\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^4} \, dx\) [170]

Optimal result

Integrand size = 33, antiderivative size = 611 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=-\frac {6 B^3 d^2 n^3 (c+d x)}{(b c-a d)^3 (a+b x)}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (b c-a d)^3 (a+b x)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{9 (b c-a d)^3 (a+b x)^3}-\frac {3 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {3 b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3} \]

[Out]

-6*B^3*d^2*n^3*(d*x+c)/(-a*d+b*c)^3/(b*x+a)+3/4*b*B^3*d*n^3*(d*x+c)^2/(-a*d+b*c)^3/(b*x+a)^2-2/27*b^2*B^3*n^3*
(d*x+c)^3/(-a*d+b*c)^3/(b*x+a)^3-6*B^2*d^2*n^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)+
3/2*b*B^2*d*n^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)^2-2/9*b^2*B^2*n^2*(d*x+c)^3*(
A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)^3-3*B*d^2*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/
(-a*d+b*c)^3/(b*x+a)+3/2*b*B*d*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^3/(b*x+a)^2-1/3*b^2*
B*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^3/(b*x+a)^3-d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x
+c)^n)))^3/(-a*d+b*c)^3/(b*x+a)+b*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*d+b*c)^3/(b*x+a)^2-1/3*b
^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*d+b*c)^3/(b*x+a)^3

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2573, 2549, 2395, 2342, 2341} \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{9 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 (a+b x)^3 (b c-a d)^3}-\frac {b^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 (a+b x)^3 (b c-a d)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^3}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 (a+b x)^2 (b c-a d)^3}-\frac {3 B d^2 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)^3}+\frac {3 b B d n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^2 (b c-a d)^3}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (a+b x)^3 (b c-a d)^3}-\frac {6 B^3 d^2 n^3 (c+d x)}{(a+b x) (b c-a d)^3}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^3} \]

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^4,x]

[Out]

(-6*B^3*d^2*n^3*(c + d*x))/((b*c - a*d)^3*(a + b*x)) + (3*b*B^3*d*n^3*(c + d*x)^2)/(4*(b*c - a*d)^3*(a + b*x)^
2) - (2*b^2*B^3*n^3*(c + d*x)^3)/(27*(b*c - a*d)^3*(a + b*x)^3) - (6*B^2*d^2*n^2*(c + d*x)*(A + B*Log[(e*(a +
b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^3*(a + b*x)) + (3*b*B^2*d*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c +
d*x)^n]))/(2*(b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))
/(9*(b*c - a*d)^3*(a + b*x)^3) - (3*B*d^2*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)
^3*(a + b*x)) + (3*b*B*d*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*c - a*d)^3*(a + b*x)^
2) - (b^2*B*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*(b*c - a*d)^3*(a + b*x)^3) - (d^2*(c
+ d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*
(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d
*x)^n])^3)/(3*(b*c - a*d)^3*(a + b*x)^3)

