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

Optimal. Leaf size=420 \[ \frac{2 b f^3 m n \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{2 b^2 f^3 m n^2 \text{PolyLog}\left (2,-\frac{e}{f x}\right )}{9 e^3}+\frac{2 b^2 f^3 m n^2 \text{PolyLog}\left (3,-\frac{e}{f x}\right )}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{f^3 m \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac{2 b f^3 m n \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac{8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}-\frac{5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}+\frac{26 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{19 b^2 f m n^2}{108 e x^2} \]

[Out]

(-19*b^2*f*m*n^2)/(108*e*x^2) + (26*b^2*f^2*m*n^2)/(27*e^2*x) + (2*b^2*f^3*m*n^2*Log[x])/(27*e^3) - (5*b*f*m*n
*(a + b*Log[c*x^n]))/(18*e*x^2) + (8*b*f^2*m*n*(a + b*Log[c*x^n]))/(9*e^2*x) - (2*b*f^3*m*n*Log[1 + e/(f*x)]*(
a + b*Log[c*x^n]))/(9*e^3) - (f*m*(a + b*Log[c*x^n])^2)/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n])^2)/(3*e^2*x) - (
f^3*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(3*e^3) - (2*b^2*f^3*m*n^2*Log[e + f*x])/(27*e^3) - (2*b^2*n^2*Lo
g[d*(e + f*x)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log
[d*(e + f*x)^m])/(3*x^3) + (2*b^2*f^3*m*n^2*PolyLog[2, -(e/(f*x))])/(9*e^3) + (2*b*f^3*m*n*(a + b*Log[c*x^n])*
PolyLog[2, -(e/(f*x))])/(3*e^3) + (2*b^2*f^3*m*n^2*PolyLog[3, -(e/(f*x))])/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.724222, antiderivative size = 462, normalized size of antiderivative = 1.1, number of steps used = 22, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589} \[ -\frac{2 b f^3 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 b^2 f^3 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{9 e^3}+\frac{2 b^2 f^3 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^3}{9 b e^3 n}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac{f^3 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac{2 b f^3 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac{8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}-\frac{5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}+\frac{26 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{19 b^2 f m n^2}{108 e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*b^2*f*m*n^2)/(108*e*x^2) + (26*b^2*f^2*m*n^2)/(27*e^2*x) + (2*b^2*f^3*m*n^2*Log[x])/(27*e^3) - (5*b*f*m*n
*(a + b*Log[c*x^n]))/(18*e*x^2) + (8*b*f^2*m*n*(a + b*Log[c*x^n]))/(9*e^2*x) + (f^3*m*(a + b*Log[c*x^n])^2)/(9
*e^3) - (f*m*(a + b*Log[c*x^n])^2)/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n])^2)/(3*e^2*x) + (f^3*m*(a + b*Log[c*x^
n])^3)/(9*b*e^3*n) - (2*b^2*f^3*m*n^2*Log[e + f*x])/(27*e^3) - (2*b^2*n^2*Log[d*(e + f*x)^m])/(27*x^3) - (2*b*
n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(3*x^3) - (2*b*f^
3*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(9*e^3) - (f^3*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(3*e^3) - (
2*b^2*f^3*m*n^2*PolyLog[2, -((f*x)/e)])/(9*e^3) - (2*b*f^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(3*e
^3) + (2*b^2*f^3*m*n^2*PolyLog[3, -((f*x)/e)])/(3*e^3)

