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

Optimal. Leaf size=602 \[ \frac{b d^2 m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac{3 b^2 d^2 m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{2 e^2}-\frac{b^2 d^2 m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e^2}+\frac{b d^2 m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{d^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac{m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{a b d m n x}{2 e}-\frac{2 b d m n x (a-b n)}{e}-\frac{2 b^2 d^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac{5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac{b^2 m n^2 (d+e x)^2}{4 e^2}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{2 b^2 d m n^2 x}{e}-\frac{1}{8} b^2 m n^2 x^2 \]

[Out]

-(a*b*d*m*n*x)/(2*e) + (2*b^2*d*m*n^2*x)/e - (2*b*d*m*n*(a - b*n)*x)/e - (b^2*m*n^2*x^2)/8 - (b^2*m*n^2*(d + e
*x)^2)/(4*e^2) - (b^2*d^2*m*n^2*Log[x])/(4*e^2) + (2*a*b*d*n*x*Log[f*x^m])/e - (2*b^2*d*n^2*x*Log[f*x^m])/e +
(b^2*n^2*(d + e*x)^2*Log[f*x^m])/(4*e^2) - (5*b^2*d*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(2*e^2) - (2*b^2*d^2*m*n
*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e^2 + (2*b^2*d*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e^2 + (b*m*n*(d
+ e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (b*d^2*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2)
- (b*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (d*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])
^2)/(2*e^2) - (m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) + (d^2*m*Log[-((e*x)/d)]*(a + b*Log[c*(d +
e*x)^n])^2)/(2*e^2) - (d*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + ((d + e*x)^2*Log[f*x^m]*(a +
b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (3*b^2*d^2*m*n^2*PolyLog[2, 1 + (e*x)/d])/(2*e^2) + (b*d^2*m*n*(a + b*Log[
c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e^2 - (b^2*d^2*m*n^2*PolyLog[3, 1 + (e*x)/d])/e^2

________________________________________________________________________________________

Rubi [A]  time = 1.29313, antiderivative size = 602, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 16, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304, 2428, 43, 2411, 2351, 2317, 2391, 2353, 2374, 6589} \[ \frac{b d^2 m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac{3 b^2 d^2 m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{2 e^2}-\frac{b^2 d^2 m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e^2}+\frac{b d^2 m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{d^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac{m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{a b d m n x}{2 e}-\frac{2 b d m n x (a-b n)}{e}-\frac{2 b^2 d^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac{5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac{b^2 m n^2 (d+e x)^2}{4 e^2}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{2 b^2 d m n^2 x}{e}-\frac{1}{8} b^2 m n^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*b*d*m*n*x)/(2*e) + (2*b^2*d*m*n^2*x)/e - (2*b*d*m*n*(a - b*n)*x)/e - (b^2*m*n^2*x^2)/8 - (b^2*m*n^2*(d + e
*x)^2)/(4*e^2) - (b^2*d^2*m*n^2*Log[x])/(4*e^2) + (2*a*b*d*n*x*Log[f*x^m])/e - (2*b^2*d*n^2*x*Log[f*x^m])/e +
(b^2*n^2*(d + e*x)^2*Log[f*x^m])/(4*e^2) - (5*b^2*d*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(2*e^2) - (2*b^2*d^2*m*n
*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e^2 + (2*b^2*d*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e^2 + (b*m*n*(d
+ e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (b*d^2*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2)
- (b*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (d*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])
^2)/(2*e^2) - (m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) + (d^2*m*Log[-((e*x)/d)]*(a + b*Log[c*(d +
e*x)^n])^2)/(2*e^2) - (d*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + ((d + e*x)^2*Log[f*x^m]*(a +
b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (3*b^2*d^2*m*n^2*PolyLog[2, 1 + (e*x)/d])/(2*e^2) + (b*d^2*m*n*(a + b*Log[
c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e^2 - (b^2*d^2*m*n^2*PolyLog[3, 1 + (e*x)/d])/e^2

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 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 2296

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

Rule 2295

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

Rule 2390

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

Rule 2305

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

Rule 2304

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 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 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 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 2351

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 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]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2353

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

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]

