3.2.63 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(a+b x)^5} \, dx\) [163]

Optimal. Leaf size=587 \[ \frac {2 B^2 d^3 n^2 (c+d x)}{(b c-a d)^4 (a+b x)}-\frac {3 b B^2 d^2 n^2 (c+d x)^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {b^3 B^2 n^2 (c+d x)^4}{32 (b c-a d)^4 (a+b x)^4}+\frac {2 B d^3 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^4 (a+b x)}-\frac {3 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 (b c-a d)^4 (a+b x)^3}-\frac {b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{8 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.31, antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2573, 2549, 2395, 2342, 2341} \begin {gather*} -\frac {b^3 (c+d x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{4 (a+b x)^4 (b c-a d)^4}-\frac {b^3 B n (c+d x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{8 (a+b x)^4 (b c-a d)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x)^3 (b c-a d)^4}+\frac {2 b^2 B d n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 (a+b x)^3 (b c-a d)^4}+\frac {d^3 (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)^4}+\frac {2 B d^3 n (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)^4}-\frac {3 b d^2 (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)^4}-\frac {3 b B d^2 n (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)^4}-\frac {b^3 B^2 n^2 (c+d x)^4}{32 (a+b x)^4 (b c-a d)^4}+\frac {2 b^2 B^2 d n^2 (c+d x)^3}{9 (a+b x)^3 (b c-a d)^4}+\frac {2 B^2 d^3 n^2 (c+d x)}{(a+b x) (b c-a d)^4}-\frac {3 b B^2 d^2 n^2 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^5,x]

[Out]

(2*B^2*d^3*n^2*(c + d*x))/((b*c - a*d)^4*(a + b*x)) - (3*b*B^2*d^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*(a + b*x)
^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3)/(9*(b*c - a*d)^4*(a + b*x)^3) - (b^3*B^2*n^2*(c + d*x)^4)/(32*(b*c - a*d)^
4*(a + b*x)^4) + (2*B*d^3*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^4*(a + b*x)) - (3
*b*B*d^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*(b*c - a*d)^4*(a + b*x)^2) + (2*b^2*B*d*n*
(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*B*n*(c + d*x)^4*(A
+ B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(8*(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])^2)/((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])^
2)/(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])^2)/((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])^2)/(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*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^5} \, dx &=\int \left (\frac {A^2}{(a+b x)^5}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}\right ) \, dx\\ &=-\frac {A^2}{4 b (a+b x)^4}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(A B (b c-a d) n) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{2 b}+\frac {\left (B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(A B (b c-a d) n) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b}+\frac {\left (B^2 (b c-a d) n\right ) \int \left (\frac {b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^5}-\frac {b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {1}{2} \left (B^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\frac {\left (B^2 d^4 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (B^2 d^5 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac {\left (B^2 d^3 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac {\left (B^2 d^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2}-\frac {\left (B^2 d n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{2 (b c-a d)}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac {B^2 d^4 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6 b}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)}+\frac {\left (B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{8 b}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {B^2 d^3 n^2}{2 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac {B^2 d^4 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac {\left (B^2 d n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{6 b}-\frac {\left (B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{2 b (b c-a d)^3}+\frac {\left (B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3}+\frac {\left (B^2 d^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{4 b (b c-a d)}+\frac {\left (B^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}-\frac {B^2 n^2}{32 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}+\frac {7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac {13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac {B^2 d^4 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {\left (B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac {\left (B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}-\frac {B^2 n^2}{32 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}+\frac {7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac {13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac {B^2 d^4 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {\left (B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac {\left (B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac {A^2}{4 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}-\frac {B^2 n^2}{32 b (a+b x)^4}+\frac {A B d n}{6 b (b c-a d) (a+b x)^3}+\frac {7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac {A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac {A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac {A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac {13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac {A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac {13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac {B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac {B^2 d^4 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {B^2 d^4 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac {B^2 d^4 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.53, size = 1011, normalized size = 1.72 \begin {gather*} -\frac {72 b B^2 n^2 \left (-4 a^3 d^3 (c+d x)+6 a^2 b d^2 \left (c^2-d^2 x^2\right )-4 a b^2 d \left (c^3+d^3 x^3\right )+b^3 \left (c^4-d^4 x^4\right )\right ) \log ^2(a+b x)+72 b B^2 n^2 \left (-4 a^3 d^3 (c+d x)+6 a^2 b d^2 \left (c^2-d^2 x^2\right )-4 a b^2 d \left (c^3+d^3 x^3\right )+b^3 \left (c^4-d^4 x^4\right )\right ) \log ^2(c+d x)-4 B d (b c-a d)^3 n (a+b x) \left (12 A+7 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+6 B d^2 (b c-a d)^2 n (a+b x)^2 \left (12 A+13 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )-12 B d^3 (b c-a d) n (a+b x)^3 \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )-12 B d^4 n (a+b x)^4 \log (a+b x) \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+12 B d^4 n (a+b x)^4 \log (c+d x) \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+9 (b c-a d)^4 \left (8 A^2+4 A B n+B^2 n^2+16 A B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+4 B^2 n \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+8 B^2 \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2\right )-12 B (b c-a d) n \log (a+b x) \left (4 B d (b c-a d)^2 n (a+b x)+6 B d^2 (-b c+a d) n (a+b x)^2+12 B d^3 n (a+b x)^3-3 (b c-a d)^3 \left (4 A+B n+4 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )\right )+12 B n \log (c+d x) \left (4 B d (b c-a d)^3 n (a+b x)-6 B d^2 (b c-a d)^2 n (a+b x)^2+12 B d^3 (b c-a d) n (a+b x)^3-12 B (b c-a d)^4 n \log (a+b x)+12 B d^4 n (a+b x)^4 \log (a+b x)-3 (b c-a d)^4 \left (4 A+B n+4 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )\right )}{288 b (b c-a d)^4 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^5,x]

