\(\int (a+b x) (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [166]

Optimal result

Integrand size = 31, antiderivative size = 376 \[ \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {3 B^2 (b c-a d)^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b d^2}-\frac {3 B (b c-a d) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b d}-\frac {3 B (b c-a d)^2 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b d^2}+\frac {(a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 b}-\frac {3 B^3 (b c-a d)^2 n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2}-\frac {3 B^2 (b c-a d)^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2}+\frac {3 B^3 (b c-a d)^2 n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2573, 2549, 2381, 2395, 2355, 2354, 2438, 2421, 6724} \[ \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {3 B^2 n^2 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^2}-\frac {3 B^2 n^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^2}-\frac {3 B n (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b d^2}-\frac {3 B n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b d}+\frac {(a+b x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 b}-\frac {3 B^3 n^3 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2}+\frac {3 B^3 n^3 (b c-a d)^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2} \]

[In]

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

[Out]

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

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 2381

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

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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 2549

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

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2984\) vs. \(2(376)=752\).

Time = 0.74 (sec) , antiderivative size = 2984, normalized size of antiderivative = 7.94 \[ \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-12*a^2*A*B^2*d^2*n^2 + 6*a*b*B^3*c*d*n^3 - 6*a^2*B^3*d^2*n^3 + 2*a*A^3*b*d^2*x - 3*A^2*b^2*B*c*d*n*x + 3*a*A
^2*b*B*d^2*n*x + A^3*b^2*d^2*x^2 + 3*a^2*A^2*B*d^2*n*Log[a + b*x] - 6*a*A*b*B^2*c*d*n^2*Log[a + b*x] + 6*a^2*A
*B^2*d^2*n^2*Log[a + b*x] + 12*a^2*B^3*d^2*n^3*Log[a + b*x] - 3*a^2*A*B^2*d^2*n^2*Log[a + b*x]^2 + 3*a*b*B^3*c
*d*n^3*Log[a + b*x]^2 - 3*a^2*B^3*d^2*n^3*Log[a + b*x]^2 + a^2*B^3*d^2*n^3*Log[a + b*x]^3 + 3*A^2*b^2*B*c^2*n*
Log[c + d*x] - 6*a*A^2*b*B*c*d*n*Log[c + d*x] + 6*A*b^2*B^2*c^2*n^2*Log[c + d*x] - 6*a*A*b*B^2*c*d*n^2*Log[c +
d*x] - 12*a^2*B^3*d^2*n^3*Log[c + d*x] - 6*A*b^2*B^2*c^2*n^2*Log[a + b*x]*Log[c + d*x] + 12*a*A*b*B^2*c*d*n^2
*Log[a + b*x]*Log[c + d*x] + 6*a^2*A*B^2*d^2*n^2*Log[a + b*x]*Log[c + d*x] - 6*b^2*B^3*c^2*n^3*Log[a + b*x]*Lo
g[c + d*x] + 6*a*b*B^3*c*d*n^3*Log[a + b*x]*Log[c + d*x] + 3*b^2*B^3*c^2*n^3*Log[a + b*x]^2*Log[c + d*x] - 6*a
*b*B^3*c*d*n^3*Log[a + b*x]^2*Log[c + d*x] - 6*a^2*B^3*d^2*n^3*Log[a + b*x]^2*Log[c + d*x] - 6*a^2*A*B^2*d^2*n
^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 6*a^2*B^3*d^2*n^3*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) +
a*d)]*Log[c + d*x] + 3*A*b^2*B^2*c^2*n^2*Log[c + d*x]^2 - 6*a*A*b*B^2*c*d*n^2*Log[c + d*x]^2 + 3*b^2*B^3*c^2*
n^3*Log[c + d*x]^2 - 3*a*b*B^3*c*d*n^3*Log[c + d*x]^2 - 6*b^2*B^3*c^2*n^3*Log[a + b*x]*Log[c + d*x]^2 + 12*a*b
*B^3*c*d*n^3*Log[a + b*x]*Log[c + d*x]^2 + 3*a^2*B^3*d^2*n^3*Log[a + b*x]*Log[c + d*x]^2 + 3*b^2*B^3*c^2*n^3*L
og[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - 6*a*b*B^3*c*d*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c +
d*x]^2 - 3*a^2*B^3*d^2*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 + b^2*B^3*c^2*n^3*Log[c + d*x]^3 -
2*a*b*B^3*c*d*n^3*Log[c + d*x]^3 + 6*A*b^2*B^2*c^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 12*a*A*b
