3.183 \(\int \frac{(c i+d i x)^3 (A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{(a g+b g x)^2} \, dx\)

Optimal. Leaf size=739 \[ \frac{6 B d i^3 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}+\frac{4 B^2 d i^3 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^4 g^2}-\frac{B^2 d i^3 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac{6 B^2 d i^3 n^2 (b c-a d)^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac{2 d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^4 g^2}-\frac{B d^2 i^3 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}+\frac{d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g^2}-\frac{i^3 (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^2 (a+b x)}-\frac{2 B i^3 n (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2 (a+b x)}+\frac{4 B d i^3 n (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac{3 d i^3 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^4 g^2}+\frac{B d i^3 n (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac{2 B^2 i^3 n^2 (c+d x) (b c-a d)^2}{b^3 g^2 (a+b x)}+\frac{B^2 d i^3 n^2 (b c-a d)^2 \log (c+d x)}{b^4 g^2} \]

[Out]

(-2*B^2*(b*c - a*d)^2*i^3*n^2*(c + d*x))/(b^3*g^2*(a + b*x)) - (B*d^2*(b*c - a*d)*i^3*n*(a + b*x)*(A + B*Log[e
*((a + b*x)/(c + d*x))^n]))/(b^4*g^2) - (2*B*(b*c - a*d)^2*i^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/(b^3*g^2*(a + b*x)) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g^2
) - ((b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d
*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g^2) + (4*B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*
x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*g^2) + (B^2*d*(b*c - a*d)^2*i^3*n^2*Log[c + d*x])/(b^4*
g^2) + (B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(
b^4*g^2) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))
])/(b^4*g^2) + (4*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^2) - (B^2*d*(b*c
- a*d)^2*i^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e
*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*n^2
*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2)

________________________________________________________________________________________

Rubi [B]  time = 4.73816, antiderivative size = 1875, normalized size of antiderivative = 2.54, number of steps used = 83, number of rules used = 23, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.511, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 2486, 31, 44, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*B*d^2*(b*c - a*d)*i^3*n*x)/(b^3*g^2)) - (2*B^2*(b*c - a*d)^3*i^3*n^2)/(b^4*g^2*(a + b*x)) - (2*B^2*d*(b*c
- a*d)^2*i^3*n^2*Log[a + b*x])/(b^4*g^2) - (3*A*B*d*(b*c - a*d)^2*i^3*n*Log[a + b*x]^2)/(b^4*g^2) + (a^2*B^2*
d^3*i^3*n^2*Log[a + b*x]^2)/(2*b^4*g^2) - (a*B^2*d^2*(3*b*c - 2*a*d)*i^3*n^2*Log[a + b*x]^2)/(b^4*g^2) + (B^2*
d*(b*c - a*d)^2*i^3*n^2*Log[a + b*x]^2)/(b^4*g^2) - (B^2*d^2*(b*c - a*d)*i^3*n*(a + b*x)*Log[e*((a + b*x)/(c +
d*x))^n])/(b^4*g^2) - (3*B^2*d*(b*c - a*d)^2*i^3*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*((a + b*x)/(c + d*x)
