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

Optimal. Leaf size=420 \[ \frac{8 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac{26 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^5}-\frac{8 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^5}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}+\frac{4 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{26 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{b^2 e n^2}{3 d^4 (d+e x)}-\frac{b^2 e n^2 \log (x)}{3 d^5}+\frac{3 b^2 e n^2 \log (d+e x)}{d^5}-\frac{2 b^2 n^2}{d^4 x} \]

[Out]

(-2*b^2*n^2)/(d^4*x) - (b^2*e*n^2)/(3*d^4*(d + e*x)) - (b^2*e*n^2*Log[x])/(3*d^5) - (2*b*n*(a + b*Log[c*x^n]))
/(d^4*x) + (b*e*n*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)^2) - (8*b*e^2*n*x*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x))
+ (4*e*(a + b*Log[c*x^n])^2)/(3*d^5) - (a + b*Log[c*x^n])^2/(d^4*x) - (e*(a + b*Log[c*x^n])^2)/(3*d^2*(d + e*
x)^3) - (e*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n])^2)/(d^5*(d + e*x)) - (4*e*(a
+ b*Log[c*x^n])^3)/(3*b*d^5*n) + (3*b^2*e*n^2*Log[d + e*x])/d^5 - (26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d
])/(3*d^5) + (4*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^5 - (26*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/(3*d^5) +
(8*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^5 - (8*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^5

________________________________________________________________________________________

Rubi [A]  time = 0.89301, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 17, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.739, Rules used = {2353, 2305, 2304, 2302, 30, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44, 2318, 2374, 6589} \[ \frac{8 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac{26 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^5}-\frac{8 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^5}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}+\frac{4 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{26 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{b^2 e n^2}{3 d^4 (d+e x)}-\frac{b^2 e n^2 \log (x)}{3 d^5}+\frac{3 b^2 e n^2 \log (d+e x)}{d^5}-\frac{2 b^2 n^2}{d^4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*n^2)/(d^4*x) - (b^2*e*n^2)/(3*d^4*(d + e*x)) - (b^2*e*n^2*Log[x])/(3*d^5) - (2*b*n*(a + b*Log[c*x^n]))
/(d^4*x) + (b*e*n*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)^2) - (8*b*e^2*n*x*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x))
+ (4*e*(a + b*Log[c*x^n])^2)/(3*d^5) - (a + b*Log[c*x^n])^2/(d^4*x) - (e*(a + b*Log[c*x^n])^2)/(3*d^2*(d + e*
x)^3) - (e*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n])^2)/(d^5*(d + e*x)) - (4*e*(a
+ b*Log[c*x^n])^3)/(3*b*d^5*n) + (3*b^2*e*n^2*Log[d + e*x])/d^5 - (26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d
])/(3*d^5) + (4*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^5 - (26*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/(3*d^5) +
(8*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^5 - (8*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^5