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^4} \, 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 (\frac {\text {Subst}\left (\int \frac {(b-d x)^2 \left (A+B \log \left (e x^n\right )\right )^3}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 {\text {Subst}\left (\int \left (\frac {b^2 \left (A+B \log \left (e x^n\right )\right )^3}{x^4}-\frac {2 b d \left (A+B \log \left (e x^n\right )\right )^3}{x^3}+\frac {d^2 \left (A+B \log \left (e x^n\right )\right )^3}{x^2}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 {b^2 \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^3}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 {(2 b d) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^3}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 {d^2 \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^3}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = -\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3}+\text {Subst}\left (\frac {\left (b^2 B n\right ) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^2}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 {(3 b B d n) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^2}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 d^2 n\right ) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = -\frac {3 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {3 b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3}+\text {Subst}\left (\frac {\left (2 b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^3},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^2 d n^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},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 (6 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = -\frac {6 B^3 d^2 n^3 (c+d x)}{(b c-a d)^3 (a+b x)}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (b c-a d)^3 (a+b x)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{9 (b c-a d)^3 (a+b x)^3}-\frac {3 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {3 b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 1003, normalized size of antiderivative = 1.64 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\frac {-36 B^3 d^3 n^3 (a+b x)^3 \log ^3(a+b x)+36 B^3 d^3 n^3 (a+b x)^3 \log ^3(c+d x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x) \left (6 A+11 B n+6 B n \log (c+d x)+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^3 n (a+b x)^3 \log (c+d x) \left (18 A^2+66 A B n+85 B^2 n^2+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (36 A^3 b^2 c^2-72 a A^3 b c d+36 a^2 A^3 d^2+36 A^2 b^2 B c^2 n-126 a A^2 b B c d n+198 a^2 A^2 B d^2 n+24 A b^2 B^2 c^2 n^2-138 a A b B^2 c d n^2+510 a^2 A B^2 d^2 n^2+8 b^2 B^3 c^2 n^3-73 a b B^3 c d n^3+575 a^2 B^3 d^2 n^3-54 A^2 b^2 B c d n x+270 a A^2 b B d^2 n x-90 A b^2 B^2 c d n^2 x+882 a A b B^2 d^2 n^2 x-57 b^2 B^3 c d n^3 x+1077 a b B^3 d^2 n^3 x+108 A^2 b^2 B d^2 n x^2+396 A b^2 B^2 d^2 n^2 x^2+510 b^2 B^3 d^2 n^3 x^2+6 B \left (18 A^2 (b c-a d)^2+6 A B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (-23 c+147 d x)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \left (6 A (b c-a d)^2+B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+36 B^3 (b c-a d)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (18 A^2+66 A B n+85 B^2 n^2+18 B^2 n^2 \log ^2(c+d x)+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 B n \log (c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{108 b (b c-a d)^3 (a+b x)^3} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^4,x]

[Out]

(-36*B^3*d^3*n^3*(a + b*x)^3*Log[a + b*x]^3 + 36*B^3*d^3*n^3*(a + b*x)^3*Log[c + d*x]^3 + 18*B^2*d^3*n^2*(a +
b*x)^3*Log[c + d*x]^2*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 18*B^2*d^3*n^2*(a + b*x)^3*Log[a
+ b*x]^2*(6*A + 11*B*n + 6*B*n*Log[c + d*x] + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B*d^3*n*(a + b*x)^3*L
og[c + d*x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 6*B*(6*A + 11*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*Log
[(e*(a + b*x)^n)/(c + d*x)^n]^2) - (b*c - a*d)*(36*A^3*b^2*c^2 - 72*a*A^3*b*c*d + 36*a^2*A^3*d^2 + 36*A^2*b^2*
B*c^2*n - 126*a*A^2*b*B*c*d*n + 198*a^2*A^2*B*d^2*n + 24*A*b^2*B^2*c^2*n^2 - 138*a*A*b*B^2*c*d*n^2 + 510*a^2*A
*B^2*d^2*n^2 + 8*b^2*B^3*c^2*n^3 - 73*a*b*B^3*c*d*n^3 + 575*a^2*B^3*d^2*n^3 - 54*A^2*b^2*B*c*d*n*x + 270*a*A^2
*b*B*d^2*n*x - 90*A*b^2*B^2*c*d*n^2*x + 882*a*A*b*B^2*d^2*n^2*x - 57*b^2*B^3*c*d*n^3*x + 1077*a*b*B^3*d^2*n^3*
x + 108*A^2*b^2*B*d^2*n*x^2 + 396*A*b^2*B^2*d^2*n^2*x^2 + 510*b^2*B^3*d^2*n^3*x^2 + 6*B*(18*A^2*(b*c - a*d)^2
+ 6*A*B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + B^2*n^2*(85*a^2*d^2 + a*b
*d*(-23*c + 147*d*x) + b^2*(4*c^2 - 15*c*d*x + 66*d^2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*(6*A*(b
*c - a*d)^2 + B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)))*Log[(e*(a + b*x)^n
)/(c + d*x)^n]^2 + 36*B^3*(b*c - a*d)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 6*B*d^3*n*(a + b*x)^3*Log[a + b*
x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*n^2*Log[c + d*x]^2 + 6*B*(6*A + 11*B*n)*Log[(e*(a + b*x)^n)/(c + d
*x)^n] + 18*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 6*B*n*Log[c + d*x]*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n
)/(c + d*x)^n])))/(108*b*(b*c - a*d)^3*(a + b*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2686\) vs. \(2(597)=1194\).

