\(\int \frac {(c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{a g+b g x} \, dx\) [182]

Optimal result

Integrand size = 45, antiderivative size = 762 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\frac {B^2 d (b c-a d)^2 i^3 n^2 x}{3 b^3 g}-\frac {5 B d (b c-a d)^2 i^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g}-\frac {B (b c-a d) i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g}+\frac {d (b c-a d)^2 i^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g}+\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g}+\frac {i^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g}+\frac {2 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b^4 g}+\frac {B^2 (b c-a d)^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \log (c+d x)}{b^4 g}+\frac {5 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^4 g}-\frac {(b c-a d)^3 i^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g}-\frac {5 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^4 g}+\frac {2 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g} \]

[Out]

1/3*B^2*d*(-a*d+b*c)^2*i^3*n^2*x/b^3/g-5/3*B*d*(-a*d+b*c)^2*i^3*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/
g-1/3*B*(-a*d+b*c)*i^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/g+d*(-a*d+b*c)^2*i^3*(b*x+a)*(A+B*ln(e*
((b*x+a)/(d*x+c))^n))^2/b^4/g+1/2*(-a*d+b*c)*i^3*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2/g+1/3*i^3*(d*
x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b/g+2*B*(-a*d+b*c)^3*i^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b
*c)/b/(d*x+c))/b^4/g+1/3*B^2*(-a*d+b*c)^3*i^3*n^2*ln((b*x+a)/(d*x+c))/b^4/g+2*B^2*(-a*d+b*c)^3*i^3*n^2*ln(d*x+
c)/b^4/g+5/3*B*(-a*d+b*c)^3*i^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g-(-a*d+b*c)^3
*i^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g+2*B^2*(-a*d+b*c)^3*i^3*n^2*polylog(2,d*
(b*x+a)/b/(d*x+c))/b^4/g-5/3*B^2*(-a*d+b*c)^3*i^3*n^2*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g+2*B*(-a*d+b*c)^3*i^
3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g+2*B^2*(-a*d+b*c)^3*i^3*n^2*polylog(3,
b*(d*x+c)/d/(b*x+a))/b^4/g

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2561, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\frac {2 B i^3 n (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}+\frac {d i^3 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^4 g}-\frac {5 B d i^3 n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^4 g}+\frac {2 B i^3 n (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}-\frac {i^3 (b c-a d)^3 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^4 g}+\frac {5 B i^3 n (b c-a d)^3 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^4 g}+\frac {i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g}-\frac {B i^3 n (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2 g}+\frac {i^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b g}+\frac {2 B^2 i^3 n^2 (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g}-\frac {5 B^2 i^3 n^2 (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^4 g}+\frac {2 B^2 i^3 n^2 (b c-a d)^3 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {B^2 i^3 n^2 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^4 g}+\frac {2 B^2 i^3 n^2 (b c-a d)^3 \log (c+d x)}{b^4 g}+\frac {B^2 d i^3 n^2 x (b c-a d)^2}{3 b^3 g} \]

[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),x]

[Out]

(B^2*d*(b*c - a*d)^2*i^3*n^2*x)/(3*b^3*g) - (5*B*d*(b*c - a*d)^2*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(3*b^4*g) - (B*(b*c - a*d)*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2*g) + (d
*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g) + ((b*c - a*d)*i^3*(c + d*x)^2*
(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g) + (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^
2)/(3*b*g) + (2*B*(b*c - a*d)^3*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) + (B^2*(b*c - a*d)^3*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*n^2*Log[c +
d*x])/(b^4*g) + (5*B*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a +
b*x))])/(3*b^4*g) - ((b*c - a*d)^3*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*B^2*(b*c - a*d)^3*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g) - (5*B^2*(b*c
- a*d)^3*i^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^4*g) + (2*B*(b*c - a*d)^3*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*B^2*(b*c - a*d)^3*i^3*n^2*PolyL
og[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g)

Rule 31

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2354

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 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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 2561

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

Rule 6724

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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4969\) vs. \(2(762)=1524\).

