3.394 \(\int (a+b \log (c (d+e x)^n))^2 (f+g \log (h (i+j x)^m)) \, dx\)

Optimal. Leaf size=649 \[ -\frac{2 b d g m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{e}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b g n x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{2 b g i m n \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{d g m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b f n x+4 a b g m n x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}+2 b^2 f n^2 x+\frac{2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-6 b^2 g m n^2 x \]

[Out]

-2*a*b*f*n*x + 4*a*b*g*m*n*x + 2*b^2*f*n^2*x - 6*b^2*g*m*n^2*x - (2*b^2*f*n*(d + e*x)*Log[c*(d + e*x)^n])/e +
(4*b^2*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e + (d*f*(a + b*Log[c*(d + e*x)^n])^2)/e - (g*m*(d + e*x)*(a + b*Lo
g[c*(d + e*x)^n])^2)/e - (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a
+ b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i +
j*x))/(e*i - d*j)])/j + (2*b^2*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/j - (2*b^2*d*g*n^2*Log[-((j*(d + e*x))/(e*
i - d*j))]*Log[h*(i + j*x)^m])/e - 2*b*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c
*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) - (2*b^2*g*
i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (2*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*
(d + e*x))/(e*i - d*j))])/e + (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))]
)/j - (2*b^2*d*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e + (2*b^2*d*g*m*n^2*PolyLog[3, -((j*(d + e*x))/
(e*i - d*j))])/e - (2*b^2*g*i*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j

________________________________________________________________________________________

Rubi [A]  time = 1.48128, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 19, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.613, Rules used = {2430, 2416, 2389, 2296, 2295, 2396, 2433, 2374, 6589, 6742, 2411, 2346, 2301, 43, 2394, 2393, 2391, 2375, 2317} \[ -\frac{2 b d g m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{e}-\frac{2 b^2 g i m n^2 \text{PolyLog}\left (3,-\frac{j (d+e x)}{e i-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b g n x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{d g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{2 b g i m n \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{d g m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b f n x+4 a b g m n x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}+2 b^2 f n^2 x+\frac{2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-6 b^2 g m n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*a*b*f*n*x + 4*a*b*g*m*n*x + 2*b^2*f*n^2*x - 6*b^2*g*m*n^2*x - (2*b^2*f*n*(d + e*x)*Log[c*(d + e*x)^n])/e +
(4*b^2*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e + (d*f*(a + b*Log[c*(d + e*x)^n])^2)/e - (g*m*(d + e*x)*(a + b*Lo
g[c*(d + e*x)^n])^2)/e - (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a
+ b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i +
j*x))/(e*i - d*j)])/j + (2*b^2*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/j - (2*b^2*d*g*n^2*Log[-((j*(d + e*x))/(e*
i - d*j))]*Log[h*(i + j*x)^m])/e - 2*b*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c
*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) - (2*b^2*g*
i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (2*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*
(d + e*x))/(e*i - d*j))])/e + (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))]
)/j - (2*b^2*d*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e + (2*b^2*d*g*m*n^2*PolyLog[3, -((j*(d + e*x))/
(e*i - d*j))])/e - (2*b^2*g*i*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j

Rule 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f
+ g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 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 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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 2375