[Out]

-1/288*(72*b*B^2*n^2*(-4*a^3*d^3*(c + d*x) + 6*a^2*b*d^2*(c^2 - d^2*x^2) - 4*a*b^2*d*(c^3 + d^3*x^3) + b^3*(c^
4 - d^4*x^4))*Log[a + b*x]^2 + 72*b*B^2*n^2*(-4*a^3*d^3*(c + d*x) + 6*a^2*b*d^2*(c^2 - d^2*x^2) - 4*a*b^2*d*(c
^3 + d^3*x^3) + b^3*(c^4 - d^4*x^4))*Log[c + d*x]^2 - 4*B*d*(b*c - a*d)^3*n*(a + b*x)*(12*A + 7*B*n + 12*B*(-(
n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n])) + 6*B*d^2*(b*c - a*d)^2*n*(a + b*x)^2*(1
2*A + 13*B*n + 12*B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n])) - 12*B*d^3*(b*c -
a*d)*n*(a + b*x)^3*(12*A + 25*B*n + 12*B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^
n])) - 12*B*d^4*n*(a + b*x)^4*Log[a + b*x]*(12*A + 25*B*n + 12*B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*
(a + b*x)^n)/(c + d*x)^n])) + 12*B*d^4*n*(a + b*x)^4*Log[c + d*x]*(12*A + 25*B*n + 12*B*(-(n*Log[a + b*x]) + n
*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n])) + 9*(b*c - a*d)^4*(8*A^2 + 4*A*B*n + B^2*n^2 + 16*A*B*(-(n*
Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 4*B^2*n*(-(n*Log[a + b*x]) + n*Log[c + d*
x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 8*B^2*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c +
d*x)^n])^2) - 12*B*(b*c - a*d)*n*Log[a + b*x]*(4*B*d*(b*c - a*d)^2*n*(a + b*x) + 6*B*d^2*(-(b*c) + a*d)*n*(a +
b*x)^2 + 12*B*d^3*n*(a + b*x)^3 - 3*(b*c - a*d)^3*(4*A + B*n + 4*B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[
(e*(a + b*x)^n)/(c + d*x)^n]))) + 12*B*n*Log[c + d*x]*(4*B*d*(b*c - a*d)^3*n*(a + b*x) - 6*B*d^2*(b*c - a*d)^2
*n*(a + b*x)^2 + 12*B*d^3*(b*c - a*d)*n*(a + b*x)^3 - 12*B*(b*c - a*d)^4*n*Log[a + b*x] + 12*B*d^4*n*(a + b*x)
^4*Log[a + b*x] - 3*(b*c - a*d)^4*(4*A + B*n + 4*B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(
c + d*x)^n]))))/(b*(b*c - a*d)^4*(a + b*x)^4)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.32, size = 33370, normalized size = 56.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(33370\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2187 vs. \(2 (579) = 1158\).
time = 0.52, size = 2187, normalized size = 3.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^5,x, algorithm="maxima")