Rubi steps

\begin{align*} \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\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 )}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-m \int \left (\frac{2 a b d n}{e}-\frac{2 b^2 d n^2}{e}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 x}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 x}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 x}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 x}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 x}\right ) \, dx\\ &=-\frac{2 b d m n (a-b n) x}{e}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\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 )}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m \int \frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac{(d m) \int \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e^2}+\frac{(b m n) \int \frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx}{2 e^2}-\frac{\left (2 b^2 d m n\right ) \int \frac{(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e^2}-\frac{\left (b^2 m n^2\right ) \int \frac{(d+e x)^2}{x} \, dx}{4 e^2}\\ &=-\frac{2 b d m n (a-b n) x}{e}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\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 )}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 e^3}+\frac{(d m) \operatorname{Subst}\left (\int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^3}+\frac{(b m n) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 e^3}-\frac{\left (2 b^2 d m n\right ) \operatorname{Subst}\left (\int \frac{x \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^3}-\frac{\left (b^2 m n^2\right ) \int \left (2 d e+\frac{d^2}{x}+e^2 x\right ) \, dx}{4 e^2}\\ &=-\frac{b^2 d m n^2 x}{2 e}-\frac{2 b d m n (a-b n) x}{e}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\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 )}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m \operatorname{Subst}\left (\int \left (d e \left (a+b \log \left (c x^n\right )\right )^2-\frac{d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}+e x \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx,x,d+e x\right )}{2 e^3}+\frac{(d m) \operatorname{Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^2-\frac{d e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}\right ) \, dx,x,d+e x\right )}{e^3}+\frac{(b m n) \operatorname{Subst}\left (\int \left (d e \left (a+b \log \left (c x^n\right )\right )-\frac{d^2 e \left (a+b \log \left (c x^n\right )\right )}{d-x}+e x \left (a+b \log \left (c x^n\right )\right )\right ) \, dx,x,d+e x\right )}{2 e^3}-\frac{\left (2 b^2 d m n\right ) \operatorname{Subst}\left (\int \left (e \log \left (c x^n\right )-\frac{d e \log \left (c x^n\right )}{d-x}\right ) \, dx,x,d+e x\right )}{e^3}\\ &=-\frac{b^2 d m n^2 x}{2 e}-\frac{2 b d m n (a-b n) x}{e}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\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 )}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac{(d m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}+\frac{(d m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2}+\frac{\left (d^2 m\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{2 e^2}-\frac{\left (d^2 m\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{e^2}+\frac{(b m n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac{(b d m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}-\frac{\left (2 b^2 d m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac{\left (b d^2 m n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{2 e^2}+\frac{\left (2 b^2 d^2 m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{a b d m n x}{2 e}+\frac{3 b^2 d m n^2 x}{2 e}-\frac{2 b d m n (a-b n) x}{e}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 m n^2 (d+e x)^2}{8 e^2}-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac{2 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac{2 b^2 d^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac{b d^2 m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\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 )}{2 e^2}+\frac{d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac{(b m n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac{(b d m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}-\frac{(2 b d m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac{\left (b^2 d m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{2 e^2}+\frac{\left (b d^2 m n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{\left (2 b d^2 m n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{\left (b^2 d^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{2 e^2}+\frac{\left (2 b^2 d^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=-\frac{a b d m n x}{2 e}+\frac{b^2 d m n^2 x}{e}-\frac{2 b d m n (a-b n) x}{e}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 m n^2 (d+e x)^2}{4 e^2}-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac{3 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac{2 b^2 d^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{b d^2 m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\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 )}{2 e^2}+\frac{d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{3 b^2 d^2 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{2 e^2}+\frac{b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^2}+\frac{\left (b^2 d m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac{\left (2 b^2 d m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}+\frac{\left (b^2 d^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac{\left (2 b^2 d^2 m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=-\frac{a b d m n x}{2 e}+\frac{2 b^2 d m n^2 x}{e}-\frac{2 b d m n (a-b n) x}{e}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 m n^2 (d+e x)^2}{4 e^2}-\frac{b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac{2 a b d n x \log \left (f x^m\right )}{e}-\frac{2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac{b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac{5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac{2 b^2 d^2 m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac{b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac{b d^2 m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\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 )}{2 e^2}+\frac{d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac{d^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac{(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac{3 b^2 d^2 m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{2 e^2}+\frac{b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^2}-\frac{b^2 d^2 m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{e^2}\\ \end{align*}

Mathematica [F]  time = 0.382877, size = 0, normalized size = 0. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 2.195, size = 0, normalized size = 0. \begin{align*} \int x\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*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

int(x*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}{4} \,{\left (b^{2}{\left (m - 2 \, \log \left (f\right )\right )} x^{2} - 2 \, b^{2} x^{2} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + \int \frac{2 \,{\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^{2} + 2 \,{\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 +{\left ({\left (4 \, a b e \log \left (f\right ) +{\left (4 \, e \log \left (c\right ) \log \left (f\right ) +{\left (m n - 2 \, n \log \left (f\right )\right )} e\right )} b^{2}\right )} x^{2} + 4 \,{\left (b^{2} d \log \left (c\right ) \log \left (f\right ) + a b d \log \left (f\right )\right )} x - 2 \,{\left ({\left ({\left (e n - 2 \, e \log \left (c\right )\right )} b^{2} - 2 \, a b e\right )} x^{2} - 2 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + 2 \,{\left ({\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{2} +{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x\right )} \log \left (x^{m}\right )}{2 \,{\left (e x + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*(m - 2*log(f))*x^2 - 2*b^2*x^2*log(x^m))*log((e*x + d)^n)^2 + integrate(1/2*(2*(b^2*e*log(c)^2*log(f
) + 2*a*b*e*log(c)*log(f) + a^2*e*log(f))*x^2 + 2*(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f
))*x + ((4*a*b*e*log(f) + (4*e*log(c)*log(f) + (m*n - 2*n*log(f))*e)*b^2)*x^2 + 4*(b^2*d*log(c)*log(f) + a*b*d
*log(f))*x - 2*(((e*n - 2*e*log(c))*b^2 - 2*a*b*e)*x^2 - 2*(b^2*d*log(c) + a*b*d)*x)*log(x^m))*log((e*x + d)^n
) + 2*((b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^2 + (b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x)*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 \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 2 \, a b x \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{2} x \log \left (f x^{m}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*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*log(f*x^m), x)