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

Rule 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]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right ) \, dx &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g j m) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{394+j x} \, dx-(2 b e n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (394+j x)^m\right )\right )}{d+e x} \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g j m) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac{394 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j (394+j x)}\right ) \, dx-(2 b e n) \int \left (\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac{g x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x}\right ) \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-(g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+(394 g m) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{394+j x} \, dx-(2 b e f n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(2 b e g n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x} \, dx\\ &=\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{(g m) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-(2 b f n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(2 b e g n) \int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{e}-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac{(788 b e g m n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{d+e x} \, dx}{j}\\ &=-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{(2 b f 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 f n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{e}-(2 b g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right ) \, dx+(2 b d g n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )}{d+e x} \, dx+\frac{(2 b g m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac{(788 b g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{394 e-d j}{e}+\frac{j x}{e}\right )}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+2 a b g m n x+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{\left (2 b^2 f n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{(2 b d g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac{394 e-d j}{e}+\frac{j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (2 b^2 g m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+(2 b g j m n) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{394+j x} \, dx+\left (2 b^2 e g n^2\right ) \int \frac{x \log \left (h (394+j x)^m\right )}{d+e x} \, dx-\frac{\left (788 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+2 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{(d g j m) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\frac{394 e-d j}{e}+\frac{j x}{e}} \, dx,x,d+e x\right )}{e^2}+(2 b g j m n) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{j}-\frac{394 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (394+j x)}\right ) \, dx+\left (2 b^2 e g n^2\right ) \int \left (\frac{\log \left (h (394+j x)^m\right )}{e}-\frac{d \log \left (h (394+j x)^m\right )}{e (d+e x)}\right ) \, dx\\ &=-2 a b f n x+2 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+(2 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-(788 b g m n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{394+j x} \, dx+\frac{(2 b d g m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\left (2 b^2 g n^2\right ) \int \log \left (h (394+j x)^m\right ) \, dx-\left (2 b^2 d g n^2\right ) \int \frac{\log \left (h (394+j x)^m\right )}{d+e x} \, dx\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-2 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\left (2 b^2 g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx+\frac{\left (2 b^2 g n^2\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,394+j x\right )}{j}+\frac{\left (2 b^2 d g m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (788 b^2 e g m n^2\right ) \int \frac{\log \left (\frac{e (394+j x)}{394 e-d j}\right )}{d+e x} \, dx}{j}+\frac{\left (2 b^2 d g j m n^2\right ) \int \frac{\log \left (\frac{j (d+e x)}{-394 e+d j}\right )}{394+j x} \, dx}{e}\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-4 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{2 b^2 g n^2 (394+j x) \log \left (h (394+j x)^m\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\frac{2 b^2 d g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}+\frac{\left (2 b^2 g 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 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-394 e+d j}\right )}{x} \, dx,x,394+j x\right )}{e}+\frac{\left (788 b^2 g m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{394 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-6 b^2 g m n^2 x-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{394 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (394+j x)}{394 e-d j}\right )}{j}+\frac{2 b^2 g n^2 (394+j x) \log \left (h (394+j x)^m\right )}{j}-\frac{2 b^2 d g n^2 \log \left (-\frac{j (d+e x)}{394 e-d j}\right ) \log \left (h (394+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (394+j x)^m\right )+\frac{d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (394+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (394+j x)^m\right )\right )-\frac{d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}+\frac{788 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}-\frac{2 b^2 d g m n^2 \text{Li}_2\left (\frac{e (394+j x)}{394 e-d j}\right )}{e}+\frac{2 b^2 d g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{e}-\frac{788 b^2 g m n^2 \text{Li}_3\left (-\frac{j (d+e x)}{394 e-d j}\right )}{j}\\ \end{align*}