Time = 95.36 (sec) , antiderivative size = 2687, normalized size of antiderivative = 4.40

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2687\)
risch	\(\text {Expression too large to display}\)	\(175812\)

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

-1/108*(-648*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c*d^3*n-216*A*B^2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c
*d^3*n-324*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*c*d^3*n-972*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c*d^3
*n^2+540*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*d^4*n+108*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c^2*d^2
*n-972*A*B^2*x*a*b^6*c*d^3*n^2-324*A^2*B*x*a*b^6*c*d^3*n-648*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3*n
+324*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c^2*d^2*n-1188*A*B^2*ln(b*x+a)*x^2*a*b^6*d^4*n^2+1188*A*B^2*ln(d*
x+c)*x^2*a*b^6*d^4*n^2-324*A^2*B*ln(b*x+a)*x^2*a*b^6*d^4*n+324*A^2*B*ln(d*x+c)*x^2*a*b^6*d^4*n-1188*A*B^2*ln(b
*x+a)*x*a^2*b^5*d^4*n^2+1188*A*B^2*ln(d*x+c)*x*a^2*b^5*d^4*n^2-324*A^2*B*ln(b*x+a)*x*a^2*b^5*d^4*n+324*A^2*B*l
n(d*x+c)*x*a^2*b^5*d^4*n+108*A^2*B*ln(d*x+c)*a^3*b^4*d^4*n-486*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*d^4
*n-108*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c*d^3*n+396*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*d^4*n^2
-396*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c*d^3*n^2-324*A*B^2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*d^4+3
96*A*B^2*x^2*a*b^6*d^4*n^2-396*A*B^2*x^2*b^7*c*d^3*n^2-324*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a^2*b^5*d^4*n-3
96*A*B^2*ln(b*x+a)*x^3*b^7*d^4*n^2+396*A*B^2*ln(d*x+c)*x^3*b^7*d^4*n^2+54*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*
b^7*c^2*d^2*n+882*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*d^4*n^2+90*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c
^2*d^2*n^2-1134*B^3*x*a*b^6*c*d^3*n^3+108*A^2*B*x^2*a*b^6*d^4*n-108*A^2*B*x^2*b^7*c*d^3*n-324*A*B^2*x*ln(e*(b*
x+a)^n/((d*x+c)^n))^2*a^2*b^5*d^4+882*A*B^2*x*a^2*b^5*d^4*n^2+90*A*B^2*x*b^7*c^2*d^2*n^2-324*B^3*ln(e*(b*x+a)^
n/((d*x+c)^n))^2*a^2*b^5*c*d^3*n+162*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*c^2*d^2*n-648*B^3*ln(e*(b*x+a)^n/
((d*x+c)^n))*a^2*b^5*c*d^3*n^2+162*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c^2*d^2*n^2+270*A^2*B*x*a^2*b^5*d^4*n
+54*A^2*B*x*b^7*c^2*d^2*n-324*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a^2*b^5*c*d^3+324*A*B^2*ln(e*(b*x+a)^n/((d*x
+c)^n))^2*a*b^6*c^2*d^2+396*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b^4*d^4*n-72*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n
))*b^7*c^3*d*n-324*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3+324*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6
*c^2*d^2+1530*B^3*ln(d*x+c)*x*a^2*b^5*d^4*n^3-396*A*B^2*ln(b*x+a)*a^3*b^4*d^4*n^2+396*A*B^2*ln(d*x+c)*a^3*b^4*
d^4*n^2-108*A^2*B*ln(b*x+a)*a^3*b^4*d^4*n-108*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^3*a*b^6*d^4+510*B^3*x^2*a*b^
6*d^4*n^3-510*B^3*x^2*b^7*c*d^3*n^3-108*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^3*a^2*b^5*d^4+1077*B^3*x*a^2*b^5*d^4
*n^3+57*B^3*x*b^7*c^2*d^2*n^3-108*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^3*a^2*b^5*c*d^3+108*B^3*ln(e*(b*x+a)^n/((d*x
+c)^n))^3*a*b^6*c^2*d^2-36*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c^3*d*n+510*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a
^3*b^4*d^4*n^2-24*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c^3*d*n^2-108*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c^
3*d+108*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b^4*d^4-108*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c^3*d+162*A^2*
B*a*b^6*c^2*d^2*n-108*A*B^2*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*d^4-1530*B^3*ln(b*x+a)*x^2*a*b^6*d^4*n^3+153
0*B^3*ln(d*x+c)*x^2*a*b^6*d^4*n^3-108*A^2*B*ln(b*x+a)*x^3*b^7*d^4*n+108*A^2*B*ln(d*x+c)*x^3*b^7*d^4*n-1530*B^3
*ln(b*x+a)*x*a^2*b^5*d^4*n^3+575*B^3*a^3*b^4*d^4*n^3-8*B^3*b^7*c^3*d*n^3-108*A^3*a^2*b^5*c*d^3-648*A*B^2*a^2*b
^5*c*d^3*n^2+162*A*B^2*a*b^6*c^2*d^2*n^2-324*A^2*B*a^2*b^5*c*d^3*n-510*B^3*ln(b*x+a)*x^3*b^7*d^4*n^3+510*B^3*l
n(d*x+c)*x^3*b^7*d^4*n^3-510*B^3*ln(b*x+a)*a^3*b^4*d^4*n^3-648*B^3*a^2*b^5*c*d^3*n^3+81*B^3*a*b^6*c^2*d^2*n^3+
510*A*B^2*a^3*b^4*d^4*n^2-24*A*B^2*b^7*c^3*d*n^2+198*A^2*B*a^3*b^4*d^4*n-36*A^2*B*b^7*c^3*d*n+510*B^3*ln(d*x+c
)*a^3*b^4*d^4*n^3-198*B^3*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*d^4*n+108*A^3*a*b^6*c^2*d^2-36*B^3*x^3*ln(e*(b
*x+a)^n/((d*x+c)^n))^3*b^7*d^4-36*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^3*b^7*c^3*d+36*A^3*a^3*b^4*d^4-36*A^3*b^7*c^
3*d+216*A*B^2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*d^4*n)/(b*x+a)^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*
c^3)/b^5/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4008 vs. \(2 (597) = 1194\).

