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

Optimal. Leaf size=303 \[ \frac {2 B g n (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}+\frac {2 B g n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}+\frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i}+\frac {g (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i}+\frac {2 B^2 g n^2 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}-\frac {2 B^2 g n^2 (b c-a d) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

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

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Rubi [B]  time = 4.03, antiderivative size = 1156, normalized size of antiderivative = 3.82, number of steps used = 65, number of rules used = 24, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ -\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{3 d^2 i}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{d^2 i}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{d^2 i}+\frac {B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{d^2 i}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{d^2 i}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{d^2 i}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{d^2 i}+\frac {2 b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}+\frac {2 A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}-\frac {2 b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 i}+\frac {2 B^2 (b c-a d) g n \log (a+b x) \log \left ((c+d x)^{-n}\right ) \log (c+d x)}{d^2 i}-\frac {2 B^2 (b c-a d) g 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 (c+d x)}{d^2 i}-\frac {a B^2 g n^2 \log ^2(a+b x)}{d i}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d i}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i}+\frac {2 a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d i}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d i}+\frac {2 a B^2 g n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d i}-\frac {2 B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac {2 b B^2 c g n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 A B (b c-a d) g n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac {2 B^2 (b c-a d) g 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 {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 B^2 (b c-a d) g n^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac {2 B^2 (b c-a d) g n^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*B^2*g*n^2*Log[a + b*x]^2)/(d*i)) + (2*a*B*g*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i)
+ (b*g*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d*i) + (2*A*B*(b*c - a*d)*g*n*Log[-((d*(a + b*x))/(b*c - a
*d))]*Log[c + d*x])/(d^2*i) + (2*b*B^2*c*g*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^2*i) + (B^2*
(b*c - a*d)*g*Log[(a + b*x)^n]^2*Log[c + d*x])/(d^2*i) - (2*b*B*c*g*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*L
og[c + d*x])/(d^2*i) - ((b*c - a*d)*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[c + d*x])/(d^2*i) - (A*B*(b
*c - a*d)*g*n*Log[c + d*x]^2)/(d^2*i) - (b*B^2*c*g*n^2*Log[c + d*x]^2)/(d^2*i) + (B^2*(b*c - a*d)*g*n^2*Log[a
+ b*x]*Log[c + d*x]^2)/(d^2*i) - (B^2*(b*c - a*d)*g*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]^2)/(d^2*i) -
(B^2*(b*c - a*d)*g*n^2*Log[c + d*x]^3)/(3*d^2*i) + (2*a*B^2*g*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]
)/(d*i) - (B^2*(b*c - a*d)*g*Log[(a + b*x)^n]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(d^2*i) + (2*B^2*(b*c - a*d)*g
*n*Log[a + b*x]*Log[c + d*x]*Log[(c + d*x)^(-n)])/(d^2*i) + (B^2*(b*c - a*d)*g*Log[a + b*x]*Log[(c + d*x)^(-n)
]^2)/(d^2*i) - (B^2*(b*c - a*d)*g*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-n)]^2)/(d^2*i) - (2*B^2*(b
*c - a*d)*g*n*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x]*(Log[(a + b*x)^n] - Log[e*((a + b*x)/(c + d*x))^n
] + Log[(c + d*x)^(-n)]))/(d^2*i) + (2*a*B^2*g*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d*i) - (2*B^2*(b
*c - a*d)*g*n*Log[(a + b*x)^n]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^2*i) + (2*A*B*(b*c - a*d)*g*n*Poly
Log[2, (b*(c + d*x))/(b*c - a*d)])/(d^2*i) + (2*b*B^2*c*g*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^2*i) +
(2*B^2*(b*c - a*d)*g*n*Log[(c + d*x)^(-n)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^2*i) - (2*B^2*(b*c - a*d
)*g*n*(Log[(a + b*x)^n] - Log[e*((a + b*x)/(c + d*x))^n] + Log[(c + d*x)^(-n)])*PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])/(d^2*i) + (2*B^2*(b*c - a*d)*g*n^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/(d^2*i) + (2*B^2*(b*c -
a*d)*g*n^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(d^2*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 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 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 {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 c+188 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{188 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{188 d}\\ &=\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {(b B g n) \int \frac {(b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{94 d}+\frac {(B (b c-a d) g 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 (c+d x)}{a+b x} \, dx}{94 d^2}\\ &=\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac {(B (b c-a d) g 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 (c+d x)}{(a+b x) (c+d x)} \, dx}{94 d^2}-\frac {(b B (b c-a d) g n) \int \frac {x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{94 d}\\ &=\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {(b B (b c-a d) g n) \int \left (-\frac {a \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{94 d}+\frac {\left (B (b c-a d)^2 g n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{94 d^2}\\ &=\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac {(a b B g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{94 d}-\frac {(b B c g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{94 d}+\frac {\left (B (b c-a d)^2 g n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{94 d^2}\\ &=\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac {(b B (b c-a d) g n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac {(B (b c-a d) g n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{94 d}+\frac {\left (b B^2 c g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac {\left (a B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{94 d}\\ &=\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac {(b B (b c-a d) g n) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{94 d^2}-\frac {(B (b c-a d) g n) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{94 d}+\frac {\left (b B^2 c g n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{94 d^2}-\frac {\left (a B^2 g n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{94 d}\\ &=\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac {(A b B (b c-a d) g n) \int \frac {\log (c+d x)}{a+b x} \, dx}{94 d^2}+\frac {\left (b B^2 (b c-a d) g n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac {(A B (b c-a d) g n) \int \frac {\log (c+d x)}{c+d x} \, dx}{94 d}-\frac {\left (B^2 (b c-a d) g n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{94 d}+\frac {1}{94} \left (a B^2 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx+\frac {\left (b^2 B^2 c g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac {\left (a b B^2 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{94 d}-\frac {\left (b B^2 c g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{94 d}\\ &=\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {(A B (b c-a d) g n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac {\left (b B^2 (b c-a d) g n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}+\frac {\left (b B^2 (b c-a d) g n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{94 d^2}-\frac {(A B (b c-a d) g n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}-\frac {\left (b B^2 c g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{94 d^2}-\frac {\left (a B^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{94 d}-\frac {\left (a b B^2 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{94 d}-\frac {\left (b B^2 c g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}+\frac {\left (b B^2 (b c-a d) g n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{188 d^2}-\frac {\left (B^2 (b c-a d) g n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{188 d}+\frac {\left (b B^2 (b c-a d) g 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 (c+d x)}{a+b x} \, dx}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}-\frac {(A B (b c-a d) g n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}+\frac {\left (B^2 (b c-a d) g n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{94 d^2}-\frac {\left (b B^2 c g n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}-\frac {\left (a B^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{94 d}-\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{188 d^2}-\frac {\left (B^2 (b c-a d) g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{94 d}-\frac {\left (B^2 (b c-a d) g 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 {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{188 b d}-\frac {\left (B^2 (b c-a d) g 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 )}{94 b d}-\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{188 d^2}-\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{94 b d}-\frac {\left (B^2 (b c-a d) g 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}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g 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 )}{94 d^2}-\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}+\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}+\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}-\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g 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 )}{94 d^2}-\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {\left (b B^2 (b c-a d) g\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 )}{188 d^3}-\frac {\left (b B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{188 d^3}+\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g 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 )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}+\frac {\left (B^2 (b c-a d) g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g 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 )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+2 \frac {\left (B^2 (b c-a d) g 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 )}{94 d^2}\\ &=-\frac {a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac {a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac {b g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac {A B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {b B^2 c g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac {b B c g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac {A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac {b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac {B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac {B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac {a B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac {B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac {B^2 (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{94 d^2}+\frac {a B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d}-\frac {B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac {A B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {b B^2 c g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac {B^2 (b c-a d) g 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 )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac {B^2 (b c-a d) g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{94 d^2}\\ \end {align*}