Mathematica [B]  time = 0.533498, size = 1355, normalized size = 2.09 \[ \frac{e f j x a^2-e g j m x a^2+e g i m \log (i+j x) a^2+e g j x \log \left (h (i+j x)^m\right ) a^2-2 b d f j n a+2 b d g j m n a-2 b e f j n x a+4 b e g j m n x a+2 b d f j n \log (d+e x) a-2 b d g j m n \log (d+e x) a+2 b e f j x \log \left (c (d+e x)^n\right ) a-2 b e g j m x \log \left (c (d+e x)^n\right ) a-2 b e g i m n \log (i+j x) a+2 b d g j m n \log (i+j x) a-2 b e g i m n \log (d+e x) \log (i+j x) a+2 b e g i m \log \left (c (d+e x)^n\right ) \log (i+j x) a+2 b e g i m n \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right ) a-2 b d g j m n \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right ) a-2 b d g j n \log \left (h (i+j x)^m\right ) a-2 b e g j n x \log \left (h (i+j x)^m\right ) a+2 b d g j n \log (d+e x) \log \left (h (i+j x)^m\right ) a+2 b e g j x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right ) a-2 b^2 d g j m n^2-b^2 d f j n^2 \log ^2(d+e x)+b^2 d g j m n^2 \log ^2(d+e x)+b^2 e f j x \log ^2\left (c (d+e x)^n\right )-b^2 e g j m x \log ^2\left (c (d+e x)^n\right )+2 b^2 e f j n^2 x-6 b^2 e g j m n^2 x+2 b^2 d g j m n^2 \log (d+e x)-2 b^2 d f j n \log \left (c (d+e x)^n\right )+2 b^2 d g j m n \log \left (c (d+e x)^n\right )-2 b^2 e f j n x \log \left (c (d+e x)^n\right )+4 b^2 e g j m n x \log \left (c (d+e x)^n\right )+2 b^2 d f j n \log (d+e x) \log \left (c (d+e x)^n\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right )+2 b^2 e g i m n^2 \log (i+j x)+b^2 e g i m n^2 \log ^2(d+e x) \log (i+j x)+b^2 e g i m \log ^2\left (c (d+e x)^n\right ) \log (i+j x)+2 b^2 e g i m n^2 \log (d+e x) \log (i+j x)-2 b^2 d g j m n^2 \log (d+e x) \log (i+j x)-2 b^2 e g i m n \log \left (c (d+e x)^n\right ) \log (i+j x)+2 b^2 d g j m n \log \left (c (d+e x)^n\right ) \log (i+j x)-2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log (i+j x)-b^2 e g i m n^2 \log ^2(d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+b^2 d g j m n^2 \log ^2(d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )-2 b^2 e g i m n^2 \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+2 b^2 d g j m n^2 \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )+2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )-b^2 d g j n^2 \log ^2(d+e x) \log \left (h (i+j x)^m\right )+b^2 e g j x \log ^2\left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b^2 e g j n^2 x \log \left (h (i+j x)^m\right )-2 b^2 d g j n \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )-2 b^2 e g j n x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b^2 d g j n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b g (e i-d j) m n \left (a-b n+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+2 b^2 g (d j-e i) m n^2 \text{PolyLog}\left (3,\frac{j (d+e x)}{d j-e i}\right )}{e j} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*d*f*j*n + 2*a*b*d*g*j*m*n - 2*b^2*d*g*j*m*n^2 + a^2*e*f*j*x - a^2*e*g*j*m*x - 2*a*b*e*f*j*n*x + 4*a*b*
e*g*j*m*n*x + 2*b^2*e*f*j*n^2*x - 6*b^2*e*g*j*m*n^2*x + 2*a*b*d*f*j*n*Log[d + e*x] - 2*a*b*d*g*j*m*n*Log[d + e
*x] + 2*b^2*d*g*j*m*n^2*Log[d + e*x] - b^2*d*f*j*n^2*Log[d + e*x]^2 + b^2*d*g*j*m*n^2*Log[d + e*x]^2 - 2*b^2*d
*f*j*n*Log[c*(d + e*x)^n] + 2*b^2*d*g*j*m*n*Log[c*(d + e*x)^n] + 2*a*b*e*f*j*x*Log[c*(d + e*x)^n] - 2*a*b*e*g*
j*m*x*Log[c*(d + e*x)^n] - 2*b^2*e*f*j*n*x*Log[c*(d + e*x)^n] + 4*b^2*e*g*j*m*n*x*Log[c*(d + e*x)^n] + 2*b^2*d
*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n] - 2*b^2*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] + b^2*e*f*j*x*Log[c*(
d + e*x)^n]^2 - b^2*e*g*j*m*x*Log[c*(d + e*x)^n]^2 + a^2*e*g*i*m*Log[i + j*x] - 2*a*b*e*g*i*m*n*Log[i + j*x] +
2*a*b*d*g*j*m*n*Log[i + j*x] + 2*b^2*e*g*i*m*n^2*Log[i + j*x] - 2*a*b*e*g*i*m*n*Log[d + e*x]*Log[i + j*x] + 2
*b^2*e*g*i*m*n^2*Log[d + e*x]*Log[i + j*x] - 2*b^2*d*g*j*m*n^2*Log[d + e*x]*Log[i + j*x] + b^2*e*g*i*m*n^2*Log
[d + e*x]^2*Log[i + j*x] + 2*a*b*e*g*i*m*Log[c*(d + e*x)^n]*Log[i + j*x] - 2*b^2*e*g*i*m*n*Log[c*(d + e*x)^n]*
Log[i + j*x] + 2*b^2*d*g*j*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] - 2*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^
n]*Log[i + j*x] + b^2*e*g*i*m*Log[c*(d + e*x)^n]^2*Log[i + j*x] + 2*a*b*e*g*i*m*n*Log[d + e*x]*Log[(e*(i + j*x
))/(e*i - d*j)] - 2*a*b*d*g*j*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 2*b^2*e*g*i*m*n^2*Log[d + e*x]
*Log[(e*(i + j*x))/(e*i - d*j)] + 2*b^2*d*g*j*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - b^2*e*g*i*m*
n^2*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] + b^2*d*g*j*m*n^2*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*
j)] + 2*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] - 2*b^2*d*g*j*m*n*Log[d +
e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] - 2*a*b*d*g*j*n*Log[h*(i + j*x)^m] + a^2*e*g*j*x*Log[h
*(i + j*x)^m] - 2*a*b*e*g*j*n*x*Log[h*(i + j*x)^m] + 2*b^2*e*g*j*n^2*x*Log[h*(i + j*x)^m] + 2*a*b*d*g*j*n*Log[
d + e*x]*Log[h*(i + j*x)^m] - b^2*d*g*j*n^2*Log[d + e*x]^2*Log[h*(i + j*x)^m] - 2*b^2*d*g*j*n*Log[c*(d + e*x)^
n]*Log[h*(i + j*x)^m] + 2*a*b*e*g*j*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] - 2*b^2*e*g*j*n*x*Log[c*(d + e*x)^
n]*Log[h*(i + j*x)^m] + 2*b^2*d*g*j*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + b^2*e*g*j*x*Log[c*(
d + e*x)^n]^2*Log[h*(i + j*x)^m] + 2*b*g*(e*i - d*j)*m*n*(a - b*n + b*Log[c*(d + e*x)^n])*PolyLog[2, (j*(d + e
*x))/(-(e*i) + d*j)] + 2*b^2*g*(-(e*i) + d*j)*m*n^2*PolyLog[3, (j*(d + e*x))/(-(e*i) + d*j)])/(e*j)

