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

Optimal. Leaf size=390 \[ -\frac{3 b B^2 n^2 (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{4 (a+b x)^2 (b c-a d)^2}+\frac{6 B^2 d n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^2}-\frac{3 b B n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{4 (a+b x)^2 (b c-a d)^2}+\frac{3 B d n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^2}-\frac{b (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 (a+b x)^2 (b c-a d)^2}+\frac{d (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)^2}-\frac{3 b B^3 n^3 (c+d x)^2}{8 (a+b x)^2 (b c-a d)^2}+\frac{6 B^3 d n^3 (c+d x)}{(a+b x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 0.803893, antiderivative size = 811, normalized size of antiderivative = 2.08, number of steps used = 21, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {6742, 2492, 44, 2491, 2490, 32, 2509, 37} \[ -\frac{A^3}{2 b (a+b x)^2}+\frac{3 B d^2 n \log (a+b x) A^2}{2 b (b c-a d)^2}-\frac{3 B d^2 n \log (c+d x) A^2}{2 b (b c-a d)^2}-\frac{3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A^2}{2 b (a+b x)^2}+\frac{3 B d n A^2}{2 b (b c-a d) (a+b x)}-\frac{3 B n A^2}{4 b (a+b x)^2}-\frac{3 b B^2 n^2 (c+d x)^2 A}{4 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) A}{2 (b c-a d)^2 (a+b x)^2}+\frac{3 B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) A}{(b c-a d)^2 (a+b x)}-\frac{3 b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A}{2 (b c-a d)^2 (a+b x)^2}+\frac{6 B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A}{(b c-a d)^2 (a+b x)}+\frac{6 B^2 d n^2 A}{b (b c-a d) (a+b x)}-\frac{b B^3 (c+d x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{B^3 d (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 b B^3 n^3 (c+d x)^2}{8 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{3 B^3 d n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 b B^3 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{6 B^3 d n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{6 B^3 d n^3}{b (b c-a d) (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2491

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)/((g_.) + (h_.)*(x_))^3
, x_Symbol] :> Dist[d/(d*g - c*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^2, x], x] - Dist[h/(d*
g - c*h), Int[((c + d*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(g + h*x)^3, 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] && NeQ[d*g - c*h, 0] && IG
tQ[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 2509

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx &=\int \left (\frac{A^3}{(a+b x)^3}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}\right ) \, dx\\ &=-\frac{A^3}{2 b (a+b x)^2}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx+\left (3 A B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx\\ &=-\frac{A^3}{2 b (a+b x)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{\left (3 A b B^2\right ) \int \frac{(c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}+\frac{\left (b B^3\right ) \int \frac{(c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}-\frac{\left (3 A B^2 d\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}-\frac{\left (B^3 d\right ) \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}+\frac{\left (3 A^2 B (b c-a d) n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b}\\ &=-\frac{A^3}{2 b (a+b x)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{3 A B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{B^3 d (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{b B^3 (c+d x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{\left (3 A b B^2 n\right ) \int \frac{(c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}+\frac{\left (3 b B^3 n\right ) \int \frac{(c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)}-\frac{\left (6 A B^2 d n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}-\frac{\left (3 B^3 d n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}+\frac{\left (3 A^2 B (b c-a d) n\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}{2 b}\\ &=-\frac{A^3}{2 b (a+b x)^2}-\frac{3 A^2 B n}{4 b (a+b x)^2}+\frac{3 A^2 B d n}{2 b (b c-a d) (a+b x)}+\frac{3 A^2 B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{6 A B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{3 B^3 d n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{B^3 d (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{b B^3 (c+d x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{\left (3 A b B^2 n^2\right ) \int \frac{c+d x}{(a+b x)^3} \, dx}{2 (b c-a d)}+\frac{\left (3 b B^3 n^2\right ) \int \frac{(c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)}-\frac{\left (6 A B^2 d n^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{b c-a d}-\frac{\left (6 B^3 d n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}\\ &=-\frac{A^3}{2 b (a+b x)^2}-\frac{3 A^2 B n}{4 b (a+b x)^2}+\frac{3 A^2 B d n}{2 b (b c-a d) (a+b x)}+\frac{6 A B^2 d n^2}{b (b c-a d) (a+b x)}-\frac{3 A b B^2 n^2 (c+d x)^2}{4 (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{6 A B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{6 B^3 d n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{3 B^3 d n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{B^3 d (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{b B^3 (c+d x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac{\left (3 b B^3 n^3\right ) \int \frac{c+d x}{(a+b x)^3} \, dx}{4 (b c-a d)}-\frac{\left (6 B^3 d n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{b c-a d}\\ &=-\frac{A^3}{2 b (a+b x)^2}-\frac{3 A^2 B n}{4 b (a+b x)^2}+\frac{3 A^2 B d n}{2 b (b c-a d) (a+b x)}+\frac{6 A B^2 d n^2}{b (b c-a d) (a+b x)}+\frac{6 B^3 d n^3}{b (b c-a d) (a+b x)}-\frac{3 A b B^2 n^2 (c+d x)^2}{4 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n^3 (c+d x)^2}{8 (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{6 A B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{6 B^3 d n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}+\frac{3 B^3 d n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{3 A b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}-\frac{3 b B^3 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac{B^3 d (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac{b B^3 (c+d x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}\\ \end{align*}

