3.162 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(a+b x)^4} \, dx\)

Optimal. Leaf size=427 \[ -\frac{b^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 (a+b x)^3 (b c-a d)^3}-\frac{2 b^2 B n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{9 (a+b x)^3 (b c-a d)^3}-\frac{d^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^3}-\frac{2 B d^2 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^3}+\frac{b d (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x)^2 (b c-a d)^3}+\frac{b B d n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x)^2 (b c-a d)^3}-\frac{2 b^2 B^2 n^2 (c+d x)^3}{27 (a+b x)^3 (b c-a d)^3}-\frac{2 B^2 d^2 n^2 (c+d x)}{(a+b x) (b c-a d)^3}+\frac{b B^2 d n^2 (c+d x)^2}{2 (a+b x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.21235, antiderivative size = 730, normalized size of antiderivative = 1.71, number of steps used = 26, 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} \[ -\frac{2 B^2 d^3 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 n^2 \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{3 b (b c-a d)^3}-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B d^2 n}{3 b (a+b x) (b c-a d)^2}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{A B d n}{3 b (a+b x)^2 (b c-a d)}-\frac{2 A B n}{9 b (a+b x)^3}+\frac{2 B^2 d^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (a+b x) (b c-a d)^3}-\frac{11 B^2 d^2 n^2}{9 b (a+b x) (b c-a d)^2}-\frac{5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac{5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^2 (b c-a d)}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{5 B^2 d n^2}{18 b (a+b x)^2 (b c-a d)}-\frac{2 B^2 n^2}{27 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^4,x]

[Out]

-A^2/(3*b*(a + b*x)^3) - (2*A*B*n)/(9*b*(a + b*x)^3) - (2*B^2*n^2)/(27*b*(a + b*x)^3) + (A*B*d*n)/(3*b*(b*c -
a*d)*(a + b*x)^2) + (5*B^2*d*n^2)/(18*b*(b*c - a*d)*(a + b*x)^2) - (2*A*B*d^2*n)/(3*b*(b*c - a*d)^2*(a + b*x))
- (11*B^2*d^2*n^2)/(9*b*(b*c - a*d)^2*(a + b*x)) - (2*A*B*d^3*n*Log[a + b*x])/(3*b*(b*c - a*d)^3) - (5*B^2*d^
3*n^2*Log[a + b*x])/(9*b*(b*c - a*d)^3) + (2*A*B*d^3*n*Log[c + d*x])/(3*b*(b*c - a*d)^3) + (5*B^2*d^3*n^2*Log[
c + d*x])/(9*b*(b*c - a*d)^3) - (2*A*B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(a + b*x)^3) - (2*B^2*n*Log[(e*(
a + b*x)^n)/(c + d*x)^n])/(9*b*(a + b*x)^3) + (B^2*d*n*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(b*c - a*d)*(a +
b*x)^2) - (2*B^2*d^2*n*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*(b*c - a*d)^3*(a + b*x)) + (2*B^2*d^3*n
*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(b*c - a*d)^3) - (2*B^2*d^3*n*Log[(b
*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(b*c - a*d)^3) - (B^2*Log[(e*(a + b*x)^n)/(c +
d*x)^n]^2)/(3*b*(a + b*x)^3) - (2*B^2*d^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*(b*c - a*d)^3) -
(2*B^2*d^3*n^2*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(3*b*(b*c - a*d)^3)

Rule 6742

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

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 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 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 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 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 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 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 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 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 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]

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)^4} \, dx &=\int \left (\frac{A^2}{(a+b x)^4}+\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac{A^2}{3 b (a+b x)^3}+(2 A B) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+B^2 \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{(2 A B (b c-a d) n) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b}+\frac{\left (2 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)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{(2 A B (b c-a d) n) \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}{3 b}+\frac{\left (2 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)^4}-\frac{b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{1}{3} \left (2 B^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac{\left (2 B^2 d^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{3 (b c-a d)^3}+\frac{\left (2 B^2 d^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{3 b (b c-a d)^3}+\frac{\left (2 B^2 d^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac{\left (2 B^2 d n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{3 (b c-a d)}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{\left (B^2 d n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b}+\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac{\left (2 B^2 d^3 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac{\left (2 B^2 d^3 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{2 B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{\left (B^2 d 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}{3 b}+\frac{\left (2 B^2 d^2 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 )}{3 b (b c-a d)^2}-\frac{\left (2 B^2 d^3 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 )}{3 b^2 (b c-a d)^2}+\frac{\left (2 B^2 (b c-a 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}{9 b}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}-\frac{2 B^2 n^2}{27 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}+\frac{5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{\left (2 B^2 d^2 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 )}{3 b (b c-a d)^2}+\frac{\left (2 B^2 d^3 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 )}{3 b^2 (b c-a d)^2}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}-\frac{2 B^2 n^2}{27 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}+\frac{5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{\left (2 B^2 d^2 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 )}{3 b (b c-a d)^2}+\frac{\left (2 B^2 d^3 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 )}{3 b^2 (b c-a d)^2}\\ &=-\frac{A^2}{3 b (a+b x)^3}-\frac{2 A B n}{9 b (a+b x)^3}-\frac{2 B^2 n^2}{27 b (a+b x)^3}+\frac{A B d n}{3 b (b c-a d) (a+b x)^2}+\frac{5 B^2 d n^2}{18 b (b c-a d) (a+b x)^2}-\frac{2 A B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 (a+b x)}-\frac{2 A B d^3 n \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3}+\frac{2 A B d^3 n \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 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 )}{3 b (b c-a d)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^2 d^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac{2 B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{3 b (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.743327, size = 432, normalized size = 1.01 \[ \frac{-(b c-a d) \left (6 B \left (B n \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+6 A (b c-a d)^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A B n \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (147 d x-23 c)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )+18 A^2 (b c-a d)^2+18 B^2 (b c-a d)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^3 n (a+b x)^3 \log (c+d x) \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+11 B n\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+6 B n \log (c+d x)+11 B n\right )+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x)}{54 b (a+b x)^3 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^4,x]