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 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 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 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 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 \log \left (d (e+f x)^m\right )}{x^4} \, dx &=-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-(f m) \int \left (-\frac{2 b^2 n^2}{27 x^3 (e+f x)}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{9 x^3 (e+f x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3 (e+f x)}\right ) \, dx\\ &=-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{1}{3} (f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (e+f x)} \, dx+\frac{1}{9} (2 b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3 (e+f x)} \, dx+\frac{1}{27} \left (2 b^2 f m n^2\right ) \int \frac{1}{x^3 (e+f x)} \, dx\\ &=-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{1}{3} (f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e x^3}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x^2}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (e+f x)}\right ) \, dx+\frac{1}{9} (2 b f m n) \int \left (\frac{a+b \log \left (c x^n\right )}{e x^3}-\frac{f \left (a+b \log \left (c x^n\right )\right )}{e^2 x^2}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (e+f x)}\right ) \, dx+\frac{1}{27} \left (2 b^2 f m n^2\right ) \int \left (\frac{1}{e x^3}-\frac{f}{e^2 x^2}+\frac{f^2}{e^3 x}-\frac{f^3}{e^3 (e+f x)}\right ) \, dx\\ &=-\frac{b^2 f m n^2}{27 e x^2}+\frac{2 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{3 e}-\frac{\left (f^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{3 e^2}+\frac{\left (f^3 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{3 e^3}-\frac{\left (f^4 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{3 e^3}+\frac{(2 b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{9 e}-\frac{\left (2 b f^2 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{9 e^2}+\frac{\left (2 b f^3 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{9 e^3}-\frac{\left (2 b f^4 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{9 e^3}\\ &=-\frac{5 b^2 f m n^2}{54 e x^2}+\frac{8 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x^2}+\frac{2 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac{2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 e^3}+\frac{\left (f^3 m\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{3 b e^3 n}+\frac{(b f m n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{3 e}-\frac{\left (2 b f^2 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{3 e^2}+\frac{\left (2 b f^3 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{3 e^3}+\frac{\left (2 b^2 f^3 m n^2\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{9 e^3}\\ &=-\frac{19 b^2 f m n^2}{108 e x^2}+\frac{26 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac{8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^3}{9 b e^3 n}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac{2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 e^3}-\frac{2 b^2 f^3 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{9 e^3}-\frac{2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{3 e^3}+\frac{\left (2 b^2 f^3 m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{3 e^3}\\ &=-\frac{19 b^2 f m n^2}{108 e x^2}+\frac{26 b^2 f^2 m n^2}{27 e^2 x}+\frac{2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac{5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac{8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^3}{9 b e^3 n}-\frac{2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac{2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac{2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 e^3}-\frac{2 b^2 f^3 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{9 e^3}-\frac{2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{3 e^3}+\frac{2 b^2 f^3 m n^2 \text{Li}_3\left (-\frac{f x}{e}\right )}{3 e^3}\\ \end{align*}

