\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^5} \, dx\) [171]

Optimal result

Integrand size = 33, antiderivative size = 830 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\frac {6 B^3 d^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac {6 B^2 d^3 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^4 (a+b x)}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{32 (b c-a d)^4 (a+b x)^4}+\frac {3 B d^3 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)}-\frac {9 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{16 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4} \]

[Out]

6*B^3*d^3*n^3*(d*x+c)/(-a*d+b*c)^4/(b*x+a)-9/8*b*B^3*d^2*n^3*(d*x+c)^2/(-a*d+b*c)^4/(b*x+a)^2+2/9*b^2*B^3*d*n^
3*(d*x+c)^3/(-a*d+b*c)^4/(b*x+a)^3-3/128*b^3*B^3*n^3*(d*x+c)^4/(-a*d+b*c)^4/(b*x+a)^4+6*B^2*d^3*n^2*(d*x+c)*(A
+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)-9/4*b*B^2*d^2*n^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^
n)))/(-a*d+b*c)^4/(b*x+a)^2+2/3*b^2*B^2*d*n^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)
^3-3/32*b^3*B^2*n^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)^4+3*B*d^3*n*(d*x+c)*(A+B*
ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^4/(b*x+a)-9/4*b*B*d^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^
2/(-a*d+b*c)^4/(b*x+a)^2+b^2*B*d*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^4/(b*x+a)^3-3/16*b
^3*B*n*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^4/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((
d*x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a
)^2+b^2*d*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^3-1/4*b^3*(d*x+c)^4*(A+B*ln(e*(b*
x+a)^n/((d*x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^4

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 830, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2573, 2549, 2395, 2342, 2341} \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=-\frac {b^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^4}{4 (b c-a d)^4 (a+b x)^4}-\frac {3 b^3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^4}{16 (b c-a d)^4 (a+b x)^4}-\frac {3 b^3 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^4}{32 (b c-a d)^4 (a+b x)^4}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac {b^2 d \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^3}{(b c-a d)^4 (a+b x)^3}+\frac {b^2 B d n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^3}{(b c-a d)^4 (a+b x)^3}+\frac {2 b^2 B^2 d n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^3}{3 (b c-a d)^4 (a+b x)^3}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {3 b d^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^2}{2 (b c-a d)^4 (a+b x)^2}-\frac {9 b B d^2 n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^2}{4 (b c-a d)^4 (a+b x)^2}-\frac {9 b B^2 d^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^2}{4 (b c-a d)^4 (a+b x)^2}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac {d^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)}{(b c-a d)^4 (a+b x)}+\frac {3 B d^3 n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)}{(b c-a d)^4 (a+b x)}+\frac {6 B^2 d^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)}{(b c-a d)^4 (a+b x)}+\frac {6 B^3 d^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)} \]

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]

[Out]

(6*B^3*d^3*n^3*(c + d*x))/((b*c - a*d)^4*(a + b*x)) - (9*b*B^3*d^2*n^3*(c + d*x)^2)/(8*(b*c - a*d)^4*(a + b*x)
^2) + (2*b^2*B^3*d*n^3*(c + d*x)^3)/(9*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^3*n^3*(c + d*x)^4)/(128*(b*c - a*
d)^4*(a + b*x)^4) + (6*B^2*d^3*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^4*(a + b*x
)) - (9*b*B^2*d^2*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*(b*c - a*d)^4*(a + b*x)^2) + (2
*b^2*B^2*d*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^
2*n^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(32*(b*c - a*d)^4*(a + b*x)^4) + (3*B*d^3*n*(c + d
*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)) - (9*b*B*d^2*n*(c + d*x)^2*(A + B*Lo
g[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*(b*c - a*d)^4*(a + b*x)^2) + (b^2*B*d*n*(c + d*x)^3*(A + B*Log[(e*(a + b
*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B*n*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d
*x)^n])^2)/(16*(b*c - a*d)^4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c -
a*d)^4*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*(b*c - a*d)^4*(a + b*
x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^4*(a + b*x)^3) - (b^3*(c +
d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(4*(b*c - a*d)^4*(a + b*x)^4)