Time = 0.41 (sec) , antiderivative size = 4008, normalized size of antiderivative = 6.56 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^4,x, algorithm="fricas")

[Out]

-1/108*(36*A^3*b^3*c^3 - 108*A^3*a*b^2*c^2*d + 108*A^3*a^2*b*c*d^2 - 36*A^3*a^3*d^3 + (8*B^3*b^3*c^3 - 81*B^3*
a*b^2*c^2*d + 648*B^3*a^2*b*c*d^2 - 575*B^3*a^3*d^3)*n^3 + 36*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 +
3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^3)*log(b*x + a)^3 - 36*(B^3*b
^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^
2*b*c*d^2)*n^3)*log(d*x + c)^3 + 36*(B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2 - B^3*a^3*d^3)*log(e)
^3 + 6*(4*A*B^2*b^3*c^3 - 27*A*B^2*a*b^2*c^2*d + 108*A*B^2*a^2*b*c*d^2 - 85*A*B^2*a^3*d^3)*n^2 + 6*(85*(B^3*b^
3*c*d^2 - B^3*a*b^2*d^3)*n^3 + 66*(A*B^2*b^3*c*d^2 - A*B^2*a*b^2*d^3)*n^2 + 18*(A^2*B*b^3*c*d^2 - A^2*B*a*b^2*
d^3)*n)*x^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2
*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n
^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*
d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3*a^2*b*d^3*n^2*x + (B^
3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(b*x + a)^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*
b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*
B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^
3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*
d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b
*c*d^2)*n^3)*log(b*x + a) + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3*a^2*b*d^3*n^2*x + (B^3*b^
3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(d*x + c)^2 + 18*(6*A*B^2*b^3*c^3 - 18*A*B^2*a*
b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2 - 6*A*B^2*a^3*d^3 + 6*(B^3*b^3*c*d^2 - B^3*a*b^2*d^3)*n*x^2 - 3*(B^3*b^3*c^2*
d - 6*B^3*a*b^2*c*d^2 + 5*B^3*a^2*b*d^3)*n*x + (2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2 - 11*B^
3*a^3*d^3)*n)*log(e)^2 + 18*(2*A^2*B*b^3*c^3 - 9*A^2*B*a*b^2*c^2*d + 18*A^2*B*a^2*b*c*d^2 - 11*A^2*B*a^3*d^3)*
n - 3*((19*B^3*b^3*c^2*d - 378*B^3*a*b^2*c*d^2 + 359*B^3*a^2*b*d^3)*n^3 + 6*(5*A*B^2*b^3*c^2*d - 54*A*B^2*a*b^
2*c*d^2 + 49*A*B^2*a^2*b*d^3)*n^2 + 18*(A^2*B*b^3*c^2*d - 6*A^2*B*a*b^2*c*d^2 + 5*A^2*B*a^2*b*d^3)*n)*x + 6*((
4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108*B^3*a^2*b*c*d^2)*n^3 + (85*B^3*b^3*d^3*n^3 + 66*A*B^2*b^3*d^3*n^2 + 1
8*A^2*B*b^3*d^3*n)*x^3 + 6*(2*A*B^2*b^3*c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2)*n^2 + 3*(18*A^2*B*a*
b^2*d^3*n + (22*B^3*b^3*c*d^2 + 63*B^3*a*b^2*d^3)*n^3 + 6*(2*A*B^2*b^3*c*d^2 + 9*A*B^2*a*b^2*d^3)*n^2)*x^2 + 1
8*(B^3*b^3*d^3*n*x^3 + 3*B^3*a*b^2*d^3*n*x^2 + 3*B^3*a^2*b*d^3*n*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*
a^2*b*c*d^2)*n)*log(e)^2 + 18*(A^2*B*b^3*c^3 - 3*A^2*B*a*b^2*c^2*d + 3*A^2*B*a^2*b*c*d^2)*n + 3*(18*A^2*B*a^2*
b*d^3*n - (5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 - 36*B^3*a^2*b*d^3)*n^3 - 6*(A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2*c
