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

Optimal. Leaf size=282 \[ -\frac{2 b B g n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^2}+\frac{2 b B^2 g n^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}-\frac{b g \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i^2}-\frac{g (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)}+\frac{2 A B g n (a+b x)}{d i^2 (c+d x)}+\frac{2 B^2 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d i^2 (c+d x)}-\frac{2 B^2 g n^2 (a+b x)}{d i^2 (c+d x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 4.17219, antiderivative size = 1157, normalized size of antiderivative = 4.1, number of steps used = 69, number of rules used = 25, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.581, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \frac{b B^2 g n^2 \log ^3(c+d x)}{3 d^2 i^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{d^2 i^2}+\frac{A b B g n \log ^2(c+d x)}{d^2 i^2}-\frac{b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{d^2 i^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{d^2 i^2}-\frac{2 b B^2 g n^2 \log (c+d x)}{d^2 i^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{d^2 i^2}+\frac{b 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^2}-\frac{2 b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i^2}-\frac{2 A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i^2}+\frac{2 b B 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^2}-\frac{2 b B^2 g n \log (a+b x) \log \left ((c+d x)^{-n}\right ) \log (c+d x)}{d^2 i^2}+\frac{2 b B^2 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^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{d^2 i^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}{d^2 i^2 (c+d x)}-\frac{b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i^2}+\frac{b B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i^2}+\frac{2 b B^2 g n^2 \log (a+b x)}{d^2 i^2}-\frac{2 b 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^2 i^2}-\frac{2 B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2 i^2 (c+d x)}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac{2 b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac{2 b B^2 g n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^2}+\frac{2 b B^2 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^2}-\frac{2 b B^2 g n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac{2 A b B g n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac{2 b B^2 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^2}+\frac{2 b B^2 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^2}-\frac{2 b B^2 g n^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^2}-\frac{2 b B^2 g n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^2}+\frac{2 B^2 (b c-a d) g n^2}{d^2 i^2 (c+d x)} \]

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)^2,x]

[Out]

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

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 2525

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

Rule 12

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

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 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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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]]

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 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 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 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 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 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]

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}{(196 c+196 d x)^2} \, dx &=\int \left (\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}{38416 d (c+d x)^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d (c+d x)}\right ) \, dx\\ &=\frac{(b 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}{38416 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)^2} \, dx}{38416 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}{38416 d^2 (c+d x)}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{(b B 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}{19208 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 )}{(a+b x) (c+d x)^2} \, dx}{19208 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}{38416 d^2 (c+d x)}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{(b B 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}{19208 d^2}-\frac{\left (B (b c-a d)^2 g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{19208 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}{38416 d^2 (c+d x)}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 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) (c+d x)} \, dx}{19208 d^2}-\frac{\left (B (b c-a d)^2 g n\right ) \int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \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)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19208 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}{38416 d^2 (c+d x)}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{\left (b^2 B g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{19208 d^2}+\frac{(b B g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{19208 d}-\frac{(b B (b c-a d) g n) \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}{19208 d^2}+\frac{(B (b c-a d) g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{19208 d}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{\left (b^2 B 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} \, dx}{19208 d^2}+\frac{(b B 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}{19208 d}+\frac{\left (b 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}{19208 d^2}-\frac{\left (b 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 (c+d x)}{a+b x} \, dx}{19208 d^2}+\frac{\left (B^2 (b c-a d) g n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{\left (b^2 B g n\right ) \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}{19208 d^2}+\frac{(b B 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}{19208 d}+\frac{\left (b 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}{19208 d^2}-\frac{\left (b B^2 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}{19208 d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac{\left (A b^2 B g n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac{\left (b^2 B^2 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}{19208 d^2}+\frac{(A b B g n) \int \frac{\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac{\left (b B^2 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}{19208 d}+\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{19208 d^2}-\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac{\left (b B^2 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{19208 d}+\frac{\left (b B^2 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{(A b B g n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac{\left (b^2 B^2 g n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac{\left (b^2 B^2 g n\right ) \int \frac{\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{19208 d^2}+\frac{(A b B g n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}+\frac{\left (b B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{19208 d^2}+\frac{\left (b B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{38416 d^2}+\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{19208 d^2}+\frac{\left (b B^2 g n^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{38416 d}+\frac{\left (b B^2 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}-\frac{\left (b^2 B^2 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}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}+\frac{(A b B 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 )}{19208 d^2}-\frac{\left (b B^2 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 )}{19208 d^2}-\frac{\left (b B^2 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 )}{19208 d^2}+\frac{\left (b B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{38416 d^2}+\frac{\left (b 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 )}{19208 d^2}+\frac{\left (b B^2 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 )}{19208 d^2}+\frac{\left (b B^2 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}{19208 d}+\frac{\left (b B^2 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}{19208 d}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{\left (B^2 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 )}{38416 d}+\frac{\left (B^2 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 )}{19208 d}+\frac{\left (b B^2 g n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{38416 d^2}+\frac{\left (b B^2 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 )}{19208 d^2}-\frac{\left (B^2 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 )}{19208 d}+\frac{\left (b B^2 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 )}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}+\frac{\left (b B^2 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 )}{19208 d^2}-\frac{\left (b B^2 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 )}{19208 d^2}-\frac{\left (b B^2 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 )}{19208 d^2}+\frac{\left (b B^2 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 )}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac{b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}+\frac{b B^2 g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{\left (b^2 B^2 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 )}{38416 d^3}+\frac{\left (b^2 B^2 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 )}{38416 d^3}-\frac{\left (b B^2 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 )}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac{b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{\left (b B^2 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 )}{19208 d^2}-\frac{\left (b B^2 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 )}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac{b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-2 \frac{\left (b B^2 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 )}{19208 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac{b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac{b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac{b 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 )}{19208 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}{38416 d^2 (c+d x)}-\frac{b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac{A b B g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac{A b B g n \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac{b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac{b B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac{b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac{b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{A b B g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac{b B^2 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 )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac{b B^2 g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19208 d^2}\\ \end{align*}