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)^5} \, 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)^3 \left (A+B \log \left (e x^n\right )\right )^3}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4},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^3 \left (A+B \log \left (e x^n\right )\right )^3}{x^5}-\frac {3 b^2 d \left (A+B \log \left (e x^n\right )\right )^3}{x^4}+\frac {3 b d^2 \left (A+B \log \left (e x^n\right )\right )^3}{x^3}-\frac {d^3 \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)^4},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^3 \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^3}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4},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 d\right ) \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)^4},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\right ) \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)^4},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^3 \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)^4},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4}+\text {Subst}\left (\frac {\left (3 b^3 B n\right ) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^2}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{4 (b c-a d)^4},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 d 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)^4},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 (9 b B d^2 n\right ) \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 )}{2 (b c-a d)^4},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^3 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)^4},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^3 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)^4 (a+b x)}-\frac {9 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{16 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4}+\text {Subst}\left (\frac {\left (3 b^3 B^2 n^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{8 (b c-a d)^4},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 (2 b^2 B^2 d 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 )}{(b c-a d)^4},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 (9 b B^2 d^2 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 )}{2 (b c-a d)^4},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^3 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)^4},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^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac {6 B^2 d^3 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)^4 (a+b x)}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{32 (b c-a d)^4 (a+b x)^4}+\frac {3 B d^3 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)^4 (a+b x)}-\frac {9 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{16 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 1370, normalized size of antiderivative = 1.65 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=-\frac {-288 B^3 d^4 n^3 (a+b x)^4 \log ^3(a+b x)+288 B^3 d^4 n^3 (a+b x)^4 \log ^3(c+d x)+72 B^2 d^4 n^2 (a+b x)^4 \log ^2(c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+72 B^2 d^4 n^2 (a+b x)^4 \log ^2(a+b x) \left (12 A+25 B n+12 B n \log (c+d x)+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+12 B d^4 n (a+b x)^4 \log (c+d x) \left (72 A^2+300 A B n+415 B^2 n^2+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+(b c-a d) \left (288 A^3 b^3 c^3-864 a A^3 b^2 c^2 d+864 a^2 A^3 b c d^2-288 a^3 A^3 d^3+216 A^2 b^3 B c^3 n-936 a A^2 b^2 B c^2 d n+1656 a^2 A^2 b B c d^2 n-1800 a^3 A^2 B d^3 n+108 A b^3 B^2 c^3 n^2-660 a A b^2 B^2 c^2 d n^2+1932 a^2 A b B^2 c d^2 n^2-4980 a^3 A B^2 d^3 n^2+27 b^3 B^3 c^3 n^3-229 a b^2 B^3 c^2 d n^3+1067 a^2 b B^3 c d^2 n^3-5845 a^3 B^3 d^3 n^3-288 A^2 b^3 B c^2 d n x+1440 a A^2 b^2 B c d^2 n x-3744 a^2 A^2 b B d^3 n x-336 A b^3 B^2 c^2 d n^2 x+2544 a A b^2 B^2 c d^2 n^2 x-13008 a^2 A b B^2 d^3 n^2 x-148 b^3 B^3 c^2 d n^3 x+1676 a b^2 B^3 c d^2 n^3 x-16468 a^2 b B^3 d^3 n^3 x+432 A^2 b^3 B c d^2 n x^2-3024 a A^2 b^2 B d^3 n x^2+936 A b^3 B^2 c d^2 n^2 x^2-11736 a A b^2 B^2 d^3 n^2 x^2+690 b^3 B^3 c d^2 n^3 x^2-15630 a b^2 B^3 d^3 n^3 x^2-864 A^2 b^3 B d^3 n x^3-3600 A b^3 B^2 d^3 n^2 x^3-4980 b^3 B^3 d^3 n^3 x^3+12 B \left (72 A^2 (b c-a d)^3+B^2 n^2 \left (-415 a^3 d^3+a^2 b d^2 (161 c-1084 d x)+a b^2 d \left (-55 c^2+212 c d x-978 d^2 x^2\right )+b^3 \left (9 c^3-28 c^2 d x+78 c d^2 x^2-300 d^3 x^3\right )\right )+12 A B n \left (-25 a^3 d^3+a^2 b d^2 (23 c-52 d x)+a b^2 d \left (-13 c^2+20 c d x-42 d^2 x^2\right )+b^3 \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \left (12 A (b c-a d)^3+B n \left (-25 a^3 d^3+a^2 b d^2 (23 c-52 d x)+a b^2 d \left (-13 c^2+20 c d x-42 d^2 x^2\right )+b^3 \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+288 B^3 (b c-a d)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-12 B d^4 n (a+b x)^4 \log (a+b x) \left (72 A^2+300 A B n+415 B^2 n^2+72 B^2 n^2 \log ^2(c+d x)+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+12 B n \log (c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{1152 b (b c-a d)^4 (a+b x)^4} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]

