3 x}+\frac{2 b^2 n^2 (d+e x)^3}{27 e^3 x}-\frac{b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 x}-\frac{b n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 x}+\frac{1}{3} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right ) \, dx\\ &=-\frac{2 b^2 d^2 m n^2 x}{e^2}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{1}{3} m \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+\frac{1}{9} (b m n) \int \frac{\left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx-\frac{\left (2 b^2 m n^2\right ) \int \frac{(d+e x)^3}{x} \, dx}{27 e^3}+\frac{\left (b^2 d m n^2\right ) \int \frac{(d+e x)^2}{x} \, dx}{2 e^3}+\frac{\left (b^2 d^3 m n^2\right ) \int \frac{\log ^2(d+e x)}{x} \, dx}{3 e^3}\\ &=-\frac{2 b^2 d^2 m n^2 x}{e^2}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{9} (b m n) \int \frac{\left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 x} \, dx+\frac{1}{9} (2 b e m n) \int \frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-\frac{\left (2 b^2 m n^2\right ) \int \left (3 d^2 e+\frac{d^3}{x}+3 d e^2 x+e^3 x^2\right ) \, dx}{27 e^3}+\frac{\left (b^2 d m n^2\right ) \int \left (2 d e+\frac{d^2}{x}+e^2 x\right ) \, dx}{2 e^3}-\frac{\left (2 b^2 d^3 m n^2\right ) \int \frac{\log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d+e x} \, dx}{3 e^2}\\ &=-\frac{11 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{9} (2 b m n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )+\frac{(b m n) \int \frac{\left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx}{9 e^3}-\frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}\\ &=-\frac{11 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{(b m n) \int \left (\frac{a \left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)\right )}{x}+\frac{b \left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)\right ) \log \left (c (d+e x)^n\right )}{x}\right ) \, dx}{9 e^3}-\frac{1}{9} \left (2 b^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x\right )-\frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}\\ &=-\frac{11 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{(a b m n) \int \frac{11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)}{x} \, dx}{9 e^3}+\frac{\left (b^2 m n\right ) \int \frac{\left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3-6 d^3 \log (d+e x)\right ) \log \left (c (d+e x)^n\right )}{x} \, dx}{9 e^3}-\frac{\left (b^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x\right )}{27 e^3}\\ &=-\frac{11 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{(a b m n) \int \left (\frac{11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3}{x}-\frac{6 d^3 \log (d+e x)}{x}\right ) \, dx}{9 e^3}+\frac{\left (b^2 m n\right ) \int \left (6 d^2 e \log \left (c (d+e x)^n\right )+\frac{11 d^3 \log \left (c (d+e x)^n\right )}{x}-3 d e^2 x \log \left (c (d+e x)^n\right )+2 e^3 x^2 \log \left (c (d+e x)^n\right )-\frac{6 d^3 \log (d+e x) \log \left (c (d+e x)^n\right )}{x}\right ) \, dx}{9 e^3}-\frac{\left (b^2 m n^2\right ) \operatorname{Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac{6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{27 e^3}\\ &=-\frac{17 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{1}{9} \left (2 b^2 m n\right ) \int x^2 \log \left (c (d+e x)^n\right ) \, dx+\frac{(a b m n) \int \frac{11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3}{x} \, dx}{9 e^3}-\frac{\left (2 a b d^3 m n\right ) \int \frac{\log (d+e x)}{x} \, dx}{3 e^3}-\frac{\left (2 b^2 d^3 m n\right ) \int \frac{\log (d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{3 e^3}+\frac{\left (11 b^2 d^3 m n\right ) \int \frac{\log \left (c (d+e x)^n\right )}{x} \, dx}{9 e^3}+\frac{\left (2 b^2 d^2 m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{3 e^2}-\frac{\left (b^2 d m n\right ) \int x \log \left (c (d+e x)^n\right ) \, dx}{3 e}+\frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x\right )}{9 e^3}\\ &=-\frac{17 b^2 d^2 m n^2 x}{9 e^2}+\frac{5 b^2 d m n^2 x^2}{36 