Mathematica [A]  time = 1.15434, size = 693, normalized size = 1.78 \[ -\frac{-6 B d^2 n (a+b x)^2 \log (a+b x) \left (2 B (2 A+3 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+3 B n\right )+2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 A^2+6 A B n+2 B^2 n^2 \log ^2(c+d x)+7 B^2 n^2\right )+6 B d^2 n (a+b x)^2 \log (c+d x) \left (2 B (2 A+3 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 A^2+6 A B n+7 B^2 n^2\right )+(b c-a d) \left (6 B \left (2 A^2 (b c-a d)+2 A B n (b (c-2 d x)-3 a d)+B^2 n^2 (b (c-6 d x)-7 a d)\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A^2 B n (b (c-2 d x)-3 a d)+4 A^3 (b c-a d)+6 B^2 (2 A (b c-a d)+B n (b (c-2 d x)-3 a d)) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 A B^2 n^2 (b (c-6 d x)-7 a d)+4 B^3 (b c-a d) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+3 B^3 n^3 (b (c-14 d x)-15 a d)\right )+6 B^2 d^2 n^2 (a+b x)^2 \log ^2(a+b x) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+2 B n \log (c+d x)+3 B n\right )+6 B^2 d^2 n^2 (a+b x)^2 \log ^2(c+d x) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+3 B n\right )+4 B^3 d^2 n^3 (a+b x)^2 \log ^3(c+d x)-4 B^3 d^2 n^3 (a+b x)^2 \log ^3(a+b x)}{8 b (a+b x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-4*B^3*d^2*n^3*(a + b*x)^2*Log[a + b*x]^3 + 4*B^3*d^2*n^3*(a + b*x)^2*Log[c + d*x]^3 + 6*B^2*d^2*n^2*(a + b*
x)^2*Log[c + d*x]^2*(2*A + 3*B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B^2*d^2*n^2*(a + b*x)^2*Log[a + b
*x]^2*(2*A + 3*B*n + 2*B*n*Log[c + d*x] + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B*d^2*n*(a + b*x)^2*Log[c
+ d*x]*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B*(2*A + 3*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*B^2*Log[(e*(a + b
*x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(4*A^3*(b*c - a*d) + 3*B^3*n^3*(-15*a*d + b*(c - 14*d*x)) + 6*A*B^2*n^2*(
-7*a*d + b*(c - 6*d*x)) + 6*A^2*B*n*(-3*a*d + b*(c - 2*d*x)) + 6*B*(2*A^2*(b*c - a*d) + B^2*n^2*(-7*a*d + b*(c
- 6*d*x)) + 2*A*B*n*(-3*a*d + b*(c - 2*d*x)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*B^2*(2*A*(b*c - a*d) + B*n
*(-3*a*d + b*(c - 2*d*x)))*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 4*B^3*(b*c - a*d)*Log[(e*(a + b*x)^n)/(c + d*x
)^n]^3) - 6*B*d^2*n*(a + b*x)^2*Log[a + b*x]*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B^2*n^2*Log[c + d*x]^2 + 2*B*(2*
A + 3*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*B*n*Log[c + d*x]*(2
*A + 3*B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n])))/(8*b*(b*c - a*d)^2*(a + b*x)^2)