[Out]

-1/1152*(-288*B^3*d^4*n^3*(a + b*x)^4*Log[a + b*x]^3 + 288*B^3*d^4*n^3*(a + b*x)^4*Log[c + d*x]^3 + 72*B^2*d^4
*n^2*(a + b*x)^4*Log[c + d*x]^2*(12*A + 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 72*B^2*d^4*n^2*(a +
b*x)^4*Log[a + b*x]^2*(12*A + 25*B*n + 12*B*n*Log[c + d*x] + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 12*B*d^4
*n*(a + b*x)^4*Log[c + d*x]*(72*A^2 + 300*A*B*n + 415*B^2*n^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c +
d*x)^n] + 72*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(288*A^3*b^3*c^3 - 864*a*A^3*b^2*c^2*d + 86
4*a^2*A^3*b*c*d^2 - 288*a^3*A^3*d^3 + 216*A^2*b^3*B*c^3*n - 936*a*A^2*b^2*B*c^2*d*n + 1656*a^2*A^2*b*B*c*d^2*n
- 1800*a^3*A^2*B*d^3*n + 108*A*b^3*B^2*c^3*n^2 - 660*a*A*b^2*B^2*c^2*d*n^2 + 1932*a^2*A*b*B^2*c*d^2*n^2 - 498
0*a^3*A*B^2*d^3*n^2 + 27*b^3*B^3*c^3*n^3 - 229*a*b^2*B^3*c^2*d*n^3 + 1067*a^2*b*B^3*c*d^2*n^3 - 5845*a^3*B^3*d
^3*n^3 - 288*A^2*b^3*B*c^2*d*n*x + 1440*a*A^2*b^2*B*c*d^2*n*x - 3744*a^2*A^2*b*B*d^3*n*x - 336*A*b^3*B^2*c^2*d
*n^2*x + 2544*a*A*b^2*B^2*c*d^2*n^2*x - 13008*a^2*A*b*B^2*d^3*n^2*x - 148*b^3*B^3*c^2*d*n^3*x + 1676*a*b^2*B^3
*c*d^2*n^3*x - 16468*a^2*b*B^3*d^3*n^3*x + 432*A^2*b^3*B*c*d^2*n*x^2 - 3024*a*A^2*b^2*B*d^3*n*x^2 + 936*A*b^3*
B^2*c*d^2*n^2*x^2 - 11736*a*A*b^2*B^2*d^3*n^2*x^2 + 690*b^3*B^3*c*d^2*n^3*x^2 - 15630*a*b^2*B^3*d^3*n^3*x^2 -
864*A^2*b^3*B*d^3*n*x^3 - 3600*A*b^3*B^2*d^3*n^2*x^3 - 4980*b^3*B^3*d^3*n^3*x^3 + 12*B*(72*A^2*(b*c - a*d)^3 +
B^2*n^2*(-415*a^3*d^3 + a^2*b*d^2*(161*c - 1084*d*x) + a*b^2*d*(-55*c^2 + 212*c*d*x - 978*d^2*x^2) + b^3*(9*c
^3 - 28*c^2*d*x + 78*c*d^2*x^2 - 300*d^3*x^3)) + 12*A*B*n*(-25*a^3*d^3 + a^2*b*d^2*(23*c - 52*d*x) + a*b^2*d*(
-13*c^2 + 20*c*d*x - 42*d^2*x^2) + b^3*(3*c^3 - 4*c^2*d*x + 6*c*d^2*x^2 - 12*d^3*x^3)))*Log[(e*(a + b*x)^n)/(c
+ d*x)^n] + 72*B^2*(12*A*(b*c - a*d)^3 + B*n*(-25*a^3*d^3 + a^2*b*d^2*(23*c - 52*d*x) + a*b^2*d*(-13*c^2 + 20
*c*d*x - 42*d^2*x^2) + b^3*(3*c^3 - 4*c^2*d*x + 6*c*d^2*x^2 - 12*d^3*x^3)))*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2
+ 288*B^3*(b*c - a*d)^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 12*B*d^4*n*(a + b*x)^4*Log[a + b*x]*(72*A^2 + 3
00*A*B*n + 415*B^2*n^2 + 72*B^2*n^2*Log[c + d*x]^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 7
2*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 12*B*n*Log[c + d*x]*(12*A + 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c +
d*x)^n])))/(b*(b*c - a*d)^4*(a + b*x)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8291\) vs. \(2(810)=1620\).