)^n]^2)/(b^4*g^2) - (3*B^2*d*(b*c - a*d)^2*i^3*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n]^2)/(b^4*g^2) - (2*B
*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^2*(a + b*x)) - (a^2*B*d^3*i^3*n*Log[a + b*
x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^2) + (2*a*B*d^2*(3*b*c - 2*a*d)*i^3*n*Log[a + b*x]*(A + B*Lo
g[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^2) - (2*B*d*(b*c - a*d)^2*i^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]))/(b^4*g^2) + (d^2*(3*b*c - 2*a*d)*i^3*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^2) + (d^3
*i^3*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g^2) - ((b*c - a*d)^3*i^3*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n])^2)/(b^4*g^2*(a + b*x)) + (3*d*(b*c - a*d)^2*i^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2)/(b^4*g^2) + (3*B^2*d*(b*c - a*d)^2*i^3*n^2*Log[c + d*x])/(b^4*g^2) - (B^2*c^2*d*i^3*n^2*Log[-((d*(a + b*
x))/(b*c - a*d))]*Log[c + d*x])/(b^2*g^2) + (2*B^2*c*d*(3*b*c - 2*a*d)*i^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d)
)]*Log[c + d*x])/(b^3*g^2) - (2*B^2*d*(b*c - a*d)^2*i^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b
^4*g^2) + (B*c^2*d*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(b^2*g^2) - (2*B*c*d*(3*b*c - 2*
a*d)*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(b^3*g^2) + (2*B*d*(b*c - a*d)^2*i^3*n*(A + B*
Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(b^4*g^2) + (B^2*c^2*d*i^3*n^2*Log[c + d*x]^2)/(2*b^2*g^2) - (B^
2*c*d*(3*b*c - 2*a*d)*i^3*n^2*Log[c + d*x]^2)/(b^3*g^2) + (B^2*d*(b*c - a*d)^2*i^3*n^2*Log[c + d*x]^2)/(b^4*g^
2) + (6*A*B*d*(b*c - a*d)^2*i^3*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^4*g^2) - (a^2*B^2*d^3*i^3*n^
2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^4*g^2) + (2*a*B^2*d^2*(3*b*c - 2*a*d)*i^3*n^2*Log[a + b*x]*L
og[(b*(c + d*x))/(b*c - a*d)])/(b^4*g^2) - (2*B^2*d*(b*c - a*d)^2*i^3*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c
- a*d)])/(b^4*g^2) + (6*A*B*d*(b*c - a*d)^2*i^3*n*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^4*g^2) - (a^2*B
^2*d^3*i^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^4*g^2) + (2*a*B^2*d^2*(3*b*c - 2*a*d)*i^3*n^2*Poly
Log[2, -((d*(a + b*x))/(b*c - a*d))])/(b^4*g^2) - (2*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[2, -((d*(a + b*x))/(b
*c - a*d))])/(b^4*g^2) - (B^2*c^2*d*i^3*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b^2*g^2) + (2*B^2*c*d*(3*b
*c - 2*a*d)*i^3*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b^3*g^2) - (2*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[
2, (b*(c + d*x))/(b*c - a*d)])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*n*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog
[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[3, 1 + (b*c - a*d)/(d*(
a + b*x))])/(b^4*g^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 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 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 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 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2344

