3.14 \(\int \frac{(A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{a g+b g x} \, dx\)

Optimal. Leaf size=138 \[ \frac{2 B n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g}+\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b g}-\frac{\log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g} \]

[Out]

-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b*g)) + (2*B*n*(A + B*Log[e
*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (2*B^2*n^2*PolyLog[3, (b*(c + d*x)
)/(d*(a + b*x))])/(b*g)

________________________________________________________________________________________

Rubi [B]  time = 3.57289, antiderivative size = 789, normalized size of antiderivative = 5.72, number of steps used = 45, number of rules used = 23, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.657, Rules used = {2524, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2500, 2440, 2434, 2433, 2375, 2317, 2374, 6589, 2499, 2302, 30, 2396} \[ \frac{2 A B n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{b g}-\frac{2 B^2 n^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{\log (a g+b g x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g}+\frac{2 A B n \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{A B n \log ^2(g (a+b x))}{b g}-\frac{B^2 n \log ^2(a g+b g x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b g}-\frac{2 B^2 n \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{b g}-\frac{B^2 n^2 \log (-c-d x) \log ^2(g (a+b x))}{b g}+\frac{B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 n^2 \log ^2(a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{B^2 \log (-c-d x) \log ^2\left ((a+b x)^n\right )}{b g}+\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n \log (-c-d x) \log (g (a+b x)) \log \left ((a+b x)^n\right )}{b g}+\frac{B^2 n^2 \log ^3(g (a+b x))}{3 b g} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x),x]

[Out]

-((A*B*n*Log[g*(a + b*x)]^2)/(b*g)) + (B^2*n^2*Log[g*(a + b*x)]^3)/(3*b*g) - (B^2*n^2*Log[g*(a + b*x)]^2*Log[-
c - d*x])/(b*g) + (2*B^2*n*Log[g*(a + b*x)]*Log[(a + b*x)^n]*Log[-c - d*x])/(b*g) - (B^2*Log[(a + b*x)^n]^2*Lo
g[-c - d*x])/(b*g) + (B^2*n^2*Log[g*(a + b*x)]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(b*g) + (B^2*Log[(a + b*x)^n]
^2*Log[(b*(c + d*x))/(b*c - a*d)])/(b*g) + (B^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-n)]^2)/(b*g)
- (B^2*Log[g*(a + b*x)]*Log[(c + d*x)^(-n)]^2)/(b*g) + ((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[a*g + b*
g*x])/(b*g) + (2*A*B*n*Log[(b*(c + d*x))/(b*c - a*d)]*Log[a*g + b*g*x])/(b*g) - (2*B^2*n*Log[(b*(c + d*x))/(b*
c - a*d)]*(Log[(a + b*x)^n] - Log[e*((a + b*x)/(c + d*x))^n] + Log[(c + d*x)^(-n)])*Log[a*g + b*g*x])/(b*g) -
(B^2*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[a*g + b*g*x]^2)/(b*g) - (B^2*n^2*Log[(b*(c + d*x))/(b*c - a*d)]*Log[
a*g + b*g*x]^2)/(b*g) + (2*A*B*n*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*g) + (2*B^2*n*Log[(a + b*x)^n]*P
olyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*g) - (2*B^2*n*(Log[(a + b*x)^n] - Log[e*((a + b*x)/(c + d*x))^n] +
Log[(c + d*x)^(-n)])*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*g) - (2*B^2*n*Log[(c + d*x)^(-n)]*PolyLog[2
, (b*(c + d*x))/(b*c - a*d)])/(b*g) - (2*B^2*n^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/(b*g) - (2*B^2*n^2*
PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(b*g)

Rule 2524

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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 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 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 6688

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

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

Rule 2500

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]

Rule 2440

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

Rule 2434

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

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 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]

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 2499

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rubi steps

\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B (b c-a d) n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B (b c-a d) n) \int \left (\frac{d \left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac{(2 B d n) \int \frac{\left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B n) \int \left (\frac{A \log (a g+b g x)}{a+b x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{g}-\frac{(2 B d n) \int \left (\frac{A \log (a g+b g x)}{-c-d x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 A B n) \int \frac{\log (a g+b g x)}{a+b x} \, dx}{g}-\frac{\left (2 B^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac{(2 A B d n) \int \frac{\log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac{\left (2 B^2 d n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-(2 A B n) \int \frac{\log \left (\frac{b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx-\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\frac{\left (2 B^2 d n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac{\left (2 B^2 d n\right ) \int \frac{\log \left ((c+d x)^{-n}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}+\frac{\left (B^2 n^2\right ) \int \frac{\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac{\left (B^2 d n^2\right ) \int \frac{\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}-\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}-\frac{(2 A B n) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (\left (\frac{-b c+a d}{d}-\frac{b x}{d}\right )^n\right ) \log \left (\frac{-b c g+a d g}{d}-\frac{b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b g}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c g+a d g}{d}+\frac{b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}+\left (2 B^2 n^2\right ) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log \left (\frac{b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{B^2 \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c g+a d g}{d}+\frac{b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (\frac{-b c+a d}{d}-\frac{b x}{d}\right )^n\right )}{\frac{-b c+a d}{d}-\frac{b x}{d}} \, dx,x,-c-d x\right )}{d g}+\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c g+a d g}{d}-\frac{b g x}{d}\right )}{\frac{-b c g+a d g}{d}-\frac{b g x}{d}} \, dx,x,-c-d x\right )}{d}+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b g \left (\frac{b c g-a d g}{b g}+\frac{d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}-\frac{\left (2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (\frac{-b c+a d}{b}-\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}+\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}+\frac{\left (B^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c g+a d g}{b g}-\frac{d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{\left (B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^n\right )}{\frac{-b c+a d}{b}-\frac{d x}{b}} \, dx,x,a+b x\right )}{b^2 g}-\frac{\left (B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c g+a d g}{b g}-\frac{d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac{B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{\left (2 B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (1-\frac{d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1-\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac{B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac{\left (2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac{A B n \log ^2(g (a+b x))}{b g}+\frac{B^2 n^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac{2 B^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b g}-\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b g}+\frac{B^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 A B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac{2 B^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b g}-\frac{B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b g}-\frac{B^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 A B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}\\ \end{align*}