Time = 191.85 (sec) , antiderivative size = 8292, normalized size of antiderivative = 9.99

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(8292\)
risch	\(\text {Expression too large to display}\)	\(236754\)

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6057 vs. \(2 (810) = 1620\).

Time = 0.53 (sec) , antiderivative size = 6057, normalized size of antiderivative = 7.30 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)**5,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5280 vs. \(2 (810) = 1620\).

Time = 0.57 (sec) , antiderivative size = 5280, normalized size of antiderivative = 6.36 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/4*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/11
52*(72*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) -
12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*
d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*
a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)*x)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*
d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3
- 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 -
a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*log((b*x + a)^n*e/(d
*x + c)^n)^2/e - (12*(9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c^2*d^2*e^2*n^2 - 576*a^3*b*c*d
^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c*d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x^3 + 6*(13*b^4*c^2*d^2*e^2*n^2
- 176*a*b^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3
+ 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a)^2 + 72*(b^4*d^4*e^2*n^2*x^
4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(d*x + c
)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2*e^2*n^2 + 324*a^2*b^2*c*d^3*e^2*n^2 - 271*a^3*b*d^4*e^2*n^2)*x
- 300*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^
4*d^4*e^2*n^2)*log(b*x + a) + 12*(25*b^4*d^4*e^2*n^2*x^4 + 100*a*b^3*d^4*e^2*n^2*x^3 + 150*a^2*b^2*d^4*e^2*n^2
*x^2 + 100*a^3*b*d^4*e^2*n^2*x + 25*a^4*d^4*e^2*n^2 - 12*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^
2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e
/(d*x + c)^n)/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4
*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3
*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a
^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7
*b^2*d^4)*x)*e) + (27*b^4*c^4*e^3*n^3 - 256*a*b^3*c^3*d*e^3*n^3 + 1296*a^2*b^2*c^2*d^2*e^3*n^3 - 6912*a^3*b*c*
d^3*e^3*n^3 + 5845*a^4*d^4*e^3*n^3 - 4980*(b^4*c*d^3*e^3*n^3 - a*b^3*d^4*e^3*n^3)*x^3 - 288*(b^4*d^4*e^3*n^3*x
^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x +
a)^3 + 288*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x
+ a^4*d^4*e^3*n^3)*log(d*x + c)^3 + 30*(23*b^4*c^2*d^2*e^3*n^3 - 544*a*b^3*c*d^3*e^3*n^3 + 521*a^2*b^2*d^4*e^3
*n^3)*x^2 + 1800*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*
n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a)^2 + 72*(25*b^4*d^4*e^3*n^3*x^4 + 100*a*b^3*d^4*e^3*n^3*x^3 + 150*a^2*b^2
*d^4*e^3*n^3*x^2 + 100*a^3*b*d^4*e^3*n^3*x + 25*a^4*d^4*e^3*n^3 - 12*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^
3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 - 4*
(37*b^4*c^3*d*e^3*n^3 - 456*a*b^3*c^2*d^2*e^3*n^3 + 4536*a^2*b^2*c*d^3*e^3*n^3 - 4117*a^3*b*d^4*e^3*n^3)*x - 4