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Maple [C]  time = 18.137, size = 120138, normalized size = 308.1 \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)))^3/(b*x+a)^3,x)

[Out]

result too large to display

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Maxima [B]  time = 1.97299, size = 3032, normalized size = 7.77 \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)))^3/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^3*x^2 + 2*a*b^2*x + a^2*b) + 1/8*(6*(2*d^2*e*n*log(b*x + a)/(b^3*
c^2 - 2*a*b^2*c*d + a^2*b*d^2) - 2*d^2*e*n*log(d*x + c)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) + (2*b*d*e*n*x - b
*c*e*n + 3*a*d*e*n)/(a^2*b^2*c - a^3*b*d + (b^4*c - a*b^3*d)*x^2 + 2*(a*b^3*c - a^2*b^2*d)*x))*log((b*x + a)^n
*e/(d*x + c)^n)^2/e - (6*(b^2*c^2*e^2*n^2 - 8*a*b*c*d*e^2*n^2 + 7*a^2*d^2*e^2*n^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2
*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(b*x + a)^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*
e^2*n^2)*log(d*x + c)^2 - 6*(b^2*c*d*e^2*n^2 - a*b*d^2*e^2*n^2)*x - 6*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2
*x + a^2*d^2*e^2*n^2)*log(b*x + a) + 2*(3*b^2*d^2*e^2*n^2*x^2 + 6*a*b*d^2*e^2*n^2*x + 3*a^2*d^2*e^2*n^2 - 2*(b
^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e/(d*x
+ c)^n)/((a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^2 + 2*(a*b^4*c^2
- 2*a^2*b^3*c*d + a^3*b^2*d^2)*x)*e) + (3*b^2*c^2*e^3*n^3 - 48*a*b*c*d*e^3*n^3 + 45*a^2*d^2*e^3*n^3 - 4*(b^2*d
^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a)^3 + 4*(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^
3*n^3*x + a^2*d^2*e^3*n^3)*log(d*x + c)^3 + 18*(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*l
og(b*x + a)^2 + 6*(3*b^2*d^2*e^3*n^3*x^2 + 6*a*b*d^2*e^3*n^3*x + 3*a^2*d^2*e^3*n^3 - 2*(b^2*d^2*e^3*n^3*x^2 +
2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 - 42*(b^2*c*d*e^3*n^3 - a*b*d^2*e^3*n^3)*x
- 42*(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a) + 6*(7*b^2*d^2*e^3*n^3*x^2 +
14*a*b*d^2*e^3*n^3*x + 7*a^2*d^2*e^3*n^3 + 2*(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log
(b*x + a)^2 - 6*(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a))*log(d*x + c))/((a^
2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c
*d + a^3*b^2*d^2)*x)*e^2))/e)*B^3 + 3/4*A*B^2*(2*(2*d^2*e*n*log(b*x + a)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) -
2*d^2*e*n*log(d*x + c)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) + (2*b*d*e*n*x - b*c*e*n + 3*a*d*e*n)/(a^2*b^2*c -
a^3*b*d + (b^4*c - a*b^3*d)*x^2 + 2*(a*b^3*c - a^2*b^2*d)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e - (b^2*c^2*e^2