Time = 7.27 (sec) , antiderivative size = 4969, normalized size of antiderivative = 6.52 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\text {Result too large to show} \]

[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),x]

[Out]

(i^3*(36*b*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c
+ d*x)])^2 + 18*b^2*d^2*(3*b*c - a*d)*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)]
)^2 + 12*b^3*d^3*x^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 36*(b*c - a*d)^
3*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 - 108*b^2*B*c^2*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*Log[a + b*x])*Log[(a + b*x)/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) -
12*B*n*(-A - B*Log[e*((a + b*x)/(c + d*x))^n] + B*n*Log[(a + b*x)/(c + d*x)])*(6*a^2*b*c*d^2 - 6*a^3*d^3 + 2*b
^3*c^2*d*x + 3*a*b^2*c*d^2*x - 5*a^2*b*d^3*x - b^3*c*d^2*x^2 + a*b^2*d^3*x^2 - 3*a^3*d^3*Log[a/b + x]^2 - 6*a^
2*b*c*d^2*Log[c/d + x] + 5*a^3*d^3*Log[a + b*x] - 6*a^3*d^3*Log[c/d + x]*Log[a + b*x] + 6*a^3*d^3*Log[a/b + x]
*(1 + Log[a + b*x]) + 6*a^3*d^3*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 6*a^2*b*d^3*x*Log[(a + b*x)/(
c + d*x)] - 3*a*b^2*d^3*x^2*Log[(a + b*x)/(c + d*x)] + 2*b^3*d^3*x^3*Log[(a + b*x)/(c + d*x)] - 6*a^3*d^3*Log[
a + b*x]*Log[(a + b*x)/(c + d*x)] - 2*b^3*c^3*Log[c + d*x] - 3*a*b^2*c^2*d*Log[c + d*x] + 6*a^3*d^3*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)]) + 36*b^3*B*c^3*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)])) + 54*b*B*c*n*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(-4*a*d^2*(a + b*x)*(-1 + Log[a/b + x]) + 2*a^2*d^2*Log[a
/b + x]^2 + 4*a*b*d*(c + d*x)*(-1 + Log[c/d + x]) + d^2*(b*x*(2*a - b*x) + 2*b^2*x^2*Log[a/b + x] - 2*a^2*Log[
a + b*x]) - 2*d^2*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x])*(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*
x)]) + b^2*(d*x*(-2*c + d*x) - 2*d^2*x^2*Log[c/d + x] + 2*c^2*Log[c + d*x]) - 4*a^2*d^2*(Log[c/d + x]*Log[(d*(
a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - 36*b^2*B^2*c^2*n^2*(a*d*Log[a/b + x]^3 -
3*d*(2*b*x - 2*(a + b*x)*Log[a/b + x] + (a + b*x)*Log[a/b + x]^2) - 3*b*(2*d*x - 2*(c + d*x)*Log[c/d + x] + (
c + d*x)*Log[c/d + x]^2) - 3*d*(b*x - a*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)]
)^2 + 6*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*x)*Lo
g[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)
]) - 3*(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)])*(-2*b*c + 2*a*d - 2*d*(a + b*x)*Log[a/b + x] +
a*d*Log[a/b + x]^2 + 2*Log[c/d + x]*(b*(c + d*x) - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) - 2*a*d*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)]) - 3*a*d*(Log[a/b + x]^2*(Log[c/d + x] - Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Log[a/