980*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d
^4*e^3*n^3)*log(b*x + a) + 12*(415*b^4*d^4*e^3*n^3*x^4 + 1660*a*b^3*d^4*e^3*n^3*x^3 + 2490*a^2*b^2*d^4*e^3*n^3
*x^2 + 1660*a^3*b*d^4*e^3*n^3*x + 415*a^4*d^4*e^3*n^3 + 72*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*
a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a)^2 - 300*(b^4*d^4*e^3*n^3*x^4 +
4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a))*
log(d*x + c))/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4
*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3
*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a
^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7
*b^2*d^4)*x)*e^2))/e)*B^3 + 1/96*A*B^2*(12*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d
^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a
^3*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25
*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e
*n)*x)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d
^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a
^6*b^2*d^3)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e - (9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c
^2*d^2*e^2*n^2 - 576*a^3*b*c*d^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c*d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x
^3 + 6*(13*b^4*c^2*d^2*e^2*n^2 - 176*a*b^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*
x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x +
a)^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x
+ a^4*d^4*e^2*n^2)*log(d*x + c)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2*e^2*n^2 + 324*a^2*b^2*c*d^3*e^2*
n^2 - 271*a^3*b*d^4*e^2*n^2)*x - 300*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^
2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a) + 12*(25*b^4*d^4*e^2*n^2*x^4 + 100*a*b^3*d^4*e^2*n^2
*x^3 + 150*a^2*b^2*d^4*e^2*n^2*x^2 + 100*a^3*b*d^4*e^2*n^2*x + 25*a^4*d^4*e^2*n^2 - 12*(b^4*d^4*e^2*n^2*x^4 +
4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a))*l
og(d*x + c))/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4*
a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*
b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^
5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*
b^2*d^4)*x)*e^2)) - 3/4*A*B^2*log((b*x + a)^n*e/(d*x + c)^n)^2/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*
b^2*x + a^4*b) + 1/16*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3
+ a^4*b*d^4) - 12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*
d^4) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3
*c*d^2*e*n - 7*a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)*x)/(a^4*b^4*c^3 -
3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4
+ 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*
a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*A^2*B
/e - 3/4*A^2*B*log((b*x + a)^n*e/(d*x + c)^n)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) -
1/4*A^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{5}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(b*x + a)^5, x)

Mupad [B] (verification not implemented)

Time = 8.17 (sec) , antiderivative size = 4257, normalized size of antiderivative = 5.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3/(a + b*x)^5,x)

[Out]