*n^2 - 8*a*b*c*d*e^2*n^2 + 7*a^2*d^2*e^2*n^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)
*log(b*x + a)^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(d*x + c)^2 - 6*(b^2*c*d*
e^2*n^2 - a*b*d^2*e^2*n^2)*x - 6*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(b*x + a) +
2*(3*b^2*d^2*e^2*n^2*x^2 + 6*a*b*d^2*e^2*n^2*x + 3*a^2*d^2*e^2*n^2 - 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^
2*x + a^2*d^2*e^2*n^2)*log(b*x + a))*log(d*x + c))/((a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2 + (b^5*c^2 - 2*a*
b^4*c*d + a^2*b^3*d^2)*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x)*e^2)) - 3/2*A*B^2*log((b*x + a)^n*
e/(d*x + c)^n)^2/(b^3*x^2 + 2*a*b^2*x + a^2*b) + 3/4*(2*d^2*e*n*log(b*x + a)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^
2) - 2*d^2*e*n*log(d*x + c)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) + (2*b*d*e*n*x - b*c*e*n + 3*a*d*e*n)/(a^2*b^2
*c - a^3*b*d + (b^4*c - a*b^3*d)*x^2 + 2*(a*b^3*c - a^2*b^2*d)*x))*A^2*B/e - 3/2*A^2*B*log((b*x + a)^n*e/(d*x
+ c)^n)/(b^3*x^2 + 2*a*b^2*x + a^2*b) - 1/2*A^3/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Fricas [B]  time = 1.49427, size = 4618, normalized size = 11.84 \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)))^3/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/8*(4*A^3*b^2*c^2 - 8*A^3*a*b*c*d + 4*A^3*a^2*d^2 + 3*(B^3*b^2*c^2 - 16*B^3*a*b*c*d + 15*B^3*a^2*d^2)*n^3 -
4*(B^3*b^2*d^2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(b*x + a)^3 + 4*(B^3*b^2*
d^2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(d*x + c)^3 + 4*(B^3*b^2*c^2 - 2*B^3
*a*b*c*d + B^3*a^2*d^2)*log(e)^3 + 6*(A*B^2*b^2*c^2 - 8*A*B^2*a*b*c*d + 7*A*B^2*a^2*d^2)*n^2 + 6*((B^3*b^2*c^2
- 4*B^3*a*b*c*d)*n^3 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n^2 - (3*B^3*b^2*d^2*n^3 + 2*A*B^2*b^2*d^2*n^2)*x^
2 - 2*(2*A*B^2*a*b*d^2*n^2 + (B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^3)*x - 2*(B^3*b^2*d^2*n^2*x^2 + 2*B^3*a*b*d^2*n^2
*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^2)*log(e))*log(b*x + a)^2 + 6*((B^3*b^2*c^2 - 4*B^3*a*b*c*d)*n^3 + 2*(A*B
^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n^2 - (3*B^3*b^2*d^2*n^3 + 2*A*B^2*b^2*d^2*n^2)*x^2 - 2*(2*A*B^2*a*b*d^2*n^2 + (
B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^3)*x - 2*(B^3*b^2*d^2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*
c*d)*n^3)*log(b*x + a) - 2*(B^3*b^2*d^2*n^2*x^2 + 2*B^3*a*b*d^2*n^2*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^2)*log
(e))*log(d*x + c)^2 + 6*(2*A*B^2*b^2*c^2 - 4*A*B^2*a*b*c*d + 2*A*B^2*a^2*d^2 - 2*(B^3*b^2*c*d - B^3*a*b*d^2)*n