[Out]

(18*B^2*d^3*n^2*(a + b*x)^3*Log[a + b*x]^2 + 18*B^2*d^3*n^2*(a + b*x)^3*Log[c + d*x]^2 + 6*B*d^3*n*(a + b*x)^3
*Log[c + d*x]*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) - 6*B*d^3*n*(a + b*x)^3*Log[a + b*x]*(6*A
+ 11*B*n + 6*B*n*Log[c + d*x] + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) - (b*c - a*d)*(18*A^2*(b*c - a*d)^2 + 6*
A*B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + B^2*n^2*(85*a^2*d^2 + a*b*d*(
-23*c + 147*d*x) + b^2*(4*c^2 - 15*c*d*x + 66*d^2*x^2)) + 6*B*(6*A*(b*c - a*d)^2 + B*n*(11*a^2*d^2 + a*b*d*(-7
*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*(b*c - a*d)^2*Log
[(e*(a + b*x)^n)/(c + d*x)^n]^2))/(54*b*(b*c - a*d)^3*(a + b*x)^3)

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Maple [C]  time = 2.658, size = 25057, normalized size = 58.7 \begin{align*} \text{output too large to display} \end{align*}

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)^4,x)

[Out]

result too large to display

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Maxima [B]  time = 1.7076, size = 2025, normalized size = 4.74 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x, algorithm="maxima")

[Out]

-1/54*B^2*(6*(6*d^3*e*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d
*x + c)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c
*d*e*n + 11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c
^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3
*b^3*c*d + a^4*b^2*d^2)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e + (4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 1
08*a^2*b*c*d^2*e^2*n^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2
*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2
*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*
e^2*n^2 - 54*a*b^2*c*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2
+ 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^
2 + 33*a^2*b*d^3*e^2*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d
^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a))*log(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2
- a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d +
3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)*e^2
)) - 1/3*B^2*log((b*x + a)^n*e/(d*x + c)^n)^2/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/9*(6*d^3*e*n*l
og(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*b^
3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n -
3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^
4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x
))*A*B/e - 2/3*A*B*log((b*x + a)^n*e/(d*x + c)^n)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/3*A^2/(b^4
*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Fricas [B]  time = 1.40228, size = 3351, normalized size = 7.85 \begin{align*} \text{result too large to display} \end{align*}

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)^4,x, algorithm="fricas")

[Out]

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a^3*d^3 + (4*B^2*b^3*c^3 - 27*B^2*a*b
^2*c^2*d + 108*B^2*a^2*b*c*d^2 - 85*B^2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 + 6*(A*B*b^3*
c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B
^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n^2)*log(b*x + a)^2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^
2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n^2)*log(d*x + c
)^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^2 + 6*(2*A*B*b^3*c^3 - 9*A
*B*a*b^2*c^2*d + 18*A*B*a^2*b*c*d^2 - 11*A*B*a^3*d^3)*n - 3*((5*B^2*b^3*c^2*d - 54*B^2*a*b^2*c*d^2 + 49*B^2*a^
2*b*d^3)*n^2 + 6*(A*B*b^3*c^2*d - 6*A*B*a*b^2*c*d^2 + 5*A*B*a^2*b*d^3)*n)*x + 6*((11*B^2*b^3*d^3*n^2 + 6*A*B*b
^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b
^3*c*d^2 + 9*B^2*a*b^2*d^3)*n^2)*x^2 + 6*(A*B*b^3*c^3 - 3*A*B*a*b^2*c^2*d + 3*A*B*a^2*b*c*d^2)*n + 3*(6*A*B*a^
2*b*d^3*n - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n^2)*x + 6*(B^2*b^3*d^3*n*x^3 + 3*B^2*a*b^2*
d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n)*log(e))*log(b*x + a
) - 6*((11*B^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n
^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 + 9*B^2*a*b^2*d^3)*n^2)*x^2 + 6*(A*B*b^3*c^3 - 3*A*B*a*b^2*c^2*d
+ 3*A*B*a^2*b*c*d^2)*n + 3*(6*A*B*a^2*b*d^3*n - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n^2)*x +
6*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d +
3*B^2*a^2*b*c*d^2)*n^2)*log(b*x + a) + 6*(B^2*b^3*d^3*n*x^3 + 3*B^2*a*b^2*d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + (
B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n)*log(e))*log(d*x + c) + 6*(6*A*B*b^3*c^3 - 18*A*B*a*b^2
*c^2*d + 18*A*B*a^2*b*c*d^2 - 6*A*B*a^3*d^3 + 6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n*x^2 - 3*(B^2*b^3*c^2*d - 6*B
^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^3)*n*x + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2 - 11*B^2*a^3*d
^3)*n)*log(e))/(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2
*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b
^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)

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

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)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{4}}\,{d x} \end{align*}

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)^4,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^2/(b*x + a)^4, x)