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

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 2507

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

Rule 2488

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

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{(183 c+183 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac{6128487 d^2 (3 b c-2 a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)^2}+\frac{18385461 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}\right ) \, dx\\ &=\frac{\left (6128487 d^3\right ) \int x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g^2}+\frac{\left (6128487 d^2 (3 b c-2 a d)\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^3 g^2}+\frac{\left (18385461 d (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{b^3 g^2}+\frac{\left (6128487 (b c-a d)^3\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{b^3 g^2}\\ &=\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}-\frac{\left (6128487 B d^3 n\right ) \int \frac{(b c-a d) x^2 \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}{b^2 g^2}-\frac{\left (12256974 B d^2 (3 b c-2 a d) n\right ) \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}{b^3 g^2}-\frac{\left (36770922 B d (b c-a d)^2 n\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) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b^4 g^2}+\frac{\left (12256974 B (b c-a d)^3 n\right ) \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)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}-\frac{\left (6128487 B d^3 (b c-a d) n\right ) \int \frac{x^2 \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}{b^2 g^2}-\frac{\left (12256974 B d^2 (3 b c-2 a d) (b c-a d) n\right ) \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}{b^3 g^2}-\frac{\left (36770922 B d (b c-a d)^2 n\right ) \int \frac{(b c-a d) \log (a+b 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}{b^4 g^2}+\frac{\left (12256974 B (b c-a d)^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}-\frac{\left (6128487 B d^3 (b c-a d) n\right ) \int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{a^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) (a+b x)}+\frac{c^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 g^2}-\frac{\left (12256974 B d^2 (3 b c-2 a d) (b c-a d) n\right ) \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}{b^3 g^2}-\frac{\left (36770922 B d (b c-a d)^3 n\right ) \int \frac{\log (a+b 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}{b^4 g^2}+\frac{\left (12256974 B (b c-a d)^4 n\right ) \int \left (\frac{b \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)^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 (a+b x)}+\frac{d^2 \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}{b^4 g^2}\\ &=\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{\left (6128487 B c^2 d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 g^2}-\frac{\left (6128487 a^2 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}+\frac{\left (12256974 a B d^2 (3 b c-2 a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B c d^2 (3 b c-2 a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 g^2}-\frac{\left (6128487 B d^2 (b c-a d) n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g^2}-\frac{\left (12256974 B d (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}+\frac{\left (12256974 B d^2 (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B (b c-a d)^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^2}-\frac{\left (36770922 B d (b c-a d)^3 n\right ) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}-\frac{\left (6128487 B^2 d^2 (b c-a d) n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g^2}-\frac{\left (36770922 A B d (b c-a d)^3 n\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^4 g^2}-\frac{\left (36770922 B^2 d (b c-a d)^3 n\right ) \int \frac{\log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}-\frac{\left (6128487 B^2 c^2 d 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}{b^2 g^2}+\frac{\left (6128487 a^2 B^2 d^3 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}{b^4 g^2}+\frac{\left (12256974 B^2 c d (3 b c-2 a d) 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}{b^3 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) 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}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 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}{b^4 g^2}-\frac{\left (12256974 B^2 d (b c-a d)^2 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}{b^4 g^2}+\frac{\left (12256974 B^2 (b c-a d)^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{6128487 B^2 d^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}+\frac{\left (18385461 B^2 d (b c-a d)^2\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}-\frac{\left (36770922 A B d (b c-a d)^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{b^5 g^2}-\frac{\left (6128487 B^2 c^2 d 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}{b^2 g^2}+\frac{\left (6128487 a^2 B^2 d^3 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}{b^4 g^2}+\frac{\left (12256974 B^2 c d (3 b c-2 a d) 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}{b^3 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) 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}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 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}{b^4 g^2}-\frac{\left (12256974 B^2 d (b c-a d)^2 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}{b^4 g^2}+\frac{\left (6128487 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B^2 (b c-a d)^4 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{6128487 B^2 d^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{6128487 B^2 d (b c-a d)^2 n^2 \log (c+d x)}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}-\frac{\left (36770922 A B d (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac{\left (36770922 A B d^2 (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^5 g^2}+\frac{\left (36770922 B^2 d (b c-a d)^3 n\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}-\frac{\left (6128487 B^2 c^2 d n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b g^2}+\frac{\left (6128487 B^2 c^2 d^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (6128487 a^2 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac{\left (6128487 a^2 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B^2 c d (3 b c-2 a