*d^2 - 6*A*B^2*a^2*b*d^3)*n^2)*x + 6*((11*B^3*b^3*d^3*n^2 + 6*A*B^2*b^3*d^3*n)*x^3 + (2*B^3*b^3*c^3 - 9*B^3*a*
b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^2)*x^2 +
6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n + 3*(6*A*B^2*a^2*b*d^3*n - (B^3*b^3*c^2*d - 6*
B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^2)*x)*log(e))*log(b*x + a) - 6*((4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108
*B^3*a^2*b*c*d^2)*n^3 + (85*B^3*b^3*d^3*n^3 + 66*A*B^2*b^3*d^3*n^2 + 18*A^2*B*b^3*d^3*n)*x^3 + 6*(2*A*B^2*b^3*
c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2)*n^2 + 3*(18*A^2*B*a*b^2*d^3*n + (22*B^3*b^3*c*d^2 + 63*B^3*a
*b^2*d^3)*n^3 + 6*(2*A*B^2*b^3*c*d^2 + 9*A*B^2*a*b^2*d^3)*n^2)*x^2 + 18*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3
*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^3)*log(b*x + a)^2 +
18*(B^3*b^3*d^3*n*x^3 + 3*B^3*a*b^2*d^3*n*x^2 + 3*B^3*a^2*b*d^3*n*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^
3*a^2*b*c*d^2)*n)*log(e)^2 + 18*(A^2*B*b^3*c^3 - 3*A^2*B*a*b^2*c^2*d + 3*A^2*B*a^2*b*c*d^2)*n + 3*(18*A^2*B*a^
2*b*d^3*n - (5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 - 36*B^3*a^2*b*d^3)*n^3 - 6*(A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2
*c*d^2 - 6*A*B^2*a^2*b*d^3)*n^2)*x + 6*((2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3
*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 +
3*(6*A*B^2*a*b^2*d^3*n^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*
c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3
*a^2*b*d^3*n^2*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(b*x + a) + 6*((11*B^
3*b^3*d^3*n^2 + 6*A*B^2*b^3*d^3*n)*x^3 + (2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^2 + 3*(6*A
*B^2*a*b^2*d^3*n + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^2)*x^2 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A
*B^2*a^2*b*c*d^2)*n + 3*(6*A*B^2*a^2*b*d^3*n - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^2)*x)*l
og(e))*log(d*x + c) + 6*(18*A^2*B*b^3*c^3 - 54*A^2*B*a*b^2*c^2*d + 54*A^2*B*a^2*b*c*d^2 - 18*A^2*B*a^3*d^3 + (
4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108*B^3*a^2*b*c*d^2 - 85*B^3*a^3*d^3)*n^2 + 6*(11*(B^3*b^3*c*d^2 - B^3*a*
b^2*d^3)*n^2 + 6*(A*B^2*b^3*c*d^2 - A*B^2*a*b^2*d^3)*n)*x^2 + 6*(2*A*B^2*b^3*c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*
B^2*a^2*b*c*d^2 - 11*A*B^2*a^3*d^3)*n - 3*((5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 + 49*B^3*a^2*b*d^3)*n^2 + 6*(
A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2*c*d^2 + 5*A*B^2*a^2*b*d^3)*n)*x)*log(e))/(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^
5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2
*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2
*d^3)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)**4,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3630 vs. \(2 (597) = 1194\).