Mathematica [B]  time = 0.42875, size = 909, normalized size = 2.16 \[ -\frac{36 a^2 \log \left (d (e+f x)^m\right ) e^3+8 b^2 n^2 \log \left (d (e+f x)^m\right ) e^3+36 b^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right ) e^3+24 a b n \log \left (d (e+f x)^m\right ) e^3+72 a b \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) e^3+24 b^2 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) e^3+18 b^2 f m x \log ^2\left (c x^n\right ) e^2+19 b^2 f m n^2 x e^2+18 a^2 f m x e^2+30 a b f m n x e^2+36 a b f m x \log \left (c x^n\right ) e^2+30 b^2 f m n x \log \left (c x^n\right ) e^2-104 b^2 f^2 m n^2 x^2 e-36 a^2 f^2 m x^2 e-96 a b f^2 m n x^2 e-36 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) e-72 a b f^2 m x^2 \log \left (c x^n\right ) e-96 b^2 f^2 m n x^2 \log \left (c x^n\right ) e-12 b^2 f^3 m n^2 x^3 \log ^3(x)+12 b^2 f^3 m n^2 x^3 \log ^2(x)+36 a b f^3 m n x^3 \log ^2(x)-36 b^2 f^3 m x^3 \log (x) \log ^2\left (c x^n\right )-8 b^2 f^3 m n^2 x^3 \log (x)-36 a^2 f^3 m x^3 \log (x)-24 a b f^3 m n x^3 \log (x)+36 b^2 f^3 m n x^3 \log ^2(x) \log \left (c x^n\right )-72 a b f^3 m x^3 \log (x) \log \left (c x^n\right )-24 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right )+8 b^2 f^3 m n^2 x^3 \log (e+f x)+36 a^2 f^3 m x^3 \log (e+f x)+24 a b f^3 m n x^3 \log (e+f x)+36 b^2 f^3 m n^2 x^3 \log ^2(x) \log (e+f x)+36 b^2 f^3 m x^3 \log ^2\left (c x^n\right ) \log (e+f x)-24 b^2 f^3 m n^2 x^3 \log (x) \log (e+f x)-72 a b f^3 m n x^3 \log (x) \log (e+f x)+72 a b f^3 m x^3 \log \left (c x^n\right ) \log (e+f x)+24 b^2 f^3 m n x^3 \log \left (c x^n\right ) \log (e+f x)-72 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right ) \log (e+f x)-36 b^2 f^3 m n^2 x^3 \log ^2(x) \log \left (\frac{f x}{e}+1\right )+24 b^2 f^3 m n^2 x^3 \log (x) \log \left (\frac{f x}{e}+1\right )+72 a b f^3 m n x^3 \log (x) \log \left (\frac{f x}{e}+1\right )+72 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+24 b f^3 m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f x}{e}\right )-72 b^2 f^3 m n^2 x^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{108 e^3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(18*a^2*e^2*f*m*x + 30*a*b*e^2*f*m*n*x + 19*b^2*e^2*f*m*n^2*x - 36*a^2*e*f^2*m*x^2 - 96*a*b*e*f^2*m*n*x^2 - 1
04*b^2*e*f^2*m*n^2*x^2 - 36*a^2*f^3*m*x^3*Log[x] - 24*a*b*f^3*m*n*x^3*Log[x] - 8*b^2*f^3*m*n^2*x^3*Log[x] + 36
*a*b*f^3*m*n*x^3*Log[x]^2 + 12*b^2*f^3*m*n^2*x^3*Log[x]^2 - 12*b^2*f^3*m*n^2*x^3*Log[x]^3 + 36*a*b*e^2*f*m*x*L
og[c*x^n] + 30*b^2*e^2*f*m*n*x*Log[c*x^n] - 72*a*b*e*f^2*m*x^2*Log[c*x^n] - 96*b^2*e*f^2*m*n*x^2*Log[c*x^n] -
72*a*b*f^3*m*x^3*Log[x]*Log[c*x^n] - 24*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n] + 36*b^2*f^3*m*n*x^3*Log[x]^2*Log[c*
x^n] + 18*b^2*e^2*f*m*x*Log[c*x^n]^2 - 36*b^2*e*f^2*m*x^2*Log[c*x^n]^2 - 36*b^2*f^3*m*x^3*Log[x]*Log[c*x^n]^2
+ 36*a^2*f^3*m*x^3*Log[e + f*x] + 24*a*b*f^3*m*n*x^3*Log[e + f*x] + 8*b^2*f^3*m*n^2*x^3*Log[e + f*x] - 72*a*b*
f^3*m*n*x^3*Log[x]*Log[e + f*x] - 24*b^2*f^3*m*n^2*x^3*Log[x]*Log[e + f*x] + 36*b^2*f^3*m*n^2*x^3*Log[x]^2*Log
[e + f*x] + 72*a*b*f^3*m*x^3*Log[c*x^n]*Log[e + f*x] + 24*b^2*f^3*m*n*x^3*Log[c*x^n]*Log[e + f*x] - 72*b^2*f^3
*m*n*x^3*Log[x]*Log[c*x^n]*Log[e + f*x] + 36*b^2*f^3*m*x^3*Log[c*x^n]^2*Log[e + f*x] + 36*a^2*e^3*Log[d*(e + f
*x)^m] + 24*a*b*e^3*n*Log[d*(e + f*x)^m] + 8*b^2*e^3*n^2*Log[d*(e + f*x)^m] + 72*a*b*e^3*Log[c*x^n]*Log[d*(e +
f*x)^m] + 24*b^2*e^3*n*Log[c*x^n]*Log[d*(e + f*x)^m] + 36*b^2*e^3*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 72*a*b*f^
3*m*n*x^3*Log[x]*Log[1 + (f*x)/e] + 24*b^2*f^3*m*n^2*x^3*Log[x]*Log[1 + (f*x)/e] - 36*b^2*f^3*m*n^2*x^3*Log[x]
^2*Log[1 + (f*x)/e] + 72*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 24*b*f^3*m*n*x^3*(3*a + b*n + 3*