d) n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B^2 c d^2 (3 b c-2 a d) n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (12256974 a B^2 d^3 (3 b c-2 a d) n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B^2 d (b c-a d)^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^4 g^2}+\frac{\left (12256974 B^2 (b c-a d)^4 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{12256974 B^2 (b c-a d)^3 n^2}{b^4 g^2 (a+b x)}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x)}{b^4 g^2}-\frac{18385461 A B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}-\frac{6128487 B^2 d^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{18385461 B^2 d (b c-a d)^2 n^2 \log (c+d x)}{b^4 g^2}-\frac{6128487 B^2 c^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{12256974 B^2 c d (3 b c-2 a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}+\frac{36770922 A B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{6128487 a^2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{12256974 a B^2 d^2 (3 b c-2 a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{36770922 B^2 d (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^2}-\frac{\left (36770922 A B d (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac{\left (6128487 B^2 c^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}+\frac{\left (6128487 B^2 c^2 d^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}+\frac{\left (6128487 a^2 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac{\left (6128487 a^2 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B^2 c d (3 b c-2 a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}-\frac{\left (12256974 B^2 c d^2 (3 b c-2 a d) n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}+\frac{\left (12256974 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 g^2}-\frac{\left (36770922 B^2 d (b c-a d)^3 n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{12256974 B^2 (b c-a d)^3 n^2}{b^4 g^2 (a+b x)}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x)}{b^4 g^2}-\frac{18385461 A B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac{6128487 a^2 B^2 d^3 n^2 \log ^2(a+b x)}{2 b^4 g^2}-\frac{6128487 a B^2 d^2 (3 b c-2 a d) n^2 \log ^2(a+b x)}{b^4 g^2}+\frac{6128487 B^2 d (b c-a d)^2 n^2 \log ^2(a+b x)}{b^4 g^2}-\frac{6128487 B^2 d^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{18385461 B^2 d (b c-a d)^2 n^2 \log (c+d x)}{b^4 g^2}-\frac{6128487 B^2 c^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{12256974 B^2 c d (3 b c-2 a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}+\frac{6128487 B^2 c^2 d n^2 \log ^2(c+d x)}{2 b^2 g^2}-\frac{6128487 B^2 c d (3 b c-2 a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac{6128487 B^2 d (b c-a d)^2 n^2 \log ^2(c+d x)}{b^4 g^2}+\frac{36770922 A B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{6128487 a^2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{12256974 a B^2 d^2 (3 b c-2 a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{36770922 A B d (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}+\frac{36770922 B^2 d (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac{36770922 B^2 d (b c-a d)^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac{\left (6128487 B^2 c^2 d 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 )}{b^2 g^2}+\frac{\left (6128487 a^2 B^2 d^3 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 )}{b^4 g^2}-\frac{\left (12256974 B^2 c d (3 b c-2 a d) 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 )}{b^3 g^2}-\frac{\left (12256974 a B^2 d^2 (3 b c-2 a d) 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 )}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 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 )}{b^4 g^2}+\frac{\left (12256974 B^2 d (b c-a d)^2 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 )}{b^4 g^2}\\ &=-\frac{6128487 A B d^2 (b c-a d) n x}{b^3 g^2}-\frac{12256974 B^2 (b c-a d)^3 n^2}{b^4 g^2 (a+b x)}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x)}{b^4 g^2}-\frac{18385461 A B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac{6128487 a^2 B^2 d^3 n^2 \log ^2(a+b x)}{2 b^4 g^2}-\frac{6128487 a B^2 d^2 (3 b c-2 a d) n^2 \log ^2(a+b x)}{b^4 g^2}+\frac{6128487 B^2 d (b c-a d)^2 n^2 \log ^2(a+b x)}{b^4 g^2}-\frac{6128487 B^2 d^2 (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{18385461 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{12256974 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}-\frac{6128487 a^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{12256974 a B d^2 (3 b c-2 a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{12256974 B d (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{6128487 d^2 (3 b c-2 a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac{6128487 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^2}-\frac{6128487 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2 (a+b x)}+\frac{18385461 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^2}+\frac{18385461 B^2 d (b c-a d)^2 n^2 \log (c+d x)}{b^4 g^2}-\frac{6128487 B^2 c^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{12256974 B^2 c d (3 b c-2 a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac{6128487 B c^2 d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{12256974 B c d (3 b c-2 a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{12256974 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^4 g^2}+\frac{6128487 B^2 c^2 d n^2 \log ^2(c+d x)}{2 b^2 g^2}-\frac{6128487 B^2 c d (3 b c-2 a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac{6128487 B^2 d (b c-a d)^2 n^2 \log ^2(c+d x)}{b^4 g^2}+\frac{36770922 A B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{6128487 a^2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{12256974 a B^2 d^2 (3 b c-2 a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{36770922 A B d (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac{6128487 a^2 B^2 d^3 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}+\frac{12256974 a B^2 d^2 (3 b c-2 a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac{6128487 B^2 c^2 d n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{12256974 B^2 c d (3 b c-2 a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{12256974 B^2 d (b c-a d)^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{36770922 B^2 d (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac{36770922 B^2 d (b c-a d)^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^2}\\ \end{align*}

