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

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

[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 [C]  time = 1.41, antiderivative size = 843, normalized size of antiderivative = 1.44, number of steps used = 29, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2492, 44, 2514, 2490, 32, 2488, 2411, 2343, 2333, 2315} \[ \frac {13 B^2 n^2 \log (a+b x) d^4}{24 b (b c-a d)^4}+\frac {A B n \log (a+b x) d^4}{2 b (b c-a d)^4}-\frac {13 B^2 n^2 \log (c+d x) d^4}{24 b (b c-a d)^4}-\frac {A B n \log (c+d x) d^4}{2 b (b c-a d)^4}-\frac {B^2 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^4}{2 b (b c-a d)^4}+\frac {B^2 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^4}{2 b (b c-a d)^4}+\frac {B^2 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) d^4}{2 b (b c-a d)^4}+\frac {B^2 n^2 \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) d^4}{2 b (b c-a d)^4}+\frac {B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^3}{2 (b c-a d)^4 (a+b x)}+\frac {25 B^2 n^2 d^3}{24 b (b c-a d)^3 (a+b x)}+\frac {A B n d^3}{2 b (b c-a d)^3 (a+b x)}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^2}{4 b (b c-a d)^2 (a+b x)^2}-\frac {13 B^2 n^2 d^2}{48 b (b c-a d)^2 (a+b x)^2}-\frac {A B n d^2}{4 b (b c-a d)^2 (a+b x)^2}+\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d}{6 b (b c-a d) (a+b x)^3}+\frac {7 B^2 n^2 d}{72 b (b c-a d) (a+b x)^3}+\frac {A B n d}{6 b (b c-a d) (a+b x)^3}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac {B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac {B^2 n^2}{32 b (a+b x)^4}-\frac {A B n}{8 b (a+b x)^4}-\frac {A^2}{4 b (a+b x)^4} \]

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]

-A^2/(4*b*(a + b*x)^4) - (A*B*n)/(8*b*(a + b*x)^4) - (B^2*n^2)/(32*b*(a + b*x)^4) + (A*B*d*n)/(6*b*(b*c - a*d)
*(a + b*x)^3) + (7*B^2*d*n^2)/(72*b*(b*c - a*d)*(a + b*x)^3) - (A*B*d^2*n)/(4*b*(b*c - a*d)^2*(a + b*x)^2) - (
13*B^2*d^2*n^2)/(48*b*(b*c - a*d)^2*(a + b*x)^2) + (A*B*d^3*n)/(2*b*(b*c - a*d)^3*(a + b*x)) + (25*B^2*d^3*n^2
)/(24*b*(b*c - a*d)^3*(a + b*x)) + (A*B*d^4*n*Log[a + b*x])/(2*b*(b*c - a*d)^4) + (13*B^2*d^4*n^2*Log[a + b*x]
)/(24*b*(b*c - a*d)^4) - (A*B*d^4*n*Log[c + d*x])/(2*b*(b*c - a*d)^4) - (13*B^2*d^4*n^2*Log[c + d*x])/(24*b*(b
*c - a*d)^4) - (A*B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*(a + b*x)^4) - (B^2*n*Log[(e*(a + b*x)^n)/(c + d*x)
^n])/(8*b*(a + b*x)^4) + (B^2*d*n*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(6*b*(b*c - a*d)*(a + b*x)^3) - (B^2*d^2*n
*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*b*(b*c - a*d)^2*(a + b*x)^2) + (B^2*d^3*n*(c + d*x)*Log[(e*(a + b*x)^n)/
(c + d*x)^n])/(2*(b*c - a*d)^4*(a + b*x)) - (B^2*d^4*n*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(
c + d*x)^n])/(2*b*(b*c - a*d)^4) + (B^2*d^4*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])
/(2*b*(b*c - a*d)^4) - (B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(4*b*(a + b*x)^4) + (B^2*d^4*n^2*PolyLog[2, (d
*(a + b*x))/(b*(c + d*x))])/(2*b*(b*c - a*d)^4) + (B^2*d^4*n^2*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(2*b
*(b*c - a*d)^4)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2343

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

Rule 2411

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

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2490

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_))^
2, x_Symbol] :> Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/((b*g - a*h)*(g + h*x)), x] - Dist[(p*
r*s*(b*c - a*d))/(b*g - a*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((c + d*x)*(g + h*x)), x], x] /
; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] &&
IGtQ[s, 0]

Rule 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*
(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && NeQ[m, -1]