*B^2*c*d*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*b^2*B^3*c^2*n^3*Log[a + b*x]*Log[(b*(c + d*x))/(b
*c - a*d)] - 12*a*b*B^3*c*d*n^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 6*a^2*B^3*d^2*n^3*Log[a + b*x]*L
og[(b*(c + d*x))/(b*c - a*d)] - 3*b^2*B^3*c^2*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 6*a*b*B^3*c*
d*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 3*a^2*B^3*d^2*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c
- a*d)] + 6*b^2*B^3*c^2*n^3*Log[a + b*x]*Log[c + d*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 12*a*b*B^3*c*d*n^3*Log[
a + b*x]*Log[c + d*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 12*a^2*B^3*d^2*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6
*a*A^2*b*B*d^2*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*A*b^2*B^2*c*d*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a
*A*b*B^2*d^2*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*A^2*b^2*B*d^2*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a
^2*A*B^2*d^2*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a*b*B^3*c*d*n^2*Log[a + b*x]*Log[(e*(a + b*x)
^n)/(c + d*x)^n] + 6*a^2*B^3*d^2*n^2*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 3*a^2*B^3*d^2*n^2*Log[a +
b*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*A*b^2*B^2*c^2*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 1
2*a*A*b*B^2*c*d*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*b^2*B^3*c^2*n^2*Log[c + d*x]*Log[(e*(a + b
*x)^n)/(c + d*x)^n] - 6*a*b*B^3*c*d*n^2*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*b^2*B^3*c^2*n^2*Log[
a + b*x]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 12*a*b*B^3*c*d*n^2*Log[a + b*x]*Log[c + d*x]*Log[(e*(
a + b*x)^n)/(c + d*x)^n] + 6*a^2*B^3*d^2*n^2*Log[a + b*x]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a^
2*B^3*d^2*n^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*b^2*B^3*c^2*
n^2*Log[c + d*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a*b*B^3*c*d*n^2*Log[c + d*x]^2*Log[(e*(a + b*x)^n)/(c
+ d*x)^n] + 6*b^2*B^3*c^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 1
2*a*b*B^3*c*d*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a*A*b*B^2*d
^2*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 3*b^2*B^3*c*d*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*a*b*B^3*d^2
*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*A*b^2*B^2*d^2*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*a^2*B^3*d
^2*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*b^2*B^3*c^2*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d
*x)^n]^2 - 6*a*b*B^3*c*d*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*a*b*B^3*d^2*x*Log[(e*(a + b*x)^
n)/(c + d*x)^n]^3 + b^2*B^3*d^2*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3 + 6*B^2*n^2*(A*b^2*c^2 - 2*a*A*b*c*d +
b^2*B*c^2*n - 2*a*b*B*c*d*n + a^2*B*d^2*n + a^2*B*d^2*n*Log[a + b*x] + b*B*c*(b*c - 2*a*d)*n*Log[c + d*x] + b^
2*B*c^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*a*b*B*c*d*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*
x))/(-(b*c) + a*d)] + 6*B^2*n^2*(a^2*B*d^2*n*Log[a + b*x] + b*B*c*(b*c - 2*a*d)*n*Log[c + d*x] - a^2*d^2*(A +
B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 6*b^2*B^3*c^2*n^3*PolyLog[3, (d*(
a + b*x))/(-(b*c) + a*d)] + 12*a*b*B^3*c*d*n^3*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*a^2*B^3*d^2*n^3*Po
lyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*b^2*B^3*c^2*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] + 12*a*b*B^3*
c*d*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 6*a^2*B^3*d^2*n^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(2*b*
d^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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