log((e*(a + b*x)^n)/(c + d*x)^n)*((x*((a*d + b*c)*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*
d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a*b*c
*d) + a*c*(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3*b^2*c*d*n^2)/2)) +
x^2*((a*d + b*c)*(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3*b^2*c*d*n^2)
/2) + b*d*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A^2
*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a*b*c*d) + 6*B^3*a*b^2*c*d^2*n^2) + x^3*(b*d*
(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3*b^2*c*d*n^2)/2) + 6*B^3*b^2*d
^2*n^2*(a*d + b*c)) + a*c*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^
2*n^2)/2 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a*b*c*d) + 6*B^3*b^3*d^3*n^2*
x^4)/(8*b*(a*d - b*c)^2*(a + b*x)^5*(c + d*x)) - (d^4*(12*A*B^2 + 25*B^3*n)*(x^3*((a*d + b*c)*(b*(b*((2*a*b*n*
(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (4*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2
) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^3*n*(a*d - b*c)^3*(4*a*d - b*c))/d^2 + (6*a*b^3*n*(a*d - b*c)^3)/d) +
b*d*(b*(a*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*
a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + a*(b*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c
))/(3*d^2)) + (4*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^2*n*(a*d - b
*c)^3*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/d^3) + (8*a*b^4*c*n*(a*d - b*c)^3)/d) + x^2*((a*d + b*c)*(b*(a*((2*a*
b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*c^
2 - 4*a*b*c*d))/(3*d^3)) + a*(b*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (4
*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^2*n*(a*d - b*c)^3*(6*a^2*d^2
+ b^2*c^2 - 4*a*b*c*d))/d^3) + a*c*(b*(b*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*
d^2)) + (4*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^3*n*(a*d - b*c)^3*
(4*a*d - b*c))/d^2 + (6*a*b^3*n*(a*d - b*c)^3)/d) + b*d*(a*(a*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^
3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + (2*b*n*(a*d - b
*c)^3*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/d^4)) + x*((a*(a*((2*a*b*n*(a*d - b*c)^3)/d + (2*
b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) +
(2*b*n*(a*d - b*c)^3*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/d^4)*(a*d + b*c) + a*c*(b*(a*((2*
a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*
c^2 - 4*a*b*c*d))/(3*d^3)) + a*(b*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) +
(4*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^2*n*(a*d - b*c)^3*(6*a^2*d
^2 + b^2*c^2 - 4*a*b*c*d))/d^3)) + x^4*(b*d*(b*(b*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b
*c))/(3*d^2)) + (4*b^2*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^3*n*(a*d -
b*c)^3*(4*a*d - b*c))/d^2 + (6*a*b^3*n*(a*d - b*c)^3)/d) + (8*b^4*n*(a*d + b*c)*(a*d - b*c)^3)/d) + a*c*(a*(a
*((2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 +
b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + (2*b*n*(a*d - b*c)^3*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/
