\(\int x \log (f x^m) (a+b \log (c (d+e x)^n))^2 \, dx\) [368]

Optimal result

Integrand size = 24, antiderivative size = 602 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=-\frac {a b d m n x}{2 e}+\frac {2 b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^2}-\frac {b^2 d^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{e^2} \]

[Out]

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

Time = 0.89 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341, 2475, 45, 2458, 2393, 2354, 2438, 2395, 2421, 6724} \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {b d^2 m n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {a b d m n x}{2 e}-\frac {2 b d m n x (a-b n)}{e}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {3 b^2 d^2 m n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{2 e^2}-\frac {b^2 d^2 m n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {2 b^2 d m n^2 x}{e}-\frac {1}{8} b^2 m n^2 x^2 \]

[In]

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

[Out]

-1/2*(a*b*d*m*n*x)/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 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 2333

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 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 2342

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 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 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 2395

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 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 2436

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 2437

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 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 2448

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 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 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*} \text {integral}& = \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 \text {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) \text {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) \text {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 ) \text {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 \text {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) \text {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) \text {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 ) \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A] (verified)

Time = 0.17 (sec) , antiderivative size = 825, normalized size of antiderivative = 1.37 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=b^2 n^2 \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log ^2(d+e x)-e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )\right )+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (\frac {1}{2} x^2 \log (d+e x)-\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+\frac {1}{2} m x^2 \log (x) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+\frac {1}{4} x^2 \left (-m+2 \left (-m \log (x)+\log \left (f x^m\right )\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )^2+b m n \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (-\frac {1}{2} x^2 \log (d+e x)+x^2 \log (x) \log (d+e x)+\frac {1}{2} e \left (-\frac {d x}{e^2}+\frac {x^2}{2 e}+\frac {d^2 \log (d+e x)}{e^3}\right )-e \left (-\frac {d x (-1+\log (x))}{e^2}+\frac {-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)}{e}+\frac {d^2 \left (\frac {\log (x) \log \left (\frac {d+e x}{d}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{e^2}\right )\right )+\frac {1}{2} b^2 m n^2 \left (-\frac {1}{2} x^2 \log ^2(d+e x)+x^2 \log (x) \log ^2(d+e x)+e \left (\frac {3 d x}{2 e^2}-\frac {x^2}{4 e}-\frac {3 d^2 \log (d+e x)}{2 e^3}-\frac {d x \log (d+e x)}{e^2}+\frac {x^2 \log (d+e x)}{2 e}+\frac {d^2 \log ^2(d+e x)}{2 e^3}\right )-2 e \left (-\frac {d \left (2 e x-d \log (d+e x)-e x \log (d+e x)+\log (x) \left (-e x+e x \log (d+e x)+d \log \left (1+\frac {e x}{d}\right )\right )+d \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )}{e^3}+\frac {-3 d e x+e^2 x^2+d^2 \log (d+e x)-e^2 x^2 \log (d+e x)+\log (x) \left (e x (2 d-e x)+2 e^2 x^2 \log (d+e x)-2 d^2 \log \left (1+\frac {e x}{d}\right )\right )-2 d^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^3}+\frac {d^2 \left (\frac {1}{2} \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log ^2(d+e x)-\log (d+e x) \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+\operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )\right )}{e^3}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int x \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}d x\]

[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)

Fricas [F]

\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]

[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)

Giac [F]

\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x \log \left (f x^{m}\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int x\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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