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

Optimal. Leaf size=137 \[ -\frac {2 B n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \]

[Out]

-(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2*ln((-a*d+b*c)/b/(d*x+c))/d/i-2*B*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*polylog(
2,d*(b*x+a)/b/(d*x+c))/d/i+2*B^2*n^2*polylog(3,d*(b*x+a)/b/(d*x+c))/d/i

________________________________________________________________________________________

Rubi [B]  time = 3.17, antiderivative size = 782, normalized size of antiderivative = 5.71, 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, 2499, 2396, 2433, 2374, 6589, 2302, 30, 2500, 2375, 2317, 2440, 2434} \[ -\frac {2 A B n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d i}+\frac {2 B^2 n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{d i}-\frac {2 B^2 n^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{d i}-\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{d i}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d i}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d i}+\frac {\log (c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i}-\frac {2 A B n \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i}+\frac {B^2 n \log ^2(c i+d i x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d i}+\frac {2 B^2 n \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((a+b x)^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{d i}-\frac {B^2 n^2 \log (a+b x) \log ^2(i (c+d x))}{d i}+\frac {B^2 n^2 \log ^2(i (c+d x)) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i}-\frac {B^2 n^2 \log ^2(c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (i (c+d x))}{d i}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d i}+\frac {B^2 \log ^2\left ((c+d x)^{-n}\right ) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d i}-\frac {2 B^2 n \log (a+b x) \log (i (c+d x)) \log \left ((c+d x)^{-n}\right )}{d i}+\frac {A B n \log ^2(i (c+d x))}{d i}+\frac {B^2 n^2 \log ^3(i (c+d x))}{3 d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 30

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

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 6688

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

Rule 6742

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

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}{189 c+189 d x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d 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 (189 c+189 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \frac {\left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (189 c+189 d x)}{c+d x} \, dx-\frac {(2 b B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \left (\frac {A \log (189 c+189 d x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x}\right ) \, dx-\frac {(2 b B n) \int \left (\frac {A \log (189 c+189 d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{a+b x}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 A B n) \int \frac {\log (189 c+189 d x)}{-c-d x} \, dx-\frac {1}{189} \left (2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x} \, dx-\frac {(2 A b B n) \int \frac {\log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+(2 A B n) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx-\frac {(2 A B n) \operatorname {Subst}\left (\int -\frac {189 \log (x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}+\frac {1}{189} \left (B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{c+d x} \, dx-\frac {\left (b B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{a+b x} \, dx}{189 d}-\frac {\left (2 b 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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {(2 A B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\frac {(2 A B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {189 b c-189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right ) \log \left (-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\left (2 B^2 n^2\right ) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right ) \log (189 c+189 d x)}{189 c+189 d x} \, dx+\frac {\left (B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {189 \log ^2(x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}+\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 {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {189 b c-189 a d}{b}+\frac {189 d x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{189 b}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {189 d \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\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 {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\frac {\left (B^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (189 (c+d x))\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{189 d^2}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}} \, dx,x,189 (c+d x)\right )}{35721 d^2}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {\left (2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {2 B^2 n \log \left (-\frac {d (a+b 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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}\\ \end {align*}

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Mathematica [B]  time = 0.28, size = 306, normalized size = 2.23 \[ \frac {-B n \left (-2 \left (\text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \log (c+d x) \left (-\log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )\right )+\log ^2\left (\frac {c}{d}+x\right )\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )+\log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )+A\right )^2-B^2 n^2 \left (-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{d i x + c i}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{d i x +c i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B^{2} \log \left (d x + c\right ) \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d i} + \frac {A^{2} \log \left (d i x + c i\right )}{d i} - \int -\frac {B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \relax (e)^{2} + 2 \, A B \log \relax (e) + 2 \, {\left (B^{2} \log \relax (e) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} n \log \left (d x + c\right ) + B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \relax (e) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d i x + c i}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{c\,i+d\,i\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{c + d x}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx}{i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(d*i*x+c*i),x)

[Out]

(Integral(A**2/(c + d*x), x) + Integral(B**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + Integr
al(2*A*B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x))/i

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