d^4) + 8*b^5*n*x^5*(a*d - b*c)^3))/(64*b^2*(a*d - b*c)^2*(a + b*x)^5*(c + d*x)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*
c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))) - log((e*(a + b*x)^n)/(c + d*x)^n)^3*(B^3/(4*b*(a^4 + b^4*x^4 + 4*a
*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)) - (B^3*d^4)/(4*b*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d
- 4*a^3*b*c*d^3))) - log((e*(a + b*x)^n)/(c + d*x)^n)^2*((3*A*B^2)/(4*(a^4*b + b^5*x^4 + 4*a^3*b^2*x + 4*a*b^
4*x^3 + 6*a^2*b^3*x^2)) - (d^4*(12*A*B^2 + 25*B^3*n))/(16*b*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c
^3*d - 4*a^3*b*c*d^3)) + (3*B^3*d^4*(x^2*(b*(b*((b*n*(a*d - b*c)*(4*a*d - b*c))/(3*d^2) + (a*b*n*(a*d - b*c))/
d) + (2*a*b^2*n*(a*d - b*c))/d + (2*b^2*n*(a*d - b*c)*(4*a*d - b*c))/(3*d^2)) + (3*a*b^3*n*(a*d - b*c))/d + (b
^3*n*(a*d - b*c)*(4*a*d - b*c))/d^2) + a*(a*((b*n*(a*d - b*c)*(4*a*d - b*c))/(3*d^2) + (a*b*n*(a*d - b*c))/d)
+ (b*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + x*(b*(a*((b*n*(a*d - b*c)*(4*a*d - b*c))/(3*d
^2) + (a*b*n*(a*d - b*c))/d) + (b*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + a*(b*((b*n*(a*d
- b*c)*(4*a*d - b*c))/(3*d^2) + (a*b*n*(a*d - b*c))/d) + (2*a*b^2*n*(a*d - b*c))/d + (2*b^2*n*(a*d - b*c)*(4*a
*d - b*c))/(3*d^2)) + (b^2*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/d^3) + (b*n*(a*d - b*c)*(4*a^3*d^3
- b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/d^4 + (4*b^4*n*x^3*(a*d - b*c))/d))/(16*b*(a^4*b + b^5*x^4 + 4*a^
3*b^2*x + 4*a*b^4*x^3 + 6*a^2*b^3*x^2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)
)) - ((288*A^3*a^3*d^3 - 288*A^3*b^3*c^3 + 5845*B^3*a^3*d^3*n^3 - 27*B^3*b^3*c^3*n^3 + 1800*A^2*B*a^3*d^3*n -
216*A^2*B*b^3*c^3*n + 864*A^3*a*b^2*c^2*d - 864*A^3*a^2*b*c*d^2 + 4980*A*B^2*a^3*d^3*n^2 - 108*A*B^2*b^3*c^3*n
^2 + 229*B^3*a*b^2*c^2*d*n^3 - 1067*B^3*a^2*b*c*d^2*n^3 + 660*A*B^2*a*b^2*c^2*d*n^2 - 1932*A*B^2*a^2*b*c*d^2*n
^2 + 936*A^2*B*a*b^2*c^2*d*n - 1656*A^2*B*a^2*b*c*d^2*n)/(12*(a*d - b*c)) + (x^2*(2605*B^3*a*b^2*d^3*n^3 - 115
*B^3*b^3*c*d^2*n^3 + 504*A^2*B*a*b^2*d^3*n - 72*A^2*B*b^3*c*d^2*n + 1956*A*B^2*a*b^2*d^3*n^2 - 156*A*B^2*b^3*c
*d^2*n^2))/(2*(a*d - b*c)) + (x*(4117*B^3*a^2*b*d^3*n^3 + 37*B^3*b^3*c^2*d*n^3 - 419*B^3*a*b^2*c*d^2*n^3 + 936
*A^2*B*a^2*b*d^3*n + 72*A^2*B*b^3*c^2*d*n + 3252*A*B^2*a^2*b*d^3*n^2 + 84*A*B^2*b^3*c^2*d*n^2 - 636*A*B^2*a*b^
2*c*d^2*n^2 - 360*A^2*B*a*b^2*c*d^2*n))/(3*(a*d - b*c)) + (x^3*(415*B^3*b^3*d^3*n^3 + 72*A^2*B*b^3*d^3*n + 300
*A*B^2*b^3*d^3*n^2))/(a*d - b*c))/(x*(384*a^3*b^4*c^2 + 384*a^5*b^2*d^2 - 768*a^4*b^3*c*d) + x^3*(384*a*b^6*c^
2 + 384*a^3*b^4*d^2 - 768*a^2*b^5*c*d) + x^4*(96*b^7*c^2 + 96*a^2*b^5*d^2 - 192*a*b^6*c*d) + x^2*(576*a^2*b^5*
c^2 + 576*a^4*b^3*d^2 - 1152*a^3*b^4*c*d) + 96*a^6*b*d^2 + 96*a^4*b^3*c^2 - 192*a^5*b^2*c*d) + (B*d^4*n*atan((
B*d^4*n*((b^5*c^4 - a^4*b*d^4 + 2*a^3*b^2*c*d^3 - 2*a*b^4*c^3*d)/(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*
b^3*c^2*d) + 2*b*d*x)*(72*A^2 + 415*B^2*n^2 + 300*A*B*n)*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*
d)*1i)/(b*(a*d - b*c)^4*(415*B^3*d^4*n^3 + 72*A^2*B*d^4*n + 300*A*B^2*d^4*n^2)))*(72*A^2 + 415*B^2*n^2 + 300*A
*B*n)*1i)/(48*b*(a*d - b*c)^4)