*x + (B^3*b^2*c^2 - 4*B^3*a*b*c*d + 3*B^3*a^2*d^2)*n)*log(e)^2 + 6*(A^2*B*b^2*c^2 - 4*A^2*B*a*b*c*d + 3*A^2*B*
a^2*d^2)*n - 6*(7*(B^3*b^2*c*d - B^3*a*b*d^2)*n^3 + 6*(A*B^2*b^2*c*d - A*B^2*a*b*d^2)*n^2 + 2*(A^2*B*b^2*c*d -
A^2*B*a*b*d^2)*n)*x + 6*((B^3*b^2*c^2 - 8*B^3*a*b*c*d)*n^3 + 2*(A*B^2*b^2*c^2 - 4*A*B^2*a*b*c*d)*n^2 - (7*B^3
*b^2*d^2*n^3 + 6*A*B^2*b^2*d^2*n^2 + 2*A^2*B*b^2*d^2*n)*x^2 - 2*(B^3*b^2*d^2*n*x^2 + 2*B^3*a*b*d^2*n*x - (B^3*
b^2*c^2 - 2*B^3*a*b*c*d)*n)*log(e)^2 + 2*(A^2*B*b^2*c^2 - 2*A^2*B*a*b*c*d)*n - 2*(2*A^2*B*a*b*d^2*n + (3*B^3*b
^2*c*d + 4*B^3*a*b*d^2)*n^3 + 2*(A*B^2*b^2*c*d + 2*A*B^2*a*b*d^2)*n^2)*x + 2*((B^3*b^2*c^2 - 4*B^3*a*b*c*d)*n^
2 - (3*B^3*b^2*d^2*n^2 + 2*A*B^2*b^2*d^2*n)*x^2 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n - 2*(2*A*B^2*a*b*d^2*n
+ (B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^2)*x)*log(e))*log(b*x + a) - 6*((B^3*b^2*c^2 - 8*B^3*a*b*c*d)*n^3 + 2*(A*B^
2*b^2*c^2 - 4*A*B^2*a*b*c*d)*n^2 - (7*B^3*b^2*d^2*n^3 + 6*A*B^2*b^2*d^2*n^2 + 2*A^2*B*b^2*d^2*n)*x^2 - 2*(B^3*
b^2*d^2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(b*x + a)^2 - 2*(B^3*b^2*d^2*n*x
^2 + 2*B^3*a*b*d^2*n*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n)*log(e)^2 + 2*(A^2*B*b^2*c^2 - 2*A^2*B*a*b*c*d)*n - 2
*(2*A^2*B*a*b*d^2*n + (3*B^3*b^2*c*d + 4*B^3*a*b*d^2)*n^3 + 2*(A*B^2*b^2*c*d + 2*A*B^2*a*b*d^2)*n^2)*x + 2*((B
^3*b^2*c^2 - 4*B^3*a*b*c*d)*n^3 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n^2 - (3*B^3*b^2*d^2*n^3 + 2*A*B^2*b^2*d
^2*n^2)*x^2 - 2*(2*A*B^2*a*b*d^2*n^2 + (B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^3)*x - 2*(B^3*b^2*d^2*n^2*x^2 + 2*B^3*a
*b*d^2*n^2*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^2)*log(e))*log(b*x + a) + 2*((B^3*b^2*c^2 - 4*B^3*a*b*c*d)*n^2
- (3*B^3*b^2*d^2*n^2 + 2*A*B^2*b^2*d^2*n)*x^2 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n - 2*(2*A*B^2*a*b*d^2*n +
(B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^2)*x)*log(e))*log(d*x + c) + 6*(2*A^2*B*b^2*c^2 - 4*A^2*B*a*b*c*d + 2*A^2*B*a
^2*d^2 + (B^3*b^2*c^2 - 8*B^3*a*b*c*d + 7*B^3*a^2*d^2)*n^2 + 2*(A*B^2*b^2*c^2 - 4*A*B^2*a*b*c*d + 3*A*B^2*a^2*
d^2)*n - 2*(3*(B^3*b^2*c*d - B^3*a*b*d^2)*n^2 + 2*(A*B^2*b^2*c*d - A*B^2*a*b*d^2)*n)*x)*log(e))/(a^2*b^3*c^2 -
2*a^3*b^2*c*d + a^4*b*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^
2*d^2)*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)))**3/(b*x+a)**3,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 )}^{3}}{{\left (b x + a\right )}^{3}}\,{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)))^3/(b*x+a)^3,x, algorithm="giac")

[Out]

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