3.4.94 \(\int (a+b \log (c (d+e x)^n))^2 (f+g \log (h (i+j x)^m)) \, dx\) [394]

Optimal. Leaf size=649 \[ -2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-6 b^2 g m n^2 x-\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {2 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-\frac {2 b^2 d g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {2 b^2 g i m n^2 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}+\frac {2 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {2 b^2 d g m n^2 \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {2 b^2 d g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}-\frac {2 b^2 g i m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j} \]

[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*(e*x+d)*ln(c*(e*x+d)^n)/e+4*b^2*g*m*n*(e*x+
d)*ln(c*(e*x+d)^n)/e+d*f*(a+b*ln(c*(e*x+d)^n))^2/e-g*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e-2*b*g*i*m*n*(a+b*ln(c
*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j-d*g*m*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/e+g*i*m*(a+b*ln
(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/j+2*b^2*g*n^2*(j*x+i)*ln(h*(j*x+i)^m)/j-2*b^2*d*g*n^2*ln(-j*(e*x+d)/
(-d*j+e*i))*ln(h*(j*x+i)^m)/e-2*b*g*n*x*(a+b*ln(c*(e*x+d)^n))*ln(h*(j*x+i)^m)+d*g*(a+b*ln(c*(e*x+d)^n))^2*ln(h
*(j*x+i)^m)/e+x*(a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m))-2*b^2*g*i*m*n^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))
/j-2*b*d*g*m*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i))/e+2*b*g*i*m*n*(a+b*ln(c*(e*x+d)^n))*poly
log(2,-j*(e*x+d)/(-d*j+e*i))/j-2*b^2*d*g*m*n^2*polylog(2,e*(j*x+i)/(-d*j+e*i))/e+2*b^2*d*g*m*n^2*polylog(3,-j*
(e*x+d)/(-d*j+e*i))/e-2*b^2*g*i*m*n^2*polylog(3,-j*(e*x+d)/(-d*j+e*i))/j

________________________________________________________________________________________

Rubi [A]
time = 1.04, antiderivative size = 649, normalized size of antiderivative = 1.00, 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 = {2479, 2463, 2436, 2333, 2332, 2443, 2481, 2421, 6724, 6874, 2458, 2388, 2338, 45, 2441, 2440, 2438, 2422, 2354} \begin {gather*} -\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 \end {gather*}

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 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 2338

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 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 2388

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 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 2422

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 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 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 2440

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 2441

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 2443

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 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 2463

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 2479

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 2481

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 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]

Rule 6874

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

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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {(2 b d g n) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1355\) vs. \(2(649)=1298\).
time = 0.31, size = 1355, normalized size = 2.09 \begin {gather*} \frac {-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 ^2(d+e x)+b^2 d g j m n^2 \log ^2(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 a b e f j x \log \left (c (d+e x)^n\right )-2 a b e g j m x \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 )+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 )+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 ^2(d+e x) \log (i+j x)+2 a b e g i m \log \left (c (d+e x)^n\right ) \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 \log ^2\left (c (d+e x)^n\right ) \log (i+j x)+2 a b e g i m n \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )-2 a b d g j m n \log (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 )-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 \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 )-2 a b d g j n \log \left (h (i+j x)^m\right )+a^2 e g j x \log \left (h (i+j x)^m\right )-2 a b e g j n x \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 a b d g j n \log (d+e x) \log \left (h (i+j x)^m\right )-b^2 d g j n^2 \log ^2(d+e 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 a b e g j x \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 )+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 g (e i-d j) m n \left (a-b n+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {j (d+e x)}{-e i+d j}\right )+2 b^2 g (-e i+d j) m n^2 \text {Li}_3\left (\frac {j (d+e x)}{-e i+d j}\right )}{e j} \end {gather*}

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 = 0.06, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )\, dx\]

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 = 0.00, 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((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

-a^2*g*j*m*(x/j - I*log(j*x + I)/j^2) + 2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a*b*f*n*e + b^2*f*x*log((x*e + d)
^n*c)^2 + 2*a*b*f*x*log((x*e + d)^n*c) + a^2*g*x*log((j*x + I)^m*h) - ((d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e
+ d))*n^2*e^(-1) - 2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c))*b^2*f + a^2*f*x - (((j*m - j*
log(h))*b^2*g*x*e - I*b^2*g*m*e*log(j*x + I))*log((x*e + d)^n)^2 + (b^2*d*g*j*n^2*log(x*e + d)^2 - b^2*g*j*x*e
*log((x*e + d)^n)^2 + (2*(g*j*n - g*j*log(c))*a*b - (2*g*j*n^2 - 2*g*j*n*log(c) + g*j*log(c)^2)*b^2)*x*e - 2*(
a*b*d*g*j*n - (d*g*j*n^2 - d*g*j*n*log(c))*b^2)*log(x*e + d) - 2*(b^2*d*g*j*n*log(x*e + d) + (a*b*g*j - (g*j*n
- g*j*log(c))*b^2)*x*e)*log((x*e + d)^n))*log((j*x + I)^m))*e^(-1)/j - integrate(-((2*(g*j^2*m*n - (j^2*m - j
^2*log(h))*g*log(c))*a*b - (2*g*j^2*m*n^2 - 2*g*j^2*m*n*log(c) + (j^2*m - j^2*log(h))*g*log(c)^2)*b^2)*x^2*e^2
+ (b^2*d*g*j^2*m*n^2*x*e + b^2*d^2*g*j^2*m*n^2)*log(x*e + d)^2 - ((-I*b^2*g*j*log(c)^2*log(h) - 2*I*a*b*g*j*l
og(c)*log(h))*e^2 - (2*(d*g*j^2*m*n - (j^2*m - j^2*log(h))*d*g*log(c))*a*b - (2*d*g*j^2*m*n^2 - 2*d*g*j^2*m*n*
log(c) + (j^2*m - j^2*log(h))*d*g*log(c)^2)*b^2)*e)*x - (-I*b^2*d*g*j*log(c)^2*log(h) - 2*I*a*b*d*g*j*log(c)*l
og(h))*e - 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*g*j^2*m*n - (d*g*j^2*m
*n^2 - d*g*j^2*m*n*log(c))*b^2)*x*e)*log(x*e + d) - 2*(((j^2*m - j^2*log(h))*a*b*g + ((j^2*m - j^2*log(h))*g*l
og(c) - (2*j^2*m*n - j^2*n*log(h))*g)*b^2)*x^2*e^2 - ((I*a*b*g*j*log(h) + (I*g*j*log(c)*log(h) + (I*j*m*n - I*
j*n*log(h))*g)*b^2)*e^2 - ((j^2*m - j^2*log(h))*a*b*d*g - (d*g*j^2*m*n - (j^2*m - j^2*log(h))*d*g*log(c))*b^2)
*e)*x - (I*b^2*d*g*j*log(c)*log(h) + I*a*b*d*g*j*log(h))*e - (-I*b^2*g*j*m*n*x*e^2 + b^2*g*m*n*e^2)*log(j*x +
I) + (b^2*d*g*j^2*m*n*x*e + b^2*d^2*g*j^2*m*n)*log(x*e + d))*log((x*e + d)^n))/(j^2*x^2*e^2 + I*d*j*e + (d*j^2
*e + I*j*e^2)*x), 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((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

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

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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((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 = 0.00, 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((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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