Mathematica [B]  time = 14.4706, size = 4942, normalized size = 6.69 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(3*b*c - 2*a*d)*i^3*x*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))^2)/(b^3*g^2)
+ (d^3*i^3*x^2*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))^2)/(2*b^2*g^2) + (3*d*(b*
c - a*d)^2*i^3*Log[a + b*x]*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))^2)/(b^4*g^2)
+ (-(A^2*b^3*c^3*i^3) + 3*a*A^2*b^2*c^2*d*i^3 - 3*a^2*A^2*b*c*d^2*i^3 + a^3*A^2*d^3*i^3 - 2*A*b^3*B*c^3*i^3*(
Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + 6*a*A*b^2*B*c^2*d*i^3*(Log[e*((a + b*x)/(c + d*
x))^n] - n*Log[(a + b*x)/(c + d*x)]) - 6*a^2*A*b*B*c*d^2*i^3*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)
/(c + d*x)]) + 2*a^3*A*B*d^3*i^3*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) - b^3*B^2*c^3*i
^3*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2 + 3*a*b^2*B^2*c^2*d*i^3*(Log[e*((a + b*x)/(
c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2 - 3*a^2*b*B^2*c*d^2*i^3*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(
a + b*x)/(c + d*x)])^2 + a^3*B^2*d^3*i^3*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2)/(b^4
*g^2*(a + b*x)) + (2*B*c^3*i^3*n*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*(-(((a/
b + x)*(Log[a/b + x] + Log[a/b + x]^2))/((a + b*x)^2*Log[a/b + x])) - ((b*(c/d + x)*Log[c/d + x])/((-a + (b*c)
/d)^2*(1 - (b*(c/d + x))/(-a + (b*c)/d))) + Log[1 - (b*(c/d + x))/(-a + (b*c)/d)]/(-a + (b*c)/d))/b - (-Log[a/
b + x] + Log[c/d + x] + Log[a/(c + d*x) + (b*x)/(c + d*x)])/(b*(a + b*x))))/g^2 + (2*B*d^3*i^3*n*(A + B*(Log[e
*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*((-2*a*(a/b + x)*(-1 + Log[a/b + x]))/b^3 + (3*a^2*Lo
g[a/b + x]^2)/(2*b^4) + (a^3*(1 + Log[a/b + x]))/(b^4*(a + b*x)) + (2*a*(c/d + x)*(-1 + Log[c/d + x]))/b^3 + (
(a*x)/(2*b) - x^2/4 + (x^2*Log[a/b + x])/2 - (a^2*Log[a + b*x])/(2*b^2))/b^2 - ((c*x)/(2*d) - x^2/4 + (x^2*Log
[c/d + x])/2 - (c^2*Log[c + d*x])/(2*d^2))/b^2 + (a^3*(-(Log[c/d + x]/(b*(a + b*x))) + (d*(Log[a + b*x]/(b*c -
a*d) - Log[c + d*x]/(b*c - a*d)))/b))/b^3 + ((-4*a*b*x + b^2*x^2 + (2*a^3)/(a + b*x) + 6*a^2*Log[a + b*x])*(-
Log[a/b + x] + Log[c/d + x] + Log[a/(c + d*x) + (b*x)/(c + d*x)]))/(2*b^4) - (3*a^2*((Log[c/d + x]*Log[(a + b*
x)/(a - (b*c)/d)])/b + PolyLog[2, (b*d*(c/d + x))/(b*c - a*d)]/b))/b^3))/g^2 + (6*B*c*d^2*i^3*n*(A + B*(Log[e*
((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*(((a/b + x)*(-1 + Log[a/b + x]))/b^2 - (a*Log[a/b + x]
^2)/b^3 - (a^2*(1 + Log[a/b + x]))/(b^3*(a + b*x)) - ((c/d + x)*(-1 + Log[c/d + x]))/b^2 - (a^2*(-(Log[c/d + x
]/(b*(a + b*x))) + (d*(Log[a + b*x]/(b*c - a*d) - Log[c + d*x]/(b*c - a*d)))/b))/b^2 + ((b*x - a^2/(a + b*x) -
2*a*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[a/(c + d*x) + (b*x)/(c + d*x)]))/b^3 + (2*a*((Log[c/d +