Mathematica [B]  time = 0.415522, size = 537, normalized size = 3.89 \[ \frac{3 B n \left (-2 \left (\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )\right )-2 \log (a+b x) \left (-\log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )\right )+\log ^2\left (\frac{a}{b}+x\right )\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )+B^2 n^2 \left (-6 \text{PolyLog}\left (3,\frac{d (a+b x)}{a d-b c}\right )-6 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )-3 \left (-\log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )\right ) \left (\log ^2\left (\frac{a}{b}+x\right )-2 \left (\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )\right )\right )+6 \log \left (\frac{a}{b}+x\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+6 \log \left (\frac{c}{d}+x\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+3 \log ^2\left (\frac{a}{b}+x\right ) \left (\log \left (\frac{b (c+d x)}{b c-a d}\right )-\log \left (\frac{c}{d}+x\right )\right )+3 \log ^2\left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )+3 \log (a+b x) \left (\log \left (\frac{a+b x}{c+d x}\right )-\log \left (\frac{a}{b}+x\right )+\log \left (\frac{c}{d}+x\right )\right )^2+\log ^3\left (\frac{a}{b}+x\right )\right )+3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )^2}{3 b g} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x),x]

[Out]

(3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 3*B*n*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x] - Log[c
/d + x] - Log[(a + b*x)/(c + d*x)]) - 2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d
*x))/(b*c - a*d)])) + B^2*n^2*(Log[a/b + x]^3 + 3*Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 3*Log[a +
b*x]*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)])^2 + 3*Log[a/b + x]^2*(-Log[c/d + x] + Log[(b*(
c + d*x))/(b*c - a*d)]) + 6*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 6*Log[c/d + x]*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)] - 3*(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)])*(Log[a/b + x]^2 - 2*(L
og[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - 6*PolyLog[3, (d*(a +
b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b*(c + d*x))/(b*c - a*d)]))/(3*b*g)

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Maple [F]  time = 0.437, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bgx+ag} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{B^{2} \log \left (b x + a\right ) \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{b g} + \frac{A^{2} \log \left (b g x + a g\right )}{b g} - \int -\frac{B^{2} b c \log \left (e\right )^{2} + 2 \, A B b c \log \left (e\right ) +{\left (B^{2} b d x + B^{2} b c\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} +{\left (B^{2} b d \log \left (e\right )^{2} + 2 \, A B b d \log \left (e\right )\right )} x + 2 \,{\left (B^{2} b c \log \left (e\right ) + A B b c +{\left (B^{2} b d \log \left (e\right ) + A B b d\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B^{2} b c \log \left (e\right ) + A B b c +{\left (B^{2} b d \log \left (e\right ) + A B b d\right )} x +{\left (B^{2} b d n x + B^{2} a d n\right )} \log \left (b x + a\right ) +{\left (B^{2} b d x + B^{2} b c\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{2} d g x^{2} + a b c g +{\left (b^{2} c g + a b d g\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x, algorithm="maxima")

[Out]

B^2*log(b*x + a)*log((d*x + c)^n)^2/(b*g) + A^2*log(b*g*x + a*g)/(b*g) - integrate(-(B^2*b*c*log(e)^2 + 2*A*B*
b*c*log(e) + (B^2*b*d*x + B^2*b*c)*log((b*x + a)^n)^2 + (B^2*b*d*log(e)^2 + 2*A*B*b*d*log(e))*x + 2*(B^2*b*c*l
og(e) + A*B*b*c + (B^2*b*d*log(e) + A*B*b*d)*x)*log((b*x + a)^n) - 2*(B^2*b*c*log(e) + A*B*b*c + (B^2*b*d*log(
e) + A*B*b*d)*x + (B^2*b*d*n*x + B^2*a*d*n)*log(b*x + a) + (B^2*b*d*x + B^2*b*c)*log((b*x + a)^n))*log((d*x +
c)^n))/(b^2*d*g*x^2 + a*b*c*g + (b^2*c*g + a*b*d*g)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x, algorithm="fricas")

[Out]

integral((B^2*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*A*B*log(e*((b*x + a)/(d*x + c))^n) + A^2)/(b*g*x + a*g), 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(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{b g x + a g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x, algorithm="giac")

[Out]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(b*g*x + a*g), x)