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

Optimal. Leaf size=306 \[ \frac{2 B i n (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac{2 B^2 i n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^2 g}+\frac{2 B^2 i n^2 (b c-a d) \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 g}+\frac{2 B i n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac{d i (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g}-\frac{i (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 2.87409, antiderivative size = 692, normalized size of antiderivative = 2.26, number of steps used = 36, number of rules used = 19, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.442, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac{2 A B i n (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{2 B^2 i n (b c-a d) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{2 a B^2 d i n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{2 B^2 i n^2 (b c-a d) \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 g}+\frac{2 B^2 c i n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 a B d i n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac{i (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g}-\frac{A B i n (b c-a d) \log ^2(a+b x)}{b^2 g}+\frac{2 A B i n (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{2 B c i n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g}+\frac{d i x \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g}-\frac{B^2 i (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac{B^2 i (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{2 a B^2 d i n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{a B^2 d i n^2 \log ^2(a+b x)}{b^2 g}+\frac{2 B^2 c i n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{B^2 c i n^2 \log ^2(c+d x)}{b g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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{(163 c+163 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac{163 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g (a+b x)}\right ) \, dx\\ &=\frac{(163 d) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}+\frac{(163 (b c-a d)) \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 g}\\ &=\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{(326 B d n) \int \frac{(b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac{(326 B (b c-a d) n) \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^2 g}\\ &=\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{(326 B (b c-a d) n) \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^2 g}-\frac{(326 B d (b c-a d) n) \int \frac{x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{(326 B d (b c-a d) n) \int \left (-\frac{a \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac{c \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b g}-\frac{\left (326 B (b c-a d)^2 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^2 g}\\ &=\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}+\frac{(326 a B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}-\frac{(326 B c d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b g}-\frac{\left (326 B (b c-a d)^2 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^2 g}\\ &=\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac{\left (326 A B (b c-a d)^2 n\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g}-\frac{\left (326 B^2 (b c-a d)^2 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^2 g}+\frac{\left (326 B^2 c 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 g}-\frac{\left (326 a B^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 (a+b x)}{a+b x} \, dx}{b^2 g}\\ &=-\frac{163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}+\frac{\left (163 B^2 (b c-a d)\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}-\frac{\left (326 A B (b c-a d)^2 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^3 g}+\frac{\left (326 B^2 c 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 g}-\frac{\left (326 a B^2 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^2 g}\\ &=-\frac{163 B^2 (b c-a d) \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^2 g}-\frac{163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac{(326 A B (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac{(326 A B d (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^3 g}+\frac{\left (326 B^2 (b c-a d)^2 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^2 g}+\frac{\left (326 B^2 c n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{g}-\frac{\left (326 a B^2 d n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g}-\frac{\left (326 B^2 c d n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b g}+\frac{\left (326 a B^2 d^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g}\\ &=-\frac{163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac{163 B^2 (b c-a d) \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^2 g}-\frac{163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}+\frac{326 B^2 c n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}+\frac{326 A B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 a B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 B^2 (b c-a d) 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^2 g}-\frac{(326 A B (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{\left (326 B^2 c n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b g}-\frac{\left (326 a B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{\left (326 a B^2 d n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}-\frac{\left (326 B^2 c d n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b g}-\frac{\left (326 B^2 (b c-a d)^2 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^2 g}\\ &=-\frac{163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac{163 a B^2 d n^2 \log ^2(a+b x)}{b^2 g}-\frac{163 B^2 (b c-a d) \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^2 g}-\frac{163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}+\frac{326 B^2 c n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac{163 B^2 c n^2 \log ^2(c+d x)}{b g}+\frac{326 A B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 a B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 A B (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{326 B^2 (b c-a d) 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^2 g}+\frac{326 B^2 (b c-a d) n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac{\left (326 B^2 c 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 g}-\frac{\left (326 a B^2 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^2 g}\\ &=-\frac{163 A B (b c-a d) n \log ^2(a+b x)}{b^2 g}-\frac{163 a B^2 d n^2 \log ^2(a+b x)}{b^2 g}-\frac{163 B^2 (b c-a d) \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^2 g}-\frac{163 B^2 (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{326 a B 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^2 g}+\frac{163 d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{163 (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 )^2}{b^2 g}+\frac{326 B^2 c n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{326 B c n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b g}-\frac{163 B^2 c n^2 \log ^2(c+d x)}{b g}+\frac{326 A B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 a B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{326 A B (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{326 a B^2 d n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{326 B^2 c n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{326 B^2 (b c-a d) 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^2 g}+\frac{326 B^2 (b c-a d) n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}\\ \end{align*}

Mathematica [B]  time = 1.86057, size = 742, normalized size = 2.42 \[ \frac{i \left (-3 B n \left (-2 a d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+2 \left (\log \left (\frac{c}{d}+x\right ) \left (-a d \log \left (\frac{d (a+b x)}{a d-b c}\right )+a d \log (a+b x)+b c\right )+(a d \log (a+b x)-b d x) \log \left (\frac{a+b x}{c+d x}\right )+a d-b c\right )+a d \log ^2\left (\frac{a}{b}+x\right )-2 a d (\log (a+b x)+1) \log \left (\frac{a}{b}+x\right )\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )+3 b B c n \left (-2 \left (\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )\right )-2 \log (a+b x) \left (-\log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )\right )+\log ^2\left (\frac{a}{b}+x\right )\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )+B^2 n^2 \left (-6 a d \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+6 \left (a d \log \left (\frac{a+b x}{c+d x}\right )-a d+b c\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{a+b x}{c+d x}\right ) \left (-a d \log ^2\left (\frac{a+b x}{c+d x}\right )+3 d \left (a \log \left (\frac{b c-a d}{b c+b d x}\right )+a+b x\right ) \log \left (\frac{a+b x}{c+d x}\right )+6 (b c-a d) \log \left (\frac{b c-a d}{b c+b d x}\right )\right )\right )-3 b B^2 c n^2 \left (-2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )-2 \log \left (\frac{a+b x}{c+d x}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )+\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{a+b x}{c+d x}\right )\right )+3 b d x \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )^2+3 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )+A\right )^2\right )}{3 b^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.527, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{bgx+ag} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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