b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] - 72*b^2*f^3*m*n^2*x^3*PolyLog[3, -((f*x)/e)])/(108*e^3*x^3)

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Maple [F]  time = 1.957, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{9 \,{\left (2 \, b^{2} f^{3} m x^{3} \log \left (f x + e\right ) - 2 \, b^{2} f^{3} m x^{3} \log \left (x\right ) - 2 \, b^{2} e f^{2} m x^{2} + b^{2} e^{2} f m x + 2 \, b^{2} e^{3} \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (9 \, b^{2} e^{3} \log \left (x^{n}\right )^{2} + 9 \, a^{2} e^{3} + 6 \,{\left (e^{3} n + 3 \, e^{3} \log \left (c\right )\right )} a b +{\left (2 \, e^{3} n^{2} + 6 \, e^{3} n \log \left (c\right ) + 9 \, e^{3} \log \left (c\right )^{2}\right )} b^{2} + 6 \,{\left (3 \, a b e^{3} +{\left (e^{3} n + 3 \, e^{3} \log \left (c\right )\right )} b^{2}\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{54 \, e^{3} x^{3}} + \int \frac{27 \, b^{2} e^{4} \log \left (c\right )^{2} \log \left (d\right ) + 54 \, a b e^{4} \log \left (c\right ) \log \left (d\right ) + 27 \, a^{2} e^{4} \log \left (d\right ) +{\left (9 \,{\left (e^{3} f m + 3 \, e^{3} f \log \left (d\right )\right )} a^{2} + 6 \,{\left (e^{3} f m n + 3 \,{\left (e^{3} f m + 3 \, e^{3} f \log \left (d\right )\right )} \log \left (c\right )\right )} a b +{\left (2 \, e^{3} f m n^{2} + 6 \, e^{3} f m n \log \left (c\right ) + 9 \,{\left (e^{3} f m + 3 \, e^{3} f \log \left (d\right )\right )} \log \left (c\right )^{2}\right )} b^{2}\right )} x - 3 \,{\left (6 \, b^{2} e f^{3} m n x^{3} + 3 \, b^{2} e^{2} f^{2} m n x^{2} - 18 \, a b e^{4} \log \left (d\right ) - 6 \,{\left (e^{4} n \log \left (d\right ) + 3 \, e^{4} \log \left (c\right ) \log \left (d\right )\right )} b^{2} -{\left (6 \,{\left (e^{3} f m + 3 \, e^{3} f \log \left (d\right )\right )} a b +{\left (5 \, e^{3} f m n + 6 \, e^{3} f n \log \left (d\right ) + 6 \,{\left (e^{3} f m + 3 \, e^{3} f \log \left (d\right )\right )} \log \left (c\right )\right )} b^{2}\right )} x - 6 \,{\left (b^{2} f^{4} m n x^{4} + b^{2} e f^{3} m n x^{3}\right )} \log \left (f x + e\right ) + 6 \,{\left (b^{2} f^{4} m n x^{4} + b^{2} e f^{3} m n x^{3}\right )} \log \left (x\right )\right )} \log \left (x^{n}\right )}{27 \,{\left (e^{3} f x^{5} + e^{4} x^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(9*(2*b^2*f^3*m*x^3*log(f*x + e) - 2*b^2*f^3*m*x^3*log(x) - 2*b^2*e*f^2*m*x^2 + b^2*e^2*f*m*x + 2*b^2*e^
3*log(d))*log(x^n)^2 + 2*(9*b^2*e^3*log(x^n)^2 + 9*a^2*e^3 + 6*(e^3*n + 3*e^3*log(c))*a*b + (2*e^3*n^2 + 6*e^3
*n*log(c) + 9*e^3*log(c)^2)*b^2 + 6*(3*a*b*e^3 + (e^3*n + 3*e^3*log(c))*b^2)*log(x^n))*log((f*x + e)^m))/(e^3*
x^3) + integrate(1/27*(27*b^2*e^4*log(c)^2*log(d) + 54*a*b*e^4*log(c)*log(d) + 27*a^2*e^4*log(d) + (9*(e^3*f*m
+ 3*e^3*f*log(d))*a^2 + 6*(e^3*f*m*n + 3*(e^3*f*m + 3*e^3*f*log(d))*log(c))*a*b + (2*e^3*f*m*n^2 + 6*e^3*f*m*
n*log(c) + 9*(e^3*f*m + 3*e^3*f*log(d))*log(c)^2)*b^2)*x - 3*(6*b^2*e*f^3*m*n*x^3 + 3*b^2*e^2*f^2*m*n*x^2 - 18
*a*b*e^4*log(d) - 6*(e^4*n*log(d) + 3*e^4*log(c)*log(d))*b^2 - (6*(e^3*f*m + 3*e^3*f*log(d))*a*b + (5*e^3*f*m*
n + 6*e^3*f*n*log(d) + 6*(e^3*f*m + 3*e^3*f*log(d))*log(c))*b^2)*x - 6*(b^2*f^4*m*n*x^4 + b^2*e*f^3*m*n*x^3)*l
og(f*x + e) + 6*(b^2*f^4*m*n*x^4 + b^2*e*f^3*m*n*x^3)*log(x))*log(x^n))/(e^3*f*x^5 + e^4*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log((f*x + e)^m*d)/x^4, 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(c*x**n))**2*ln(d*(f*x+e)**m)/x**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 (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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