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Maple [F]  time = 2.82, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m)),x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b*e*f*n*(x/e - d*log(e*x + d)/e^2) - a^2*g*j*m*(x/j - i*log(j*x + i)/j^2) + b^2*f*x*log((e*x + d)^n*c)^2
+ 2*a*b*f*x*log((e*x + d)^n*c) + a^2*g*x*log((j*x + i)^m*h) - (2*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)^
n*c) + (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*b^2*f + a^2*f*x + ((b^2*e*g*i*m*log(j*x + i) - (j*
m - j*log(h))*b^2*e*g*x)*log((e*x + d)^n)^2 - (b^2*d*g*j*n^2*log(e*x + d)^2 - b^2*e*g*j*x*log((e*x + d)^n)^2 +
(2*(e*g*j*n - e*g*j*log(c))*a*b - (2*e*g*j*n^2 - 2*e*g*j*n*log(c) + e*g*j*log(c)^2)*b^2)*x - 2*(a*b*d*g*j*n -
(d*g*j*n^2 - d*g*j*n*log(c))*b^2)*log(e*x + d) - 2*(b^2*d*g*j*n*log(e*x + d) + (a*b*e*g*j - (e*g*j*n - e*g*j*
log(c))*b^2)*x)*log((e*x + d)^n))*log((j*x + i)^m))/(e*j) - integrate(-(b^2*d*e*g*i*j*log(c)^2*log(h) + 2*a*b*
d*e*g*i*j*log(c)*log(h) + (2*(e^2*g*j^2*m*n - (j^2*m - j^2*log(h))*e^2*g*log(c))*a*b - (2*e^2*g*j^2*m*n^2 - 2*
e^2*g*j^2*m*n*log(c) + (j^2*m - j^2*log(h))*e^2*g*log(c)^2)*b^2)*x^2 + (b^2*d*e*g*j^2*m*n^2*x + b^2*d^2*g*j^2*
m*n^2)*log(e*x + d)^2 + (2*(d*e*g*j^2*m*n + (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*a*b - (2*d
*e*g*j^2*m*n^2 - 2*d*e*g*j^2*m*n*log(c) - (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c)^2)*b^2)*x - 2
*(a*b*d^2*g*j^2*m*n - (d^2*g*j^2*m*n^2 - d^2*g*j^2*m*n*log(c))*b^2 + (a*b*d*e*g*j^2*m*n - (d*e*g*j^2*m*n^2 - d
*e*g*j^2*m*n*log(c))*b^2)*x)*log(e*x + d) + 2*(b^2*d*e*g*i*j*log(c)*log(h) + a*b*d*e*g*i*j*log(h) - ((j^2*m -
j^2*log(h))*a*b*e^2*g + ((j^2*m - j^2*log(h))*e^2*g*log(c) - (2*j^2*m*n - j^2*n*log(h))*e^2*g)*b^2)*x^2 + ((e^
2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*a*b + (d*e*g*j^2*m*n + (i*j*m*n - i*j*n*log(h))*e^2*g + (e^2*g*i*
j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*b^2)*x - (b^2*d*e*g*j^2*m*n*x + b^2*d^2*g*j^2*m*n)*log(e*x + d)
- (b^2*e^2*g*i*j*m*n*x + b^2*e^2*g*i^2*m*n)*log(j*x + i))*log((e*x + d)^n))/(e^2*j^2*x^2 + d*e*i*j + (e^2*i*j
+ d*e*j^2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} f \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b f \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} f +{\left (b^{2} g \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b g \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} g\right )} \log \left ({\left (j x + i\right )}^{m} h\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^2*(g*log((j*x + i)^m*h) + f), x)