Rule 2353

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 2305

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 2304

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

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

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2374

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

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x^2}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2}{d^5 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)^4}+\frac{2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)^2}+\frac{4 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^4}-\frac{(4 e) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^5}+\frac{\left (4 e^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^5}+\frac{\left (3 e^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^4}+\frac{\left (2 e^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{d^3}+\frac{e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{d^2}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}-\frac{(4 e) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^5 n}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^4}-\frac{(8 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^5}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^3}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 d^2}-\frac{\left (6 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}\\ &=-\frac{2 b^2 n^2}{d^4 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}-\frac{6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^5}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}+\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^4}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 d^3}-\frac{\left (2 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^4}-\frac{\left (2 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 d^3}+\frac{\left (6 b^2 e n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^5}-\frac{\left (8 b^2 e n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^5}\\ &=-\frac{2 b^2 n^2}{d^4 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{2 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}-\frac{6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^5}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}-\frac{6 b^2 e n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}+\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}-\frac{8 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^5}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^5}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d^4}-\frac{\left (2 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}-\frac{\left (2 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d^4}-\frac{\left (b^2 e n^2\right ) \int \frac{1}{x (d+e x)^2} \, dx}{3 d^3}+\frac{\left (2 b^2 e^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{d^5}\\ &=-\frac{2 b^2 n^2}{d^4 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac{2 b^2 e n^2 \log (d+e x)}{d^5}-\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^5}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}-\frac{6 b^2 e n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}+\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}-\frac{8 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^5}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{3 d^5}-\frac{\left (2 b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^5}+\frac{\left (2 b^2 e n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^5}-\frac{\left (b^2 e n^2\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{3 d^3}+\frac{\left (2 b^2 e^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{3 d^5}\\ &=-\frac{2 b^2 n^2}{d^4 x}-\frac{b^2 e n^2}{3 d^4 (d+e x)}-\frac{b^2 e n^2 \log (x)}{3 d^5}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac{3 b^2 e n^2 \log (d+e x)}{d^5}-\frac{26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^5}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}-\frac{8 b^2 e n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}+\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}-\frac{8 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^5}+\frac{\left (2 b^2 e n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 d^5}\\ &=-\frac{2 b^2 n^2}{d^4 x}-\frac{b^2 e n^2}{3 d^4 (d+e x)}-\frac{b^2 e n^2 \log (x)}{3 d^5}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac{8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac{3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac{3 b^2 e n^2 \log (d+e x)}{d^5}-\frac{26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^5}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^5}-\frac{26 b^2 e n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{3 d^5}+\frac{8 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^5}-\frac{8 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.657879, size = 378, normalized size = 0.9 \[ -\frac{-24 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+26 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+24 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{d^3 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac{b d^2 e n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac{9 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-\frac{8 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-12 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+26 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}-13 e \left (a+b \log \left (c x^n\right )\right )^2+8 b^2 e n^2 (\log (x)-\log (d+e x))+\frac{b^2 e n^2 (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}+\frac{6 b^2 d n^2}{x}}{3 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*b^2*d*n^2)/x + (6*b*d*n*(a + b*Log[c*x^n]))/x - (b*d^2*e*n*(a + b*Log[c*x^n]))/(d + e*x)^2 - (8*b*d*e*n*(
a + b*Log[c*x^n]))/(d + e*x) - 13*e*(a + b*Log[c*x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/x + (d^3*e*(a + b*Log[c*
x^n])^2)/(d + e*x)^3 + (3*d^2*e*(a + b*Log[c*x^n])^2)/(d + e*x)^2 + (9*d*e*(a + b*Log[c*x^n])^2)/(d + e*x) + (
4*e*(a + b*Log[c*x^n])^3)/(b*n) + 8*b^2*e*n^2*(Log[x] - Log[d + e*x]) + (b^2*e*n^2*(d + (d + e*x)*Log[x] - (d
+ e*x)*Log[d + e*x]))/(d + e*x) + 26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 12*e*(a + b*Log[c*x^n])^2*Log
[1 + (e*x)/d] + 26*b^2*e*n^2*PolyLog[2, -((e*x)/d)] - 24*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 24*
b^2*e*n^2*PolyLog[3, -((e*x)/d)])/(3*d^5)

________________________________________________________________________________________

Maple [F]  time = 0.846, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{2} \left ( ex+d \right ) ^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2/x^2/(e*x+d)^4,x)

[Out]

int((a+b*ln(c*x^n))^2/x^2/(e*x+d)^4,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a^{2}{\left (\frac{12 \, e^{3} x^{3} + 30 \, d e^{2} x^{2} + 22 \, d^{2} e x + 3 \, d^{3}}{d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{3} + 3 \, d^{6} e x^{2} + d^{7} x} - \frac{12 \, e \log \left (e x + d\right )}{d^{5}} + \frac{12 \, e \log \left (x\right )}{d^{5}}\right )} + \int \frac{b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*a^2*((12*e^3*x^3 + 30*d*e^2*x^2 + 22*d^2*e*x + 3*d^3)/(d^4*e^3*x^4 + 3*d^5*e^2*x^3 + 3*d^6*e*x^2 + d^7*x)
- 12*e*log(e*x + d)/d^5 + 12*e*log(x)/d^5) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2
*log(c) + a*b)*log(x^n))/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3 + d^4*x^2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3 + d^
4*x^2), x)

________________________________________________________________________________________

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(c*x**n))**2/x**2/(e*x+d)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="giac")

[Out]

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