e}-\frac{2}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{(a b m n) \int \left (6 d^2 e+\frac{11 d^3}{x}-3 d e^2 x+2 e^3 x^2\right ) \, dx}{9 e^3}+\frac{\left (2 b^2 d^2 m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{3 e^3}+\frac{\left (2 a b d^3 m n\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{3 e^2}+\frac{\left (2 b^2 d^3 m n\right ) \int \frac{\log (x) \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{3 e^2}+\frac{1}{6} \left (b^2 d m n^2\right ) \int \frac{x^2}{d+e x} \, dx+\frac{\left (2 b^2 d^3 m n^2\right ) \int \frac{\log (x) \log (d+e x)}{d+e x} \, dx}{3 e^2}-\frac{\left (11 b^2 d^3 m n^2\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{9 e^2}-\frac{1}{27} \left (2 b^2 e m n^2\right ) \int \frac{x^3}{d+e x} \, dx\\ &=\frac{2 a b d^2 m n x}{3 e^2}-\frac{23 b^2 d^2 m n^2 x}{9 e^2}-\frac{a b d m n x^2}{6 e}+\frac{5 b^2 d m n^2 x^2}{36 e}+\frac{2}{27} a b m n x^3-\frac{2}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{11 a b d^3 m n \log (x)}{9 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{2 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 a b d^3 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{9 e^3}+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{\left (2 b^2 d^3 m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \log \left (-\frac{d}{e}+\frac{x}{e}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}+\frac{1}{6} \left (b^2 d m n^2\right ) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx+\frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{d}{e}+\frac{x}{e}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}-\frac{1}{27} \left (2 b^2 e m n^2\right ) \int \left (\frac{d^2}{e^3}-\frac{d x}{e^2}+\frac{x^2}{e}-\frac{d^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{2 a b d^2 m n x}{3 e^2}-\frac{151 b^2 d^2 m n^2 x}{54 e^2}-\frac{a b d m n x^2}{6 e}+\frac{7 b^2 d m n^2 x^2}{27 e}+\frac{2}{27} a b m n x^3-\frac{4}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{11 a b d^3 m n \log (x)}{9 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{13 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log (x) \log ^2(d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{2 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{b^2 d^3 m \log (x) \log ^2\left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 a b d^3 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{9 e^3}+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{\left (b^2 d^3 m\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{3 e^4}-\frac{\left (b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{3 e^4}\\ &=\frac{2 a b d^2 m n x}{3 e^2}-\frac{151 b^2 d^2 m n^2 x}{54 e^2}-\frac{a b d m n x^2}{6 e}+\frac{7 b^2 d m n^2 x^2}{27 e}+\frac{2}{27} a b m n x^3-\frac{4}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{11 a b d^3 m n \log (x)}{9 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{13 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log (x) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{2 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{b^2 d^3 m \log (x) \log ^2\left (c (d+e x)^n\right )}{3 e^3}-\frac{b^2 d^3 m \log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 a b d^3 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{9 e^3}+\frac{2 b^2 d^3 m n^2 \log (d+e x) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{\left (2 b^2 d^3 m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}+\frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}\\ &=\frac{2 a b d^2 m n x}{3 e^2}-\frac{151 b^2 d^2 m n^2 x}{54 e^2}-\frac{a b d m n x^2}{6 e}+\frac{7 b^2 d m n^2 x^2}{27 e}+\frac{2}{27} a b m n x^3-\frac{4}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{11 a b d^3 m n \log (x)}{9 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{13 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log (x) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{2 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{b^2 d^3 m \log (x) \log ^2\left (c (d+e x)^n\right )}{3 e^3}-\frac{b^2 d^3 m \log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 a b d^3 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log \left (c (d+e x)^n\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}-\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}+2 \frac{\left (2 b^2 d^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 e^3}\\ &=\frac{2 a b d^2 m n x}{3 e^2}-\frac{151 b^2 d^2 m n^2 x}{54 e^2}-\frac{a b d m n x^2}{6 e}+\frac{7 b^2 d m n^2 x^2}{27 e}+\frac{2}{27} a b m n x^3-\frac{4}{81} b^2 m n^2 x^3+\frac{b^2 d m n^2 (d+e x)^2}{6 e^3}-\frac{2 b^2 m n^2 (d+e x)^3}{81 e^3}+\frac{11 a b d^3 m n \log (x)}{9 e^3}+\frac{23 b^2 d^3 m n^2 \log (x)}{54 e^3}+\frac{2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac{b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac{2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}+\frac{13 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}-\frac{2 a b d^3 m n \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{3 e^3}+\frac{b^2 d^3 m n^2 \log ^2(d+e x)}{9 e^3}+\frac{b^2 d^3 m n^2 \log (x) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d^3 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{3 e^3}-\frac{b^2 d m n x^2 \log \left (c (d+e x)^n\right )}{6 e}+\frac{2}{27} b^2 m n x^3 \log \left (c (d+e x)^n\right )+\frac{2 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{11 b^2 d^3 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log (x) \log (d+e x) \log \left (c (d+e x)^n\right )}{3 e^3}+\frac{b^2 d^3 m \log (x) \log ^2\left (c (d+e x)^n\right )}{3 e^3}-\frac{b^2 d^3 m \log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^n\right )}{3 e^3}+\frac{1}{27} b m n \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} b n \log \left (f x^m\right ) \left (\frac{18 d^2 (d+e x)}{e^3}-\frac{9 d (d+e x)^2}{e^3}+\frac{2 (d+e x)^3}{e^3}-\frac{6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac{1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac{2 a b d^3 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{11 b^2 d^3 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{9 e^3}-\frac{2 b^2 d^3 m n \log \left (c (d+e x)^n\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 e^3}+\frac{2 b^2 d^3 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 1.60643, size = 735, normalized size = 1.04 \[ \frac{b m n \left (-36 d^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-6 \log (x) \left (e x \left (-6 d^2+3 d e x-2 e^2 x^2\right )+6 d^3 \log \left (\frac{e x}{d}+1\right )+6 e^3 x^3 \log (d+e x)\right )-48 d^2 e x+12 d^3 \log (d+e x)+15 d e^2 x^2+12 e^3 x^3 \log (d+e x)-8 e^3 x^3\right ) \left (-a-b \log \left (c (d+e x)^n\right )+b n \log (d+e x)\right )-b^2 n^2 \left (66 d^3 m \text{PolyLog}\left (2,-\frac{e x}{d}\right )-36 d^3 m \text{PolyLog}\left (3,\frac{e x}{d}+1\right )+36 d^3 m \log (d+e x) \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-18 d^3 \log ^2(d+e x) \log \left (f x^m\right )+66 d^3 \log (d+e x) \log \left (f x^m\right )-66 d^2 e x \log \left (f x^m\right )+36 d^2 e x \log (d+e x) \log \left (f x^m\right )+137 d^2 e m x+6 d^3 m \log ^2(d+e x)+18 d^3 m \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)-71 d^3 m \log (d+e x)-66 d^3 m \log (x) \log (d+e x)+66 d^3 m \log (x) \log \left (\frac{e x}{d}+1\right )-48 d^2 e m x \log (d+e x)-36 d^3 \log \left (f x^m\right )+36 d^3 m \log (x)-18 e^3 x^3 \log ^2(d+e x) \log \left (f x^m\right )+15 d e^2 x^2 \log \left (f x^m\right )-18 d e^2 x^2 \log (d+e x) \log \left (f x^m\right )+12 e^3 x^3 \log (d+e x) \log \left (f x^m\right )-19 d e^2 m x^2+6 e^3 m x^3 \log ^2(d+e x)+15 d e^2 m x^2 \log (d+e x)-8 e^3 m x^3 \log (d+e x)-4 e^3 x^3 \log \left (f x^m\right )+4 e^3 m x^3\right )+6 b n \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 \left (d^3+e^3 x^3\right ) \log (d+e x)\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-6 e^3 x^3 \left (-3 \log \left (f x^m\right )+3 m \log (x)+m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+18 e^3 m x^3 \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{54 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*n*(m*Log[x] - Log[f*x^m])*(e*x*(6*d^2 - 3*d*e*x + 2*e^2*x^2) - 6*(d^3 + e^3*x^3)*Log[d + e*x])*(a - b*n*L