x]*Log[(a + b*x)/(a - (b*c)/d)])/b + PolyLog[2, (b*d*(c/d + x))/(b*c - a*d)]/b))/b^2))/g^2 + (6*B*c^2*d*i^3*n
*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*(Log[a/b + x]^2/(2*b^2) + (a*(1 + Log[a
/b + x]))/(b^2*(a + b*x)) + (a*(-(Log[c/d + x]/(b*(a + b*x))) + (d*(Log[a + b*x]/(b*c - a*d) - Log[c + d*x]/(b
*c - a*d)))/b))/b + ((a/(a + b*x) + Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[a/(c + d*x) + (b*x)/(c +
d*x)]))/b^2 - ((Log[c/d + x]*Log[(a + b*x)/(a - (b*c)/d)])/b + PolyLog[2, (b*d*(c/d + x))/(b*c - a*d)]/b)/b))
/g^2 + (B^2*c^3*i^3*n^2*(-2*b*c + 2*a*d - 2*d*(a + b*x)*Log[a + b*x] - 2*(b*c - a*d)*Log[(a + b*x)/(c + d*x)]
- 2*d*(a + b*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*d*(a + b*x)
*Log[c + d*x] - 2*d*(a + b*x)*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + d*(a + b*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)]) + d*(a + b
*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*(b*c - a*d)*g^2*(a + b*x)) + (B^2*d^3*i^3*n^2*((-2*b^2*c^2*(b*c -
a*d)*Log[(b*c - a*d)/(c + d*x)])/d^2 + 2*a^2*(-(b*c) + a*d)*Log[(b*c - a*d)/(c + d*x)] - (4*a*b*c*(-(b*c) + a
*d)*Log[(b*c - a*d)/(c + d*x)])/d + (b^3*c^3*Log[(a + b*x)/(c + d*x)]^2)/d^2 - (a*b^2*c^2*Log[(a + b*x)/(c + d
*x)]^2)/d - ((b*c - a*d)^2*(b*c + 5*a*d)*Log[(a + b*x)/(c + d*x)]^2)/d^2 + (b*c - a*d)*(a + b*x)^2*Log[(a + b*
x)/(c + d*x)]^2 + 2*a^2*b*c*Log[(a + b*x)/(c + d*x)]^3 - 2*a^3*d*Log[(a + b*x)/(c + d*x)]^3 + (2*(-(b*c) + a*d
)*(a + b*x)*Log[(a + b*x)/(c + d*x)]*(b*c - a*d + 3*a*d*Log[(a + b*x)/(c + d*x)]))/d + (2*a^3*b*(c + d*x)*(2 +
2*Log[(a + b*x)/(c + d*x)] + Log[(a + b*x)/(c + d*x)]^2))/(a + b*x) - 9*a^2*b*c*(Log[(a + b*x)/(c + d*x)]*(Lo
g[(a + b*x)/(c + d*x)] - 2*Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))]) + (4*a
*b^2*c^2*(Log[(a + b*x)/(c + d*x)]*(Log[(a + b*x)/(c + d*x)] - 2*Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2
, (d*(a + b*x))/(b*(c + d*x))]))/d + 5*a^3*d*(Log[(a + b*x)/(c + d*x)]*(Log[(a + b*x)/(c + d*x)] - 2*Log[(b*c
- a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))]) - (b^2*c^2*(b*c - a*d)*(Log[(a + b*x)/(c +
d*x)]*(2*Log[(-(b*c) + a*d)/(d*(a + b*x))] + Log[(a + b*x)/(c + d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(d*(a + b
*x))]))/d^2 + 2*a^2*(-(b*c) + a*d)*(Log[(a + b*x)/(c + d*x)]^2*(3*Log[(-(b*c) + a*d)/(d*(a + b*x))] + Log[(a +
b*x)/(c + d*x)]) - 6*Log[(a + b*x)/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 6*PolyLog[3, (b*(c +
d*x))/(d*(a + b*x))])))/(2*b^4*(b*c - a*d)*g^2) + (B^2*c^2*d*i^3*n^2*(6*b*c - 6*a*d - (6*b^2*c*x)/(a + b*x) +
(6*a*b*d*x)/(a + b*x) + 6*a*d*Log[a/b + x] + 3*b*c*Log[a/b + x]^2 - 3*a*d*Log[a/b + x]^2 - 6*b*c*Log[c/d + x]
+ 6*b*c*Log[a + b*x] - 6*a*d*Log[a + b*x] - 6*b*c*Log[a/b + x]*Log[a + b*x] + 6*a*d*Log[a/b + x]*Log[a + b*x]
