3.4.67 \(\int x^2 \log (f x^m) (a+b \log (c (d+e x)^n))^2 \, dx\) [367]

Optimal. Leaf size=705 \[ \frac {2 a b d^2 m n x}{9 e^2}-\frac {71 b^2 d^2 m n^2 x}{54 e^2}+\frac {b d^2 m n (6 a-11 b n) x}{9 e^2}+\frac {19 b^2 d m n^2 x^2}{54 e}-\frac {2}{27} b^2 m n^2 x^3-\frac {2 a b d^2 n x \log \left (f x^m\right )}{3 e^2}+\frac {11 b^2 d^2 n^2 x \log \left (f x^m\right )}{9 e^2}-\frac {5 b^2 d n^2 x^2 \log \left (f x^m\right )}{18 e}+\frac {2}{27} b^2 n^2 x^3 \log \left (f x^m\right )+\frac {23 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}+\frac {5 b^2 d^3 m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{9 e^3}-\frac {5 b^2 d^3 n^2 \log \left (f x^m\right ) \log (d+e x)}{9 e^3}+\frac {8 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{9 e^3}+\frac {2 b^2 d^3 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{3 e^3}-\frac {2 b^2 d^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{3 e^3}-\frac {5 b d m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{18 e}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2}{9} b n x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {d^3 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{9 e^3}-\frac {1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {d^3 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {d^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {11 b^2 d^3 m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{9 e^3}-\frac {2 b d^3 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{3 e^3}+\frac {2 b^2 d^3 m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{3 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.29, antiderivative size = 863, normalized size of antiderivative = 1.22, number of steps used = 52, number of rules used = 19, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {2445, 2458, 45, 2372, 12, 14, 2338, 2475, 2443, 2481, 2421, 6724, 2393, 2332, 2354, 2438, 2341, 2484, 2422} \begin {gather*} \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 {m \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d^3}{3 e^3}-\frac {m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d^3}{3 e^3}+\frac {23 b^2 m n^2 \log (x) d^3}{54 e^3}+\frac {11 b m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{9 e^3}-\frac {2 b m n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{9 e^3}-\frac {2 b m n \log (x) \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{3 e^3}+\frac {2 b n \log \left (f x^m\right ) \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\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 b m n \left (a+b \log \left (c (d+e x)^n\right )\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 {28 b^2 m n^2 x d^2}{9 e^2}+\frac {11 a b m n x d^2}{9 e^2}+\frac {2 b^2 n^2 x \log \left (f x^m\right ) d^2}{e^2}+\frac {11 b^2 m n (d+e x) \log \left (c (d+e x)^n\right ) d^2}{9 e^3}+\frac {2 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{3 e^3}-\frac {2 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{e^3}+\frac {5 b^2 m n^2 x^2 d}{36 e}+\frac {13 b^2 m n^2 (d+e x)^2 d}{36 e^3}-\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right ) d}{2 e^3}-\frac {13 b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{18 e^3}+\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{e^3}-\frac {2}{81} b^2 m n^2 x^3-\frac {4 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 {4 b m n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{27 e^3}-\frac {2 b n (d+e x)^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*a*b*d^2*m*n*x)/(9*e^2) - (28*b^2*d^2*m*n^2*x)/(9*e^2) + (5*b^2*d*m*n^2*x^2)/(36*e) - (2*b^2*m*n^2*x^3)/81
+ (13*b^2*d*m*n^2*(d + e*x)^2)/(36*e^3) - (4*b^2*m*n^2*(d + e*x)^3)/(81*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*L
og[f*x^m])/(27*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) + (11*b^2*d^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(9*e^3) + (2*
b*d^2*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) - (13*b*d*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/
(18*e^3) + (4*b*m*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(27*e^3) + (11*b*d^3*m*n*Log[-((e*x)/d)]*(a + b*Lo
g[c*(d + e*x)^n]))/(9*e^3) - (2*b*d^2*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/e^3 + (b*d*n*(d + e*x
)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/e^3 - (2*b*n*(d + e*x)^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(9*
e^3) - (2*b*d^3*m*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*d^3*m*n*Log[x]*Log[d + e*x]*(a + b
*Log[c*(d + e*x)^n]))/(3*e^3) + (2*b*d^3*n*Log[f*x^m]*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) - (m*x^
3*(a + b*Log[c*(d + e*x)^n])^2)/9 + (d^3*m*Log[x]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) - (d^3*m*Log[-((e*x)/d
)]*(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)

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 45

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 2332

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

Rule 2338

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2354

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]

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2422

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 2438

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 2443

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 2445

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 2458

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 2475

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 2481

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 2484

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 6724

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]

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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.10, size = 735, normalized size = 1.04 \begin {gather*} \frac {6 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \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 (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )+18 e^3 m x^3 \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-6 e^3 x^3 \left (m+3 m \log (x)-3 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+b m n \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (-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 \log (d+e x)-6 \log (x) \left (e x \left (-6 d^2+3 d e x-2 e^2 x^2\right )+6 e^3 x^3 \log (d+e x)+6 d^3 \log \left (1+\frac {e x}{d}\right )\right )-36 d^3 \text {Li}_2\left (-\frac {e x}{d}\right )\right )-b^2 n^2 \left (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 \left (f x^m\right )-66 d^2 e x \log \left (f x^m\right )+15 d e^2 x^2 \log \left (f x^m\right )-4 e^3 x^3 \log \left (f x^m\right )-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 \left (f x^m\right ) \log (d+e x)+36 d^2 e x \log \left (f x^m\right ) \log (d+e x)-18 d e^2 x^2 \log \left (f x^m\right ) \log (d+e x)+12 e^3 x^3 \log \left (f x^m\right ) \log (d+e x)+6 d^3 m \log ^2(d+e x)+6 e^3 m x^3 \log ^2(d+e x)+18 d^3 m \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-18 d^3 \log \left (f x^m\right ) \log ^2(d+e x)-18 e^3 x^3 \log \left (f x^m\right ) \log ^2(d+e x)+66 d^3 m \log (x) \log \left (1+\frac {e x}{d}\right )+66 d^3 m \text {Li}_2\left (-\frac {e x}{d}\right )+36 d^3 m \log (d+e x) \text {Li}_2\left (1+\frac {e x}{d}\right )-36 d^3 m \text {Li}_3\left (1+\frac {e x}{d}\right )\right )}{54 e^3} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int x^{2} \ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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