[Out]

1/24*(12*d^4*n*e*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*n*e*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*n*x^3*e - 6*(b^3*c*d^2*n - 7*a*b^2*d^3*n)*x^2*e + 4*(b^3*c^2*d*n - 5*a*b^2*c*d^2*n + 13*a^2*b*d^3*n)*x*e - (
3*b^3*c^3*n - 13*a*b^2*c^2*d*n + 23*a^2*b*c*d^2*n - 25*a^3*d^3*n)*e)/(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*B*e^(-1) + 1/288*(12*(12*d^4*n
*e*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*n*e*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*n*x^3*e - 6*
(b^3*c*d^2*n - 7*a*b^2*d^3*n)*x^2*e + 4*(b^3*c^2*d*n - 5*a*b^2*c*d^2*n + 13*a^2*b*d^3*n)*x*e - (3*b^3*c^3*n -
13*a*b^2*c^2*d*n + 23*a^2*b*c*d^2*n - 25*a^3*d^3*n)*e)/(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))*e^(-1)*log((b*x + a)^n*e/(d*x + c)^n) + (300*(
b^4*c*d^3*n^2 - a*b^3*d^4*n^2)*x^3*e^2 - 6*(13*b^4*c^2*d^2*n^2 - 176*a*b^3*c*d^3*n^2 + 163*a^2*b^2*d^4*n^2)*x^
2*e^2 + 4*(7*b^4*c^3*d*n^2 - 60*a*b^3*c^2*d^2*n^2 + 324*a^2*b^2*c*d^3*n^2 - 271*a^3*b*d^4*n^2)*x*e^2 - 72*(b^4
*d^4*n^2*x^4*e^2 + 4*a*b^3*d^4*n^2*x^3*e^2 + 6*a^2*b^2*d^4*n^2*x^2*e^2 + 4*a^3*b*d^4*n^2*x*e^2 + a^4*d^4*n^2*e
^2)*log(b*x + a)^2 - 72*(b^4*d^4*n^2*x^4*e^2 + 4*a*b^3*d^4*n^2*x^3*e^2 + 6*a^2*b^2*d^4*n^2*x^2*e^2 + 4*a^3*b*d
^4*n^2*x*e^2 + a^4*d^4*n^2*e^2)*log(d*x + c)^2 - (9*b^4*c^4*n^2 - 64*a*b^3*c^3*d*n^2 + 216*a^2*b^2*c^2*d^2*n^2
- 576*a^3*b*c*d^3*n^2 + 415*a^4*d^4*n^2)*e^2 + 300*(b^4*d^4*n^2*x^4*e^2 + 4*a*b^3*d^4*n^2*x^3*e^2 + 6*a^2*b^2
*d^4*n^2*x^2*e^2 + 4*a^3*b*d^4*n^2*x*e^2 + a^4*d^4*n^2*e^2)*log(b*x + a) - 12*(25*b^4*d^4*n^2*x^4*e^2 + 100*a*
b^3*d^4*n^2*x^3*e^2 + 150*a^2*b^2*d^4*n^2*x^2*e^2 + 100*a^3*b*d^4*n^2*x*e^2 + 25*a^4*d^4*n^2*e^2 - 12*(b^4*d^4
*n^2*x^4*e^2 + 4*a*b^3*d^4*n^2*x^3*e^2 + 6*a^2*b^2*d^4*n^2*x^2*e^2 + 4*a^3*b*d^4*n^2*x*e^2 + a^4*d^4*n^2*e^2)*
log(b*x + a))*log(d*x + c))*e^(-2)/(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))*B^2 - 1/4*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) - 1/2*A*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^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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2062 vs. \(2 (579) = 1158\).
time = 0.40, size = 2062, normalized size = 3.51 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/288*(72*(A^2 + 2*A*B + B^2)*b^4*c^4 - 288*(A^2 + 2*A*B + B^2)*a*b^3*c^3*d + 432*(A^2 + 2*A*B + B^2)*a^2*b^2
*c^2*d^2 - 288*(A^2 + 2*A*B + B^2)*a^3*b*c*d^3 + 72*(A^2 + 2*A*B + B^2)*a^4*d^4 - 12*(25*(B^2*b^4*c*d^3 - B^2*
a*b^3*d^4)*n^2 + 12*((A*B + B^2)*b^4*c*d^3 - (A*B + B^2)*a*b^3*d^4)*n)*x^3 + (9*B^2*b^4*c^4 - 64*B^2*a*b^3*c^3
