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

Optimal. Leaf size=309 \[ -\frac{2 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x \log \left (f x^m\right )+2 a b m n x+2 b m n x (a-b n)-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \log \left (f x^m\right )-4 b^2 m n^2 x \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.452542, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2389, 2296, 2295, 2423, 2411, 43, 2351, 2317, 2391, 2353, 2374, 6589} \[ -\frac{2 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x \log \left (f x^m\right )+2 a b m n x+2 b m n x (a-b n)-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \log \left (f x^m\right )-4 b^2 m n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2423

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_), x_Symbol] :> With[{u = In
tHide[(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]] /; Free
Q[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 1]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A]  time = 0.473595, size = 549, normalized size = 1.78 \[ -2 b e m n \left (\frac{x (\log (x)-1)}{e}-\frac{d \left (\frac{\text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e}+\frac{\log (x) \log \left (\frac{d+e x}{d}\right )}{e}\right )}{e}\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+b^2 m n^2 \left (-2 e \left (\frac{d \text{PolyLog}\left (2,-\frac{e x}{d}\right )-e x \log (d+e x)-d \log (d+e x)+\log (x) \left (e x \log (d+e x)+d \log \left (\frac{e x}{d}+1\right )-e x\right )+2 e x}{e^2}-\frac{d \left (\text{PolyLog}\left (3,\frac{d+e x}{d}\right )-\log (d+e x) \text{PolyLog}\left (2,\frac{d+e x}{d}\right )+\frac{1}{2} \left (\log (x)-\log \left (-\frac{e x}{d}\right )\right ) \log ^2(d+e x)\right )}{e^2}\right )+2 e \left (-\frac{d \log ^2(d+e x)}{2 e^2}+\frac{d \log (d+e x)}{e^2}+\frac{x \log (d+e x)}{e}-\frac{x}{e}\right )-x \log ^2(d+e x)+x \log (x) \log ^2(d+e x)\right )-2 b n x \left (\log \left (f x^m\right )+m (-\log (x))-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+2 b n x \log (d+e x) \left (\log \left (f x^m\right )-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+\frac{2 b d n \log (d+e x) \left (\log \left (f x^m\right )+m (-\log (x))-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 n^2 \left (x \log ^2(d+e x)-2 e \left (-\frac{d \log ^2(d+e x)}{2 e^2}+\frac{d \log (d+e x)}{e^2}+\frac{x \log (d+e x)}{e}-\frac{x}{e}\right )\right ) \left (\log \left (f x^m\right )-m \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 1.871, size = 0, normalized size = 0. \begin{align*} \int \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(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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} \log \left (f x^{m}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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*log(f*x^m), x)