Time = 0.43 (sec) , antiderivative size = 3630, normalized size of antiderivative = 5.94 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/108*(18*(6*d^3*e*n
*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*
b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n
- 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*
b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)
*x))*log((b*x + a)^n*e/(d*x + c)^n)^2/e + (6*(4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2
*n^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2
*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2
*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*e^2*n^2 - 54*a*b^2
*c*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2
*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^2 + 33*a^2*b*d^3*e
^2*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3
*d^3*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e/(d*x + c)^n)/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a
^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^
2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^
2*d^3)*x)*e) + (8*b^3*c^3*e^3*n^3 - 81*a*b^2*c^2*d*e^3*n^3 + 648*a^2*b*c*d^2*e^3*n^3 - 575*a^3*d^3*e^3*n^3 + 3
6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(b*x + a)^3 - 3
6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(d*x + c)^3 + 5
10*(b^3*c*d^2*e^3*n^3 - a*b^2*d^3*e^3*n^3)*x^2 - 198*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*
d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(b*x + a)^2 - 18*(11*b^3*d^3*e^3*n^3*x^3 + 33*a*b^2*d^3*e^3*n^3*x^2 + 33*a
^2*b*d^3*e^3*n^3*x + 11*a^3*d^3*e^3*n^3 - 6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n
^3*x + a^3*d^3*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 - 3*(19*b^3*c^2*d*e^3*n^3 - 378*a*b^2*c*d^2*e^3*n^3 + 359
*a^2*b*d^3*e^3*n^3)*x + 510*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e
^3*n^3)*log(b*x + a) - 6*(85*b^3*d^3*e^3*n^3*x^3 + 255*a*b^2*d^3*e^3*n^3*x^2 + 255*a^2*b*d^3*e^3*n^3*x + 85*a^
3*d^3*e^3*n^3 + 18*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*l
og(b*x + a)^2 - 66*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*l
og(b*x + a))*log(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*
c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x
^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)*e^2))/e)*B^3 - 1/18*A*B^2*(6*(6*d^3*
e*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3
*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*
e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a
^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d
^2)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e + (4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2*n
^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d
^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d
^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*e^2*n^2 - 54*a*b^2*c
*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n
^2*x + a^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^2 + 33*a^2*b*d^3*e^2
*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d
^3*e^2*n^2)*log(b*x + a))*log(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c
^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a
^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)*e^2)) - A*B^2*log((b*x
+ a)^n*e/(d*x + c)^n)^2/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/6*(6*d^3*e*n*log(b*x + a)/(b^4*c^3 -
3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*
d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*
b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5*
c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x))*A^2*B/e - A^2*B*log
((b*x + a)^n*e/(d*x + c)^n)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/3*A^3/(b^4*x^3 + 3*a*b^3*x^2 + 3
*a^2*b^2*x + a^3*b)

Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{4}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^4,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(b*x + a)^4, x)

Mupad [B] (verification not implemented)

Time = 7.73 (sec) , antiderivative size = 2069, normalized size of antiderivative = 3.39 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3/(a + b*x)^4,x)

[Out]

((36*A^3*a^2*d^2 + 36*A^3*b^2*c^2 + 575*B^3*a^2*d^2*n^3 + 8*B^3*b^2*c^2*n^3 + 198*A^2*B*a^2*d^2*n + 36*A^2*B*b
^2*c^2*n - 72*A^3*a*b*c*d + 510*A*B^2*a^2*d^2*n^2 + 24*A*B^2*b^2*c^2*n^2 - 73*B^3*a*b*c*d*n^3 - 126*A^2*B*a*b*
c*d*n - 138*A*B^2*a*b*c*d*n^2)/(6*(a*d - b*c)) + (x*(359*B^3*a*b*d^2*n^3 - 19*B^3*b^2*c*d*n^3 + 90*A^2*B*a*b*d
^2*n - 18*A^2*B*b^2*c*d*n + 294*A*B^2*a*b*d^2*n^2 - 30*A*B^2*b^2*c*d*n^2))/(2*(a*d - b*c)) + (x^2*(85*B^3*b^2*
d^2*n^3 + 18*A^2*B*b^2*d^2*n + 66*A*B^2*b^2*d^2*n^2))/(a*d - b*c))/(x^3*(18*b^5*c - 18*a*b^4*d) + x*(54*a^2*b^
3*c - 54*a^3*b^2*d) - x^2*(54*a^2*b^3*d - 54*a*b^4*c) + 18*a^3*b^2*c - 18*a^4*b*d) - log((e*(a + b*x)^n)/(c +
d*x)^n)^3*(B^3/(3*b*(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)) - (B^3*d^3)/(3*b*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c
^2*d - 3*a^2*b*c*d^2))) - log((e*(a + b*x)^n)/(c + d*x)^n)^2*((A*B^2)/(a^3*b + b^4*x^3 + 3*a^2*b^2*x + 3*a*b^3
*x^2) - (d^3*(6*A*B^2 + 11*B^3*n))/(6*b*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^3*d^3*(a*((b
*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*b*n*(a*d - b*c))/(3*d)) + x*(b*((b*n*(a*d - b*c)*(3*a*d - b*c))/(6*
d^2) + (a*b*n*(a*d - b*c))/(3*d)) + (2*a*b^2*n*(a*d - b*c))/(3*d) + (b^2*n*(a*d - b*c)*(3*a*d - b*c))/(3*d^2))
+ (b*n*(a*d - b*c)*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*d^3) + (b^3*n*x^2*(a*d - b*c))/d))/(b*(a^3*d^3 - b^3
*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*b + b^4*x^3 + 3*a^2*b^2*x + 3*a*b^3*x^2))) - log((e*(a + b*x)^n)/(c