Mathematica [B]  time = 2.24838, size = 1261, normalized size = 4.47 \[ \text{result too large to display} \]

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)^2,x]

[Out]

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

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Maple [F]  time = 0.551, size = 0, normalized size = 0. \begin{align*} \int{\frac{bgx+ag}{ \left ( dix+ci \right ) ^{2}} \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((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, A B a g n{\left (\frac{1}{d^{2} i^{2} x + c d i^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} + A^{2} b g{\left (\frac{c}{d^{3} i^{2} x + c d^{2} i^{2}} + \frac{\log \left (d x + c\right )}{d^{2} i^{2}}\right )} - \frac{2 \, A B a g \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac{A^{2} a g}{d^{2} i^{2} x + c d i^{2}} + \frac{{\left ({\left (b c g - a d g\right )} B^{2} +{\left (B^{2} b d g x + B^{2} b c g\right )} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d^{3} i^{2} x + c d^{2} i^{2}} - \int -\frac{B^{2} a d g \log \left (e\right )^{2} +{\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 \left (e\right )^{2} + 2 \, A B b d g \log \left (e\right )\right )} x + 2 \,{\left (B^{2} a d g \log \left (e\right ) +{\left (B^{2} b d g \log \left (e\right ) + A B b d g\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left ({\left (b c g n -{\left (g n - g \log \left (e\right )\right )} a d\right )} B^{2} +{\left (B^{2} b d g \log \left (e\right ) + A B b d g\right )} x +{\left (B^{2} b d g n x + B^{2} b c g n\right )} \log \left (d x + c\right ) +{\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^{3} i^{2} x^{2} + 2 \, c d^{2} i^{2} x + c^{2} d i^{2}}\,{d x} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

2*A*B*a*g*n*(1/(d^2*i^2*x + c*d*i^2) + b*log(b*x + a)/((b*c*d - a*d^2)*i^2) - b*log(d*x + c)/((b*c*d - a*d^2)*
i^2)) + A^2*b*g*(c/(d^3*i^2*x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - 2*A*B*a*g*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n)/(d^2*i^2*x + c*d*i^2) - A^2*a*g/(d^2*i^2*x + c*d*i^2) + ((b*c*g - a*d*g)*B^2 + (B^2*b*d*g*x + B^2*b*
c*g)*log(d*x + c))*log((d*x + c)^n)^2/(d^3*i^2*x + c*d^2*i^2) - integrate(-(B^2*a*d*g*log(e)^2 + (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) + (B^2*b*d
*g*log(e) + A*B*b*d*g)*x)*log((b*x + a)^n) - 2*((b*c*g*n - (g*n - g*log(e))*a*d)*B^2 + (B^2*b*d*g*log(e) + A*B
*b*d*g)*x + (B^2*b*d*g*n*x + B^2*b*c*g*n)*log(d*x + c) + (B^2*b*d*g*x + B^2*a*d*g)*log((b*x + a)^n))*log((d*x
+ c)^n))/(d^3*i^2*x^2 + 2*c*d^2*i^2*x + c^2*d*i^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, x\right ) \end{align*}

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

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

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