b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)]) + 3*a*d*(Log[c/d
+ x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*PolyLog[3
, (b*(c + d*x))/(b*c - a*d)])) + 9*b*B^2*c*n^2*(4*a^2*d^2*Log[a/b + x]^3 - 12*a*d^2*(2*b*x - 2*(a + b*x)*Log[a
/b + x] + (a + b*x)*Log[a/b + x]^2) - 3*d^2*(b*x*(6*a - b*x) + (-6*a^2 - 4*a*b*x + 2*b^2*x^2)*Log[a/b + x] + 2
*(a^2 - b^2*x^2)*Log[a/b + x]^2) - 12*a*b*d*(2*d*x - 2*(c + d*x)*Log[c/d + x] + (c + d*x)*Log[c/d + x]^2) - 3*
b^2*(d*x*(6*c - d*x) + (-6*c^2 - 4*c*d*x + 2*d^2*x^2)*Log[c/d + x] + 2*(c^2 - d^2*x^2)*Log[c/d + x]^2) + 6*d^2
*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)])^2 - 6*(Log[
a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)])*(-4*a*d^2*(a + b*x)*(-1 + Log[a/b + x]) + 2*a^2*d^2*Log[a/
b + x]^2 + 4*a*b*d*(c + d*x)*(-1 + Log[c/d + x]) + d^2*(b*x*(2*a - b*x) + 2*b^2*x^2*Log[a/b + x] - 2*a^2*Log[a
+ b*x]) + b^2*(d*x*(-2*c + d*x) - 2*d^2*x^2*Log[c/d + x] + 2*c^2*Log[c + d*x]) - 4*a^2*d^2*(Log[c/d + x]*Log[
(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 6*(2*a*b*c*d + 3*b^2*c*d*x + 3*a*b*d
^2*x - b^2*d^2*x^2 - 2*a*b*d^2*x*Log[c/d + x] + b^2*d^2*x^2*Log[c/d + x] - a^2*d^2*Log[a + b*x] - b^2*c^2*Log[
c + d*x] - 2*a*b*c*d*Log[c + d*x] - Log[a/b + x]*(b*d*(2*a*c + b*x*(2*c - d*x)) - 2*d^2*(a^2 - b^2*x^2)*Log[c/
d + x] + (-2*b^2*c^2 + 2*a^2*d^2)*Log[(b*(c + d*x))/(b*c - a*d)]) + 2*(b^2*c^2 - a^2*d^2)*PolyLog[2, (d*(a + b
*x))/(-(b*c) + a*d)] + 4*a*d*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b
*x)) + d*(a + b*x)*Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a +
b*x))/(-(b*c) + a*d)]) - 2*a^2*d^2*(Log[a/b + x]^2*(Log[c/d + x] - Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Log[a/
b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)])) + 12*a^2*d^2*(L
og[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*Pol
yLog[3, (b*(c + d*x))/(b*c - a*d)])) + 12*b^3*B^2*c^3*n^2*(Log[a/b + x]^3 + 3*Log[c/d + x]^2*Log[(d*(a + b*x))
/(-(b*c) + a*d)] + 3*Log[a + b*x]*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)])^2 + 3*Log[a/b + x]
^2*(-Log[c/d + x] + Log[(b*(c + d*x))/(b*c - a*d)]) + 6*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]
+ 6*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 3*(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d
*x)])*(Log[a/b + x]^2 - 2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*
d)])) - 6*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b*(c + d*x))/(b*c - a*d)]) - 2*B^2*n^2*(12*
a*b^2*c^2*d + 18*a^2*b*c*d^2 + 36*a^3*d^3 - 6*b^3*c^2*d*x + 12*a*b^2*c*d^2*x - 6*a^2*b*d^3*x - 12*a*b^2*c^2*d*
Log[a/b + x] + 18*a^2*b*c*d^2*Log[a/b + x] + 13*a^3*d^3*Log[a/b + x] + 3*a^3*d^3*Log[a/b + x]^2 - 12*a^3*d^3*L
