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

Optimal. Leaf size=602 \[ -\frac {a b d m n x}{2 e}+\frac {2 b^2 d m n^2 x}{e}-\frac {2 b d m n (a-b n) x}{e}-\frac {1}{8} b^2 m n^2 x^2-\frac {b^2 m n^2 (d+e x)^2}{4 e^2}-\frac {b^2 d^2 m n^2 \log (x)}{4 e^2}+\frac {2 a b d n x \log \left (f x^m\right )}{e}-\frac {2 b^2 d n^2 x \log \left (f x^m\right )}{e}+\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right )}{4 e^2}-\frac {5 b^2 d m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2}-\frac {2 b^2 d^2 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {2 b^2 d n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b d^2 m n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {d m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {d (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}+\frac {(d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}-\frac {3 b^2 d^2 m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}-\frac {b^2 d^2 m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e^2} \]

[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]
time = 0.88, antiderivative size = 602, normalized size of antiderivative = 1.00, 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 = {2448, 2436, 2333, 2332, 2437, 2342, 2341, 2475, 45, 2458, 2393, 2354, 2438, 2395, 2421, 6724} \begin {gather*} \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 \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \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*}

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Mathematica [F]
time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx \end {gather*}

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 = 0.08, size = 0, normalized size = 0.00 \[\int x \ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*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 = 0.00, 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*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((x*e + d)^n)^2 + integrate(1/2*(2*(b^2*log(c)^2*log(f)
+ 2*a*b*log(c)*log(f) + a^2*log(f))*x^2*e + 2*(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f))*x
+ (((m*n - 2*n*log(f) + 4*log(c)*log(f))*b^2 + 4*a*b*log(f))*x^2*e + 4*(b^2*d*log(c)*log(f) + a*b*d*log(f))*x
- 2*((b^2*(n - 2*log(c)) - 2*a*b)*x^2*e - 2*(b^2*d*log(c) + a*b*d)*x)*log(x^m))*log((x*e + d)^n) + 2*((b^2*lo
g(c)^2 + 2*a*b*log(c) + a^2)*x^2*e + (b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x)*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*log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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