+ 6*b*c*Log[c/d + x]*Log[a + b*x] - 6*a*d*Log[c/d + x]*Log[a + b*x] - 6*b*c*Log[c/d + x]*Log[(d*(a + b*x))/(-(
b*c) + a*d)] + 6*a*d*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] - (6*b*(b*c - a*d)*x*Log[(a + b*x)/(c + d*
x)])/(a + b*x) + 6*b*c*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] - 6*a*d*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] + 3
*a*d*Log[(a + b*x)/(c + d*x)]^2 + 3*b*d*x*Log[(a + b*x)/(c + d*x)]^2 - (3*b^2*x*(c + d*x)*Log[(a + b*x)/(c + d
*x)]^2)/(a + b*x) - 3*b*c*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)]^2 - a*d*Log[(a + b*x)/(c
+ d*x)]^3 + 6*b*c*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] - 6*a*d*Log[(a + b*x)/(c + d*x)]*Log
[(b*c - a*d)/(b*c + b*d*x)] + 3*a*d*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*(b*c - a*d +
a*d*Log[(a + b*x)/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 6*(b*c - a*d)*PolyLog[2, (b*(c + d*x)
)/(b*c - a*d)] + 6*b*c*Log[(a + b*x)/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 6*a*d*PolyLog[3, (d*
(a + b*x))/(b*(c + d*x))] + 6*b*c*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))]))/(b^2*(b*c - a*d)*g^2) - (B^2*c*d*i
^3*n^2*(6*a^2*b*c*d + 6*a^2*b*d^2*x + 6*a^2*b*c*d*Log[(a + b*x)/(c + d*x)] + 6*a^2*b*d^2*x*Log[(a + b*x)/(c +
d*x)] + 12*a^2*b*c*d*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)] + 12*a*b^2*c*d*x*Log[(-(b*c) +
a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)] + 6*a^2*b*c*d*Log[(a + b*x)/(c + d*x)]^2 + 3*a^3*d^2*Log[(a + b*
x)/(c + d*x)]^2 + 9*a^2*b*d^2*x*Log[(a + b*x)/(c + d*x)]^2 - 3*b^3*c*d*x^2*Log[(a + b*x)/(c + d*x)]^2 + 3*a*b^
2*d^2*x^2*Log[(a + b*x)/(c + d*x)]^2 - 6*a^2*b*c*d*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)]^
2 - 6*a*b^2*c*d*x*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)]^2 - 2*a^3*d^2*Log[(a + b*x)/(c +
d*x)]^3 - 2*a^2*b*d^2*x*Log[(a + b*x)/(c + d*x)]^3 - 6*a*b^2*c^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c
+ b*d*x)] - 6*a^3*d^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] - 6*b^3*c^2*x*Log[(a + b*x)/(c
+ d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] - 6*a^2*b*d^2*x*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)]
+ 6*a^3*d^2*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*a^2*b*d^2*x*Log[(a + b*x)/(c + d*x)
]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*(a + b*x)*(-(b^2*c^2) - a^2*d^2 + 2*a^2*d^2*Log[(a + b*x)/(c + d*x)])*P
olyLog[2, (d*(a + b*x))/(b*(c + d*x))] + 12*a*b*c*d*(a + b*x)*(-1 + Log[(a + b*x)/(c + d*x)])*PolyLog[2, (b*(c
+ d*x))/(d*(a + b*x))] - 12*a^3*d^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] - 12*a^2*b*d^2*x*PolyLog[3, (d*(a
+ b*x))/(b*(c + d*x))] + 12*a^2*b*c*d*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))] + 12*a*b^2*c*d*x*PolyLog[3, (b*
(c + d*x))/(d*(a + b*x))]))/(b^3*(b*c - a*d)*g^2*(a + b*x))