*d + 216*B^2*a^2*b^2*c^2*d^2 - 576*B^2*a^3*b*c*d^3 + 415*B^2*a^4*d^4)*n^2 + 6*((13*B^2*b^4*c^2*d^2 - 176*B^2*a
*b^3*c*d^3 + 163*B^2*a^2*b^2*d^4)*n^2 + 12*((A*B + B^2)*b^4*c^2*d^2 - 8*(A*B + B^2)*a*b^3*c*d^3 + 7*(A*B + B^2
)*a^2*b^2*d^4)*n)*x^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*
a^3*b*d^4*n^2*x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n^2)*log(b*x +
a)^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*a^3*b*d^4*n^2*x
- (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n^2)*log(d*x + c)^2 + 12*(3*(A
*B + B^2)*b^4*c^4 - 16*(A*B + B^2)*a*b^3*c^3*d + 36*(A*B + B^2)*a^2*b^2*c^2*d^2 - 48*(A*B + B^2)*a^3*b*c*d^3 +
25*(A*B + B^2)*a^4*d^4)*n - 4*((7*B^2*b^4*c^3*d - 60*B^2*a*b^3*c^2*d^2 + 324*B^2*a^2*b^2*c*d^3 - 271*B^2*a^3*
b*d^4)*n^2 + 12*((A*B + B^2)*b^4*c^3*d - 6*(A*B + B^2)*a*b^3*c^2*d^2 + 18*(A*B + B^2)*a^2*b^2*c*d^3 - 13*(A*B
+ B^2)*a^3*b*d^4)*n)*x - 12*((25*B^2*b^4*d^4*n^2 + 12*(A*B + B^2)*b^4*d^4*n)*x^4 + 4*(12*(A*B + B^2)*a*b^3*d^4
*n + (3*B^2*b^4*c*d^3 + 22*B^2*a*b^3*d^4)*n^2)*x^3 - (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*
d^2 - 48*B^2*a^3*b*c*d^3)*n^2 + 6*(12*(A*B + B^2)*a^2*b^2*d^4*n - (B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 - 18*B^
2*a^2*b^2*d^4)*n^2)*x^2 - 12*((A*B + B^2)*b^4*c^4 - 4*(A*B + B^2)*a*b^3*c^3*d + 6*(A*B + B^2)*a^2*b^2*c^2*d^2
- 4*(A*B + B^2)*a^3*b*c*d^3)*n + 4*(12*(A*B + B^2)*a^3*b*d^4*n + (B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2
*a^2*b^2*c*d^3 + 12*B^2*a^3*b*d^4)*n^2)*x)*log(b*x + a) + 12*((25*B^2*b^4*d^4*n^2 + 12*(A*B + B^2)*b^4*d^4*n)*
x^4 + 4*(12*(A*B + B^2)*a*b^3*d^4*n + (3*B^2*b^4*c*d^3 + 22*B^2*a*b^3*d^4)*n^2)*x^3 - (3*B^2*b^4*c^4 - 16*B^2*
a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^2 - 48*B^2*a^3*b*c*d^3)*n^2 + 6*(12*(A*B + B^2)*a^2*b^2*d^4*n - (B^2*b^4*c^
2*d^2 - 8*B^2*a*b^3*c*d^3 - 18*B^2*a^2*b^2*d^4)*n^2)*x^2 - 12*((A*B + B^2)*b^4*c^4 - 4*(A*B + B^2)*a*b^3*c^3*d
+ 6*(A*B + B^2)*a^2*b^2*c^2*d^2 - 4*(A*B + B^2)*a^3*b*c*d^3)*n + 4*(12*(A*B + B^2)*a^3*b*d^4*n + (B^2*b^4*c^3
*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 12*B^2*a^3*b*d^4)*n^2)*x + 12*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a
*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*a^3*b*d^4*n^2*x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^
2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n^2)*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)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**2/(b*x+a)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^5,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^2/(b*x + a)^5, x)