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Mathematica [B]  time = 0.70, size = 802, normalized size = 2.65 \[ -\frac {g \left (-B^2 \left (d (a+b x) \log ^2\left (\frac {a+b x}{c+d x}\right )+b c \log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )-(b c-a d) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-2 \log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b c \left (\log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right ) n^2+a B^2 d \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {a+b x}{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 )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right ) n^2+a B d \left (A+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 )\right ) \left (\log ^2\left (\frac {c}{d}+x\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )\right ) n+B \left (A+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 )\right ) \left (-b c \log ^2\left (\frac {c}{d}+x\right )-2 d (a+b x) \left (\log \left (\frac {a}{b}+x\right )-1\right )+2 b (c+d x) \left (\log \left (\frac {c}{d}+x\right )-1\right )+2 b \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) (d x-c \log (c+d x))+2 b c \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )\right ) n-b d x \left (A+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 )\right )^2+(b c-a d) \left (A+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 )\right )^2 \log (c+d x)\right )}{d^2 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((A^2*b*g*x + A^2*a*g + (B^2*b*g*x + B^2*a*g)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b*g*x + A*B*a*
g)*log(e*((b*x + a)/(d*x + c))^n))/(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((b*g*x+a*g)*(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 = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{d i x +c i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*g*x+a*g)*(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 \[ A^{2} b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A^{2} a g \log \left (d i x + c i\right )}{d i} + \frac {{\left (B^{2} b d g x - {\left (b c g - a d g\right )} B^{2} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d^{2} i} - \int -\frac {B^{2} a d g \log \relax (e)^{2} + 2 \, A B a d g \log \relax (e) + {\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b d g \log \relax (e)^{2} + 2 \, A B b d g \log \relax (e)\right )} x + 2 \, {\left (B^{2} a d g \log \relax (e) + A B a d g + {\left (B^{2} b d g \log \relax (e) + A B b d g\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} a d g \log \relax (e) + A B a d g - {\left (b c g n - a d g n\right )} B^{2} \log \left (d x + c\right ) + {\left ({\left (g n + g \log \relax (e)\right )} B^{2} b d + A B b d g\right )} x + {\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d^{2} i x + c d i}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((a*g + b*g*x)*(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 \[ \frac {g \left (\int \frac {A^{2} a}{c + d x}\, dx + \int \frac {A^{2} b x}{c + d x}\, dx + \int \frac {B^{2} a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {B^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx\right )}{i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*(Integral(A**2*a/(c + d*x), x) + Integral(A**2*b*x/(c + d*x), x) + Integral(B**2*a*log(e*(a/(c + d*x) + b*x/
(c + d*x))**n)**2/(c + d*x), x) + Integral(2*A*B*a*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) + Int
egral(B**2*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + Integral(2*A*B*b*x*log(e*(a/(c + d*x
) + b*x/(c + d*x))**n)/(c + d*x), x))/i

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