________________________________________________________________________________________

Maple [F]  time = 0.686, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{3}}{ \left ( bgx+ag \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((d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

-2*A*B*c^3*i^3*n*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*
b*d)*g^2)) - 3*A^2*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^2))*c*d^2*i^3 + 1/2*(2
*a^3/(b^5*g^2*x + a*b^4*g^2) + 6*a^2*log(b*x + a)/(b^4*g^2) + (b*x^2 - 4*a*x)/(b^3*g^2))*A^2*d^3*i^3 + 3*A^2*c
^2*d*i^3*(a/(b^3*g^2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - 2*A*B*c^3*i^3*log(e*(b*x/(d*x + c) + a/(d*x +
c))^n)/(b^2*g^2*x + a*b*g^2) - A^2*c^3*i^3/(b^2*g^2*x + a*b*g^2) + 1/2*(B^2*b^3*d^3*i^3*x^3 + 3*(2*b^3*c*d^2*i
^3 - a*b^2*d^3*i^3)*B^2*x^2 + 2*(3*a*b^2*c*d^2*i^3 - 2*a^2*b*d^3*i^3)*B^2*x - 2*(b^3*c^3*i^3 - 3*a*b^2*c^2*d*i
^3 + 3*a^2*b*c*d^2*i^3 - a^3*d^3*i^3)*B^2 + 6*((b^3*c^2*d*i^3 - 2*a*b^2*c*d^2*i^3 + a^2*b*d^3*i^3)*B^2*x + (a*
b^2*c^2*d*i^3 - 2*a^2*b*c*d^2*i^3 + a^3*d^3*i^3)*B^2)*log(b*x + a))*log((d*x + c)^n)^2/(b^5*g^2*x + a*b^4*g^2)
- integrate(-(B^2*b^4*c^4*i^3*log(e)^2 + (B^2*b^4*d^4*i^3*log(e)^2 + 2*A*B*b^4*d^4*i^3*log(e))*x^4 + 4*(B^2*b
^4*c*d^3*i^3*log(e)^2 + 2*A*B*b^4*c*d^3*i^3*log(e))*x^3 + 6*(B^2*b^4*c^2*d^2*i^3*log(e)^2 + 2*A*B*b^4*c^2*d^2*
i^3*log(e))*x^2 + (B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d
*i^3*x + B^2*b^4*c^4*i^3)*log((b*x + a)^n)^2 + 2*(2*B^2*b^4*c^3*d*i^3*log(e)^2 + 3*A*B*b^4*c^3*d*i^3*log(e))*x
+ 2*(B^2*b^4*c^4*i^3*log(e) + (B^2*b^4*d^4*i^3*log(e) + A*B*b^4*d^4*i^3)*x^4 + 4*(B^2*b^4*c*d^3*i^3*log(e) +
A*B*b^4*c*d^3*i^3)*x^3 + 6*(B^2*b^4*c^2*d^2*i^3*log(e) + A*B*b^4*c^2*d^2*i^3)*x^2 + (4*B^2*b^4*c^3*d*i^3*log(e
) + 3*A*B*b^4*c^3*d*i^3)*x)*log((b*x + a)^n) - ((2*A*B*b^4*d^4*i^3 + (i^3*n + 2*i^3*log(e))*B^2*b^4*d^4)*x^4 +
2*(4*A*B*b^4*c*d^3*i^3 - (a*b^3*d^4*i^3*n - (3*i^3*n + 4*i^3*log(e))*b^4*c*d^3)*B^2)*x^3 - 2*(a*b^3*c^3*d*i^3
*n - 3*a^2*b^2*c^2*d^2*i^3*n + 3*a^3*b*c*d^3*i^3*n - a^4*d^4*i^3*n - b^4*c^4*i^3*log(e))*B^2 + (12*A*B*b^4*c^2
*d^2*i^3 + (12*a*b^3*c*d^3*i^3*n - 7*a^2*b^2*d^4*i^3*n + 12*b^4*c^2*d^2*i^3*log(e))*B^2)*x^2 + 2*(3*A*B*b^4*c^
3*d*i^3 + (3*a*b^3*c^2*d^2*i^3*n - a^3*b*d^4*i^3*n - (i^3*n - 4*i^3*log(e))*b^4*c^3*d)*B^2)*x + 6*((b^4*c^2*d^
2*i^3*n - 2*a*b^3*c*d^3*i^3*n + a^2*b^2*d^4*i^3*n)*B^2*x^2 + 2*(a*b^3*c^2*d^2*i^3*n - 2*a^2*b^2*c*d^3*i^3*n +
a^3*b*d^4*i^3*n)*B^2*x + (a^2*b^2*c^2*d^2*i^3*n - 2*a^3*b*c*d^3*i^3*n + a^4*d^4*i^3*n)*B^2)*log(b*x + a) + 2*(
B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B^2*b^4*c^
4*i^3)*log((b*x + a)^n))*log((d*x + c)^n))/(b^6*d*g^2*x^3 + a^2*b^4*c*g^2 + (b^6*c*g^2 + 2*a*b^5*d*g^2)*x^2 +
(2*a*b^5*c*g^2 + a^2*b^4*d*g^2)*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2,x, algorithm="giac")

[Out]

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