og[a/b + x]^3 + 22*b^3*c^3*Log[c/d + x] + 27*a*b^2*c^2*d*Log[c/d + x] + 36*a^3*d^3*Log[c/d + x] - 36*a^2*b*c*d
^2*Log[a/b + x]*Log[c/d + x] + 30*a^3*d^3*Log[a/b + x]*Log[c/d + x] - 6*b^3*c^3*Log[c/d + x]^2 - 9*a*b^2*c^2*d
*Log[c/d + x]^2 + 18*a^2*b*c*d^2*Log[c/d + x]^2 - 6*a^2*b*c*d^2*Log[a + b*x] - 13*a^3*d^3*Log[a + b*x] + 30*a^
3*d^3*Log[a/b + x]*Log[a + b*x] + 18*a^3*d^3*Log[a/b + x]^2*Log[a + b*x] - 30*a^3*d^3*Log[c/d + x]*Log[a + b*x
] - 36*a^3*d^3*Log[a/b + x]*Log[c/d + x]*Log[a + b*x] + 18*a^3*d^3*Log[c/d + x]^2*Log[a + b*x] + 36*a^3*d^3*Lo
g[a/b + x]*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 18*a^3*d^3*Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c
) + a*d)] - 36*a^2*b*c*d^2*Log[(a + b*x)/(c + d*x)] + 36*a^3*d^3*Log[(a + b*x)/(c + d*x)] - 12*b^3*c^2*d*x*Log
[(a + b*x)/(c + d*x)] - 18*a*b^2*c*d^2*x*Log[(a + b*x)/(c + d*x)] + 30*a^2*b*d^3*x*Log[(a + b*x)/(c + d*x)] +
6*b^3*c*d^2*x^2*Log[(a + b*x)/(c + d*x)] - 6*a*b^2*d^3*x^2*Log[(a + b*x)/(c + d*x)] - 36*a^3*d^3*Log[a/b + x]*
Log[(a + b*x)/(c + d*x)] + 18*a^3*d^3*Log[a/b + x]^2*Log[(a + b*x)/(c + d*x)] + 36*a^2*b*c*d^2*Log[c/d + x]*Lo
g[(a + b*x)/(c + d*x)] - 30*a^3*d^3*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] - 36*a^3*d^3*Log[a/b + x]*Log[a + b*
x]*Log[(a + b*x)/(c + d*x)] + 36*a^3*d^3*Log[c/d + x]*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] - 36*a^3*d^3*Log[c
/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[(a + b*x)/(c + d*x)] - 18*a^2*b*d^3*x*Log[(a + b*x)/(c + d*x)]^2
+ 9*a*b^2*d^3*x^2*Log[(a + b*x)/(c + d*x)]^2 - 6*b^3*d^3*x^3*Log[(a + b*x)/(c + d*x)]^2 + 18*a^3*d^3*Log[a +
b*x]*Log[(a + b*x)/(c + d*x)]^2 - 4*b^3*c^3*Log[c + d*x] - 15*a*b^2*c^2*d*Log[c + d*x] - 66*a^2*b*c*d^2*Log[c
+ d*x] - 12*b^3*c^3*Log[a/b + x]*Log[c + d*x] - 18*a*b^2*c^2*d*Log[a/b + x]*Log[c + d*x] + 12*b^3*c^3*Log[c/d
+ x]*Log[c + d*x] + 18*a*b^2*c^2*d*Log[c/d + x]*Log[c + d*x] + 12*b^3*c^3*Log[(a + b*x)/(c + d*x)]*Log[c + d*x
] + 18*a*b^2*c^2*d*Log[(a + b*x)/(c + d*x)]*Log[c + d*x] + 12*b^3*c^3*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*
d)] + 18*a*b^2*c^2*d*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + 36*a^2*b*c*d^2*Log[a/b + x]*Log[(b*(c + d*x
))/(b*c - a*d)] - 66*a^3*d^3*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + 18*a^3*d^3*Log[a/b + x]^2*Log[(b*(c
+ d*x))/(b*c - a*d)] + 6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 6*a^3*d^3*Log[a/b + x])*Po
lyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*a^3*d^3*(Log[a/b + x] - Log[(a + b*x)/(c + d*x)])*PolyLog[2, (b*(c
+ d*x))/(b*c - a*d)] - 36*a^3*d^3*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 36*a^3*d^3*PolyLog[3, (b*(c + d*
x))/(b*c - a*d)])))/(36*b^4*g)