og[d + e*x] + b*Log[c*(d + e*x)^n]) + 18*e^3*m*x^3*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 6*
e^3*x^3*(m + 3*m*Log[x] - 3*Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + b*m*n*(-a + b*n*Log[
d + e*x] - b*Log[c*(d + e*x)^n])*(-48*d^2*e*x + 15*d*e^2*x^2 - 8*e^3*x^3 + 12*d^3*Log[d + e*x] + 12*e^3*x^3*Lo
g[d + e*x] - 6*Log[x]*(e*x*(-6*d^2 + 3*d*e*x - 2*e^2*x^2) + 6*e^3*x^3*Log[d + e*x] + 6*d^3*Log[1 + (e*x)/d]) -
36*d^3*PolyLog[2, -((e*x)/d)]) - b^2*n^2*(137*d^2*e*m*x - 19*d*e^2*m*x^2 + 4*e^3*m*x^3 + 36*d^3*m*Log[x] - 36
*d^3*Log[f*x^m] - 66*d^2*e*x*Log[f*x^m] + 15*d*e^2*x^2*Log[f*x^m] - 4*e^3*x^3*Log[f*x^m] - 71*d^3*m*Log[d + e*
x] - 48*d^2*e*m*x*Log[d + e*x] + 15*d*e^2*m*x^2*Log[d + e*x] - 8*e^3*m*x^3*Log[d + e*x] - 66*d^3*m*Log[x]*Log[
d + e*x] + 66*d^3*Log[f*x^m]*Log[d + e*x] + 36*d^2*e*x*Log[f*x^m]*Log[d + e*x] - 18*d*e^2*x^2*Log[f*x^m]*Log[d
+ e*x] + 12*e^3*x^3*Log[f*x^m]*Log[d + e*x] + 6*d^3*m*Log[d + e*x]^2 + 6*e^3*m*x^3*Log[d + e*x]^2 + 18*d^3*m*
Log[-((e*x)/d)]*Log[d + e*x]^2 - 18*d^3*Log[f*x^m]*Log[d + e*x]^2 - 18*e^3*x^3*Log[f*x^m]*Log[d + e*x]^2 + 66*
d^3*m*Log[x]*Log[1 + (e*x)/d] + 66*d^3*m*PolyLog[2, -((e*x)/d)] + 36*d^3*m*Log[d + e*x]*PolyLog[2, 1 + (e*x)/d
] - 36*d^3*m*PolyLog[3, 1 + (e*x)/d]))/(54*e^3)

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Maple [F]  time = 1.685, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\ln \left ( f{x}^{m} \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{9} \,{\left (b^{2}{\left (m - 3 \, \log \left (f\right )\right )} x^{3} - 3 \, b^{2} x^{3} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + \int \frac{9 \,{\left (b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) + a^{2} e \log \left (f\right )\right )} x^{3} + 9 \,{\left (b^{2} d \log \left (c\right )^{2} \log \left (f\right ) + 2 \, a b d \log \left (c\right ) \log \left (f\right ) + a^{2} d \log \left (f\right )\right )} x^{2} + 2 \,{\left ({\left (9 \, a b e \log \left (f\right ) +{\left (9 \, e \log \left (c\right ) \log \left (f\right ) +{\left (m n - 3 \, n \log \left (f\right )\right )} e\right )} b^{2}\right )} x^{3} + 9 \,{\left (b^{2} d \log \left (c\right ) \log \left (f\right ) + a b d \log \left (f\right )\right )} x^{2} - 3 \,{\left ({\left ({\left (e n - 3 \, e \log \left (c\right )\right )} b^{2} - 3 \, a b e\right )} x^{3} - 3 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2}\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + 9 \,{\left ({\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{3} +{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{2}\right )} \log \left (x^{m}\right )}{9 \,{\left (e x + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(b^2*(m - 3*log(f))*x^3 - 3*b^2*x^3*log(x^m))*log((e*x + d)^n)^2 + integrate(1/9*(9*(b^2*e*log(c)^2*log(f
) + 2*a*b*e*log(c)*log(f) + a^2*e*log(f))*x^3 + 9*(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f
))*x^2 + 2*((9*a*b*e*log(f) + (9*e*log(c)*log(f) + (m*n - 3*n*log(f))*e)*b^2)*x^3 + 9*(b^2*d*log(c)*log(f) + a
*b*d*log(f))*x^2 - 3*(((e*n - 3*e*log(c))*b^2 - 3*a*b*e)*x^3 - 3*(b^2*d*log(c) + a*b*d)*x^2)*log(x^m))*log((e*
x + d)^n) + 9*((b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^3 + (b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^2)*
log(x^m))/(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x^2*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*x^2*log((e*x + d)^n*c)*log(f*x^m) + a^2*x^2*log(f*x^m
), 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(x**2*ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**2,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 )}^{2} x^{2} \log \left (f x^{m}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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