+ d*x)^n)*((x*((a*d + b*c)*(3*A^2*B*a*d - 3*A^2*B*b*c - 6*B^3*a*d*n^2 + 3*B^3*b*c*n^2) - 3*B^3*a*b*c*d*n^2) +
x^2*(b*d*(3*A^2*B*a*d - 3*A^2*B*b*c - 6*B^3*a*d*n^2 + 3*B^3*b*c*n^2) - 3*B^3*b*d*n^2*(a*d + b*c)) + a*c*(3*A^
2*B*a*d - 3*A^2*B*b*c - 6*B^3*a*d*n^2 + 3*B^3*b*c*n^2) - 3*B^3*b^2*d^2*n^2*x^3)/(3*b*(a*d - b*c)*(a + b*x)^4*(
c + d*x)) + (d^3*(6*A*B^2 + 11*B^3*n)*(x*((a*((a*b*n*(a*d - b*c)^2)/d + (b*n*(a*d - b*c)^2*(3*a*d - b*c))/(2*d
^2)) + (b*n*(a*d - b*c)^2*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3)*(a*d + b*c) + a*c*(b*((a*b*n*(a*d - b*c)^2)/
d + (b*n*(a*d - b*c)^2*(3*a*d - b*c))/(2*d^2)) + (b^2*n*(a*d - b*c)^2*(3*a*d - b*c))/d^2 + (2*a*b^2*n*(a*d - b
*c)^2)/d)) + x^2*((a*d + b*c)*(b*((a*b*n*(a*d - b*c)^2)/d + (b*n*(a*d - b*c)^2*(3*a*d - b*c))/(2*d^2)) + (b^2*
n*(a*d - b*c)^2*(3*a*d - b*c))/d^2 + (2*a*b^2*n*(a*d - b*c)^2)/d) + b*d*(a*((a*b*n*(a*d - b*c)^2)/d + (b*n*(a*
d - b*c)^2*(3*a*d - b*c))/(2*d^2)) + (b*n*(a*d - b*c)^2*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3) + (3*a*b^3*c*n
*(a*d - b*c)^2)/d) + x^3*(b*d*(b*((a*b*n*(a*d - b*c)^2)/d + (b*n*(a*d - b*c)^2*(3*a*d - b*c))/(2*d^2)) + (b^2*
n*(a*d - b*c)^2*(3*a*d - b*c))/d^2 + (2*a*b^2*n*(a*d - b*c)^2)/d) + (3*b^3*n*(a*d + b*c)*(a*d - b*c)^2)/d) + a
*c*(a*((a*b*n*(a*d - b*c)^2)/d + (b*n*(a*d - b*c)^2*(3*a*d - b*c))/(2*d^2)) + (b*n*(a*d - b*c)^2*(3*a^2*d^2 +
b^2*c^2 - 3*a*b*c*d))/d^3) + 3*b^4*n*x^4*(a*d - b*c)^2))/(9*b^2*(a*d - b*c)*(a + b*x)^4*(c + d*x)*(a^3*d^3 - b
^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (B*d^3*n*atan((B*d^3*n*((b^4*c^3 + a^3*b*d^3 - a^2*b^2*c*d^2 - a*b
^3*c^2*d)/(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d) + 2*b*d*x)*(18*A^2 + 85*B^2*n^2 + 66*A*B*n)*(b^3*c^2 + a^2*b*d^2
- 2*a*b^2*c*d)*1i)/(b*(a*d - b*c)^3*(85*B^3*d^3*n^3 + 18*A^2*B*d^3*n + 66*A*B^2*d^3*n^2)))*(18*A^2 + 85*B^2*n
^2 + 66*A*B*n)*1i)/(9*b*(a*d - b*c)^3)