Maple [F]

\[\int \frac {\left (d i x +c i \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}{b g x +a g}d x\]

[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),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),x)

Fricas [F]

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

[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),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*g*x + a*g), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\text {Timed out} \]

[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),x)

[Out]

Timed out

Maxima [F]

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

[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),x, algorithm="maxima")

[Out]

3*A^2*c^2*d*i^3*(x/(b*g) - a*log(b*x + a)/(b^2*g)) - 1/6*A^2*d^3*i^3*(6*a^3*log(b*x + a)/(b^4*g) - (2*b^2*x^3
- 3*a*b*x^2 + 6*a^2*x)/(b^3*g)) + 3/2*A^2*c*d^2*i^3*(2*a^2*log(b*x + a)/(b^3*g) + (b*x^2 - 2*a*x)/(b^2*g)) + A
^2*c^3*i^3*log(b*g*x + a*g)/(b*g) + 1/6*(2*B^2*b^3*d^3*i^3*x^3 + 3*(3*b^3*c*d^2*i^3 - a*b^2*d^3*i^3)*B^2*x^2 +
6*(3*b^3*c^2*d*i^3 - 3*a*b^2*c*d^2*i^3 + a^2*b*d^3*i^3)*B^2*x + 6*(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*log(b*x + a))*log((d*x + c)^n)^2/(b^4*g) - integrate(-1/3*(3*B^2*b^4*c^4*i^3*log(
e)^2 + 6*A*B*b^4*c^4*i^3*log(e) + 3*(B^2*b^4*d^4*i^3*log(e)^2 + 2*A*B*b^4*d^4*i^3*log(e))*x^4 + 12*(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 + 18*(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 + 3*(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 + 12*(B^2*b^4*c^3*d*i^3*log(e)^2 + 2*A*B*b^4*c^3*d*i^3*log(e))*x +
6*(B^2*b^4*c^4*i^3*log(e) + A*B*b^4*c^4*i^3 + (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) + A*B*b^4*c^3*d*i^3)*x)*log((b*x + a)^n) - (6*B^2*b^4*c^4*i^3*log(e) + 6*A*B*b^4*c^4*i^3 + 2*
(3*A*B*b^4*d^4*i^3 + (i^3*n + 3*i^3*log(e))*B^2*b^4*d^4)*x^4 + (24*A*B*b^4*c*d^3*i^3 - (a*b^3*d^4*i^3*n - 3*(3
*i^3*n + 8*i^3*log(e))*b^4*c*d^3)*B^2)*x^3 + 3*(12*A*B*b^4*c^2*d^2*i^3 - (3*a*b^3*c*d^3*i^3*n - a^2*b^2*d^4*i^
3*n - 6*(i^3*n + 2*i^3*log(e))*b^4*c^2*d^2)*B^2)*x^2 + 6*(4*A*B*b^4*c^3*d*i^3 + (3*a*b^3*c^2*d^2*i^3*n - 3*a^2
*b^2*c*d^3*i^3*n + a^3*b*d^4*i^3*n + 4*b^4*c^3*d*i^3*log(e))*B^2)*x + 6*((b^4*c^3*d*i^3*n - 3*a*b^3*c^2*d^2*i^
3*n + 3*a^2*b^2*c*d^3*i^3*n - a^3*b*d^4*i^3*n)*B^2*x + (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^2)*log(b*x + a) + 6*(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^5*d*g*x^2 +
a*b^4*c*g + (b^5*c*g + a*b^4*d*g)*x), x)

Giac [F]

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

[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),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), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{a\,g+b\,g\,x} \,d x \]

[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),x)

[Out]

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