Rule 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [A]  time = 0.95, size = 1011, normalized size = 1.72 \[ -\frac {9 \left (8 A^2+4 B n A+16 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 ) A+B^2 n^2+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+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 )\right ) (b c-a d)^4-4 B d 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 ) (b c-a d)^3+6 B 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 ) (b c-a d)^2-12 B d^3 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 ) (b c-a d)-12 B n \log (a+b x) \left (-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 ) (b c-a d)^3+4 B d n (a+b x) (b c-a d)^2+12 B d^3 n (a+b x)^3+6 B d^2 (a d-b c) n (a+b x)^2\right ) (b c-a d)+72 b B^2 n^2 \left (\left (c^4-d^4 x^4\right ) b^3-4 a d \left (c^3+d^3 x^3\right ) b^2+6 a^2 d^2 \left (c^2-d^2 x^2\right ) b-4 a^3 d^3 (c+d x)\right ) \log ^2(a+b x)+72 b B^2 n^2 \left (\left (c^4-d^4 x^4\right ) b^3-4 a d \left (c^3+d^3 x^3\right ) b^2+6 a^2 d^2 \left (c^2-d^2 x^2\right ) b-4 a^3 d^3 (c+d x)\right ) \log ^2(c+d x)-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 )+12 B n \log (c+d x) \left (-12 B n \log (a+b x) (b c-a d)^4-3 \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 ) (b c-a d)^4+4 B d n (a+b x) (b c-a d)^3-6 B d^2 n (a+b x)^2 (b c-a d)^2+12 B d^3 n (a+b x)^3 (b c-a d)+12 B d^4 n (a+b x)^4 \log (a+b x)\right )}{288 b (b c-a d)^4 (a+b x)^4} \]

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)

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fricas [B]  time = 0.84, size = 2458, normalized size = 4.19 \[ \text {result too large to display} \]

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*b^4*c^4 - 288*A^2*a*b^3*c^3*d + 432*A^2*a^2*b^2*c^2*d^2 - 288*A^2*a^3*b*c*d^3 + 72*A^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^4*c*d^3 - A*B*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^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + 7*A*B*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 + 72*(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 + B^2*a^4*d^4)*log(e)^2 + 12*(3*A*B*b^4*c^4 - 16
*A*B*a*b^3*c^3*d + 36*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 25*A*B*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^4*c^3*d - 6*A*B*a*b^3*c^2*d^2 +
18*A*B*a^2*b^2*c*d^3 - 13*A*B*a^3*b*d^4)*n)*x - 12*((25*B^2*b^4*d^4*n^2 + 12*A*B*b^4*d^4*n)*x^4 + 4*(12*A*B*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*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^4*c^4 - 4*A*B*a*b^3*c^3*d + 6*A*B*a^2*b^2*c^2*d^2 - 4*A*B*a^3*b*c*d^3)*n +
4*(12*A*B*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*x^4 + 4*B^2*a*b^3*d^4*n*x^3 + 6*B^2*a^2*b^2*d^4*n*x^2 + 4*B^2*a^3*b*d^4*n*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)*log(e))*log(b*x + a) + 12*((25*B^2*b^4*
d^4*n^2 + 12*A*B*b^4*d^4*n)*x^4 + 4*(12*A*B*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*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^4*c^4 - 4*A*B*a*b^3*c^3*d + 6
*A*B*a^2*b^2*c^2*d^2 - 4*A*B*a^3*b*c*d^3)*n + 4*(12*A*B*a^3*b*d^4*n + (B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 1
8*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) + 12*(B^2*b^4*d^4*n*x^4 + 4*B^2*a*b^3*d^4*n*x^3 + 6*B^2*a^2*b^2*d^4*n*x^2 + 4*B^2
*a^3*b*d^4*n*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)*log(e))*log(
d*x + c) + 12*(12*A*B*b^4*c^4 - 48*A*B*a*b^3*c^3*d + 72*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 12*A*B*a^4*
d^4 - 12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n*x^3 + 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*n
*x^2 - 4*(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 - 13*B^2*a^3*b*d^4)*n*x + (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 + 25*B^2*a^4*d^4)*n)*log(e))/(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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{5}}\,{d x} \]

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)

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maple [C]  time = 5.84, size = 33370, normalized size = 56.85 \[ \text {output too large to display} \]

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)

[Out]

result too large to display

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maxima [B]  time = 3.00, size = 2238, normalized size = 3.81 \[ \text {result too large to display} \]

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]

1/288*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))*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)) - 1/4*B^2*l
og((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) + 1/24*(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*B/e - 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)

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mupad [B]  time = 9.61, size = 1579, normalized size = 2.69 \[ \text {result too large to display} \]

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)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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