________________________________________________________________________________________

Mupad [B]
time = 9.61, size = 1579, normalized size = 2.69 \begin {gather*} -\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {A\,B}{2\,\left (a^4\,b+4\,a^3\,b^2\,x+6\,a^2\,b^3\,x^2+4\,a\,b^4\,x^3+b^5\,x^4\right )}+\frac {B^2\,d^4\,\left (x^2\,\left (b\,\left (b\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )+\frac {a\,b^2\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{3\,d^2}\right )+\frac {3\,a\,b^3\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+a\,\left (a\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )+\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{6\,d^3}\right )+x\,\left (b\,\left (a\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )+\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{6\,d^3}\right )+a\,\left (b\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )+\frac {a\,b^2\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{3\,d^2}\right )+\frac {b^2\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3}\right )+\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (4\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,d^4}+\frac {2\,b^4\,n\,x^3\,\left (a\,d-b\,c\right )}{d}\right )}{4\,b\,\left (a^4\,b+4\,a^3\,b^2\,x+6\,a^2\,b^3\,x^2+4\,a\,b^4\,x^3+b^5\,x^4\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {B^2}{4\,b\,\left (a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4\right )}-\frac {B^2\,d^4}{4\,b\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )-\frac {\frac {72\,A^2\,a^3\,d^3-216\,A^2\,a^2\,b\,c\,d^2+216\,A^2\,a\,b^2\,c^2\,d-72\,A^2\,b^3\,c^3+300\,A\,B\,a^3\,d^3\,n-276\,A\,B\,a^2\,b\,c\,d^2\,n+156\,A\,B\,a\,b^2\,c^2\,d\,n-36\,A\,B\,b^3\,c^3\,n+415\,B^2\,a^3\,d^3\,n^2-161\,B^2\,a^2\,b\,c\,d^2\,n^2+55\,B^2\,a\,b^2\,c^2\,d\,n^2-9\,B^2\,b^3\,c^3\,n^2}{12\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (-13\,c\,B^2\,b^3\,d^2\,n^2+163\,a\,B^2\,b^2\,d^3\,n^2-12\,A\,c\,B\,b^3\,d^2\,n+84\,A\,a\,B\,b^2\,d^3\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (271\,B^2\,a^2\,b\,d^3\,n^2-53\,B^2\,a\,b^2\,c\,d^2\,n^2+7\,B^2\,b^3\,c^2\,d\,n^2+156\,A\,B\,a^2\,b\,d^3\,n-60\,A\,B\,a\,b^2\,c\,d^2\,n+12\,A\,B\,b^3\,c^2\,d\,n\right )}{3\,\left (a\,d-b\,c\right )}+\frac {d\,x^3\,\left (25\,B^2\,b^3\,d^2\,n^2+12\,A\,B\,b^3\,d^2\,n\right )}{a\,d-b\,c}}{x\,\left (96\,a^5\,b^2\,d^2-192\,a^4\,b^3\,c\,d+96\,a^3\,b^4\,c^2\right )+x^3\,\left (96\,a^3\,b^4\,d^2-192\,a^2\,b^5\,c\,d+96\,a\,b^6\,c^2\right )+x^4\,\left (24\,a^2\,b^5\,d^2-48\,a\,b^6\,c\,d+24\,b^7\,c^2\right )+x^2\,\left (144\,a^4\,b^3\,d^2-288\,a^3\,b^4\,c\,d+144\,a^2\,b^5\,c^2\right )+24\,a^6\,b\,d^2+24\,a^4\,b^3\,c^2-48\,a^5\,b^2\,c\,d}+\frac {B\,d^4\,n\,\mathrm {atan}\left (\frac {B\,d^4\,n\,\left (12\,A+25\,B\,n\right )\,\left (\frac {-a^4\,b\,d^4+2\,a^3\,b^2\,c\,d^3-2\,a\,b^4\,c^3\,d+b^5\,c^4}{-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}+2\,b\,d\,x\right )\,\left (-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3\right )\,1{}\mathrm {i}}{b\,\left (25\,B^2\,d^4\,n^2+12\,A\,B\,d^4\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (12\,A+25\,B\,n\right )\,1{}\mathrm {i}}{12\,b\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^2/(a + b*x)^5,x)

