3.89 \(\int \frac{(a+b \log (c x^n))^3 \log (d (e+f x)^m)}{x^3} \, dx\)

Optimal. Leaf size=555 \[ -\frac{3 b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{3 b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{3 b f^2 m n \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (2,-\frac{e}{f x}\right )}{4 e^2}-\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (3,-\frac{e}{f x}\right )}{2 e^2}-\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (4,-\frac{e}{f x}\right )}{e^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}+\frac{3 b^2 f^2 m n^2 \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{3 b f^2 m n \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac{f^2 m \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{45 b^3 f m n^3}{8 e x} \]

[Out]

(-45*b^3*f*m*n^3)/(8*e*x) - (3*b^3*f^2*m*n^3*Log[x])/(8*e^2) - (21*b^2*f*m*n^2*(a + b*Log[c*x^n]))/(4*e*x) + (
3*b^2*f^2*m*n^2*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/(4*e^2) - (9*b*f*m*n*(a + b*Log[c*x^n])^2)/(4*e*x) + (3*b
*f^2*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(4*e^2) - (f*m*(a + b*Log[c*x^n])^3)/(2*e*x) + (f^2*m*Log[1 +
e/(f*x)]*(a + b*Log[c*x^n])^3)/(2*e^2) + (3*b^3*f^2*m*n^3*Log[e + f*x])/(8*e^2) - (3*b^3*n^3*Log[d*(e + f*x)^m
])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e
+ f*x)^m])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/(2*x^2) - (3*b^3*f^2*m*n^3*PolyLog[2, -(e/(f*x)
)])/(4*e^2) - (3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/(2*e^2) - (3*b*f^2*m*n*(a + b*Log[c*
x^n])^2*PolyLog[2, -(e/(f*x))])/(2*e^2) - (3*b^3*f^2*m*n^3*PolyLog[3, -(e/(f*x))])/(2*e^2) - (3*b^2*f^2*m*n^2*
(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x))])/e^2 - (3*b^3*f^2*m*n^3*PolyLog[4, -(e/(f*x))])/e^2

________________________________________________________________________________________

Rubi [A]  time = 1.01468, antiderivative size = 614, normalized size of antiderivative = 1.11, number of steps used = 30, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589, 2383} \[ \frac{3 b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{3 b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{3 b f^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{4 e^2}-\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{2 e^2}+\frac{3 b^3 f^2 m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{e^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}+\frac{3 b^2 f^2 m n^2 \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}+\frac{f^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{3 b f^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac{9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{45 b^3 f m n^3}{8 e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*f*m*n^3)/(8*e*x) - (3*b^3*f^2*m*n^3*Log[x])/(8*e^2) - (21*b^2*f*m*n^2*(a + b*Log[c*x^n]))/(4*e*x) - (
3*b*f^2*m*n*(a + b*Log[c*x^n])^2)/(8*e^2) - (9*b*f*m*n*(a + b*Log[c*x^n])^2)/(4*e*x) - (f^2*m*(a + b*Log[c*x^n
])^3)/(4*e^2) - (f*m*(a + b*Log[c*x^n])^3)/(2*e*x) - (f^2*m*(a + b*Log[c*x^n])^4)/(8*b*e^2*n) + (3*b^3*f^2*m*n
^3*Log[e + f*x])/(8*e^2) - (3*b^3*n^3*Log[d*(e + f*x)^m])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f
*x)^m])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[d*(e + f
*x)^m])/(2*x^2) + (3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(4*e^2) + (3*b*f^2*m*n*(a + b*Log[c*x^
n])^2*Log[1 + (f*x)/e])/(4*e^2) + (f^2*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/(2*e^2) + (3*b^3*f^2*m*n^3*Pol
yLog[2, -((f*x)/e)])/(4*e^2) + (3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(2*e^2) + (3*b*f^2*
m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)])/(2*e^2) - (3*b^3*f^2*m*n^3*PolyLog[3, -((f*x)/e)])/(2*e^2) -
(3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/e^2 + (3*b^3*f^2*m*n^3*PolyLog[4, -((f*x)/e)])/e^2

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]