[Out]

(B*d^4*n*atan((B*d^4*n*(12*A + 25*B*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)*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)*1
i)/(b*(25*B^2*d^4*n^2 + 12*A*B*d^4*n)*(a*d - b*c)^4))*(12*A + 25*B*n)*1i)/(12*b*(a*d - b*c)^4) - log((e*(a + b
*x)^n)/(c + d*x)^n)^2*(B^2/(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^2*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))) - ((72*A^2*a^3*d^3 - 72*A^2*b^3*c^3 +
415*B^2*a^3*d^3*n^2 - 9*B^2*b^3*c^3*n^2 + 216*A^2*a*b^2*c^2*d - 216*A^2*a^2*b*c*d^2 + 300*A*B*a^3*d^3*n - 36*A
*B*b^3*c^3*n + 55*B^2*a*b^2*c^2*d*n^2 - 161*B^2*a^2*b*c*d^2*n^2 + 156*A*B*a*b^2*c^2*d*n - 276*A*B*a^2*b*c*d^2*
n)/(12*(a*d - b*c)) + (x^2*(163*B^2*a*b^2*d^3*n^2 - 13*B^2*b^3*c*d^2*n^2 + 84*A*B*a*b^2*d^3*n - 12*A*B*b^3*c*d
^2*n))/(2*(a*d - b*c)) + (x*(271*B^2*a^2*b*d^3*n^2 + 7*B^2*b^3*c^2*d*n^2 - 53*B^2*a*b^2*c*d^2*n^2 + 156*A*B*a^
2*b*d^3*n + 12*A*B*b^3*c^2*d*n - 60*A*B*a*b^2*c*d^2*n))/(3*(a*d - b*c)) + (d*x^3*(25*B^2*b^3*d^2*n^2 + 12*A*B*
b^3*d^2*n))/(a*d - b*c))/(x*(96*a^3*b^4*c^2 + 96*a^5*b^2*d^2 - 192*a^4*b^3*c*d) + x^3*(96*a*b^6*c^2 + 96*a^3*b
^4*d^2 - 192*a^2*b^5*c*d) + x^4*(24*b^7*c^2 + 24*a^2*b^5*d^2 - 48*a*b^6*c*d) + x^2*(144*a^2*b^5*c^2 + 144*a^4*
b^3*d^2 - 288*a^3*b^4*c*d) + 24*a^6*b*d^2 + 24*a^4*b^3*c^2 - 48*a^5*b^2*c*d) - log((e*(a + b*x)^n)/(c + d*x)^n
)*((A*B)/(2*(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)) + (B^2*d^4*(x^2*(b*(b*((b*n*(a*d -
b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*n*(a*d - b*c))/(2*d)) + (a*b^2*n*(a*d - b*c))/d + (b^2*n*(a*d - b*c)*(4*a*d
- b*c))/(3*d^2)) + (3*a*b^3*n*(a*d - b*c))/(2*d) + (b^3*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + a*(a*((b*n*(a
*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*n*(a*d - b*c))/(2*d)) + (b*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*
c*d))/(6*d^3)) + x*(b*(a*((b*n*(a*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*n*(a*d - b*c))/(2*d)) + (b*n*(a*d - b
*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(6*d^3)) + a*(b*((b*n*(a*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*n*(a*d
- b*c))/(2*d)) + (a*b^2*n*(a*d - b*c))/d + (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))/(2*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))/(2*d^4) + (2*b^4*n*x^3*(a*d - b*c))/d))/(4*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)))

________________________________________________________________________________________