Rule 2383

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^3} \, dx &=-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}-(f m) \int \left (-\frac{3 b^3 n^3}{8 x^2 (e+f x)}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2 (e+f x)}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2 (e+f x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (e+f x)}\right ) \, dx\\ &=-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{1}{2} (f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2 (e+f x)} \, dx+\frac{1}{4} (3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (e+f x)} \, dx+\frac{1}{4} \left (3 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2 (e+f x)} \, dx+\frac{1}{8} \left (3 b^3 f m n^3\right ) \int \frac{1}{x^2 (e+f x)} \, dx\\ &=-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{1}{2} (f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{e x^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e^2 x}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{e^2 (e+f x)}\right ) \, dx+\frac{1}{4} (3 b f m n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e x^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (e+f x)}\right ) \, dx+\frac{1}{4} \left (3 b^2 f m n^2\right ) \int \left (\frac{a+b \log \left (c x^n\right )}{e x^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 (e+f x)}\right ) \, dx+\frac{1}{8} \left (3 b^3 f m n^3\right ) \int \left (\frac{1}{e x^2}-\frac{f}{e^2 x}+\frac{f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac{3 b^3 f m n^3}{8 e x}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx}{2 e}-\frac{\left (f^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{2 e^2}+\frac{\left (f^3 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{2 e^2}+\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{4 e}-\frac{\left (3 b f^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{4 e^2}+\frac{\left (3 b f^3 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{4 e^2}+\frac{\left (3 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{4 e}-\frac{\left (3 b^2 f^2 m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{4 e^2}+\frac{\left (3 b^2 f^3 m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{4 e^2}\\ &=-\frac{9 b^3 f m n^3}{8 e x}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac{3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 e^2}-\frac{\left (3 f^2 m\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{\left (f^2 m\right ) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{2 e}-\frac{\left (3 b f^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{2 e^2}+\frac{\left (3 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e}-\frac{\left (3 b^2 f^2 m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{2 e^2}-\frac{\left (3 b^3 f^2 m n^3\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{4 e^2}\\ &=-\frac{21 b^3 f m n^3}{8 e x}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac{9 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 e^2}+\frac{3 b^3 f^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 e^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}+\frac{\left (3 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{e}-\frac{\left (3 b^2 f^2 m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{e^2}-\frac{\left (3 b^3 f^2 m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{2 e^2}\\ &=-\frac{45 b^3 f m n^3}{8 e x}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac{21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 e^2}+\frac{3 b^3 f^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 e^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}-\frac{3 b^3 f^2 m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{2 e^2}-\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{e^2}+\frac{\left (3 b^3 f^2 m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac{45 b^3 f m n^3}{8 e x}-\frac{3 b^3 f^2 m n^3 \log (x)}{8 e^2}-\frac{21 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e x}-\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac{9 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{4 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{2 e x}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{3 b^3 f^2 m n^3 \log (e+f x)}{8 e^2}-\frac{3 b^3 n^3 \log \left (d (e+f x)^m\right )}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 e^2}+\frac{3 b^3 f^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 e^2}+\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}+\frac{3 b f^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 e^2}-\frac{3 b^3 f^2 m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{2 e^2}-\frac{3 b^2 f^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{e^2}+\frac{3 b^3 f^2 m n^3 \text{Li}_4\left (-\frac{f x}{e}\right )}{e^2}\\ \end{align*}

Mathematica [B]  time = 0.762791, size = 1736, normalized size = 3.13 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*a^3*e*f*m*x + 18*a^2*b*e*f*m*n*x + 42*a*b^2*e*f*m*n^2*x + 45*b^3*e*f*m*n^3*x + 4*a^3*f^2*m*x^2*Log[x] + 6*
a^2*b*f^2*m*n*x^2*Log[x] + 6*a*b^2*f^2*m*n^2*x^2*Log[x] + 3*b^3*f^2*m*n^3*x^2*Log[x] - 6*a^2*b*f^2*m*n*x^2*Log
[x]^2 - 6*a*b^2*f^2*m*n^2*x^2*Log[x]^2 - 3*b^3*f^2*m*n^3*x^2*Log[x]^2 + 4*a*b^2*f^2*m*n^2*x^2*Log[x]^3 + 2*b^3
*f^2*m*n^3*x^2*Log[x]^3 - b^3*f^2*m*n^3*x^2*Log[x]^4 + 12*a^2*b*e*f*m*x*Log[c*x^n] + 36*a*b^2*e*f*m*n*x*Log[c*
x^n] + 42*b^3*e*f*m*n^2*x*Log[c*x^n] + 12*a^2*b*f^2*m*x^2*Log[x]*Log[c*x^n] + 12*a*b^2*f^2*m*n*x^2*Log[x]*Log[
c*x^n] + 6*b^3*f^2*m*n^2*x^2*Log[x]*Log[c*x^n] - 12*a*b^2*f^2*m*n*x^2*Log[x]^2*Log[c*x^n] - 6*b^3*f^2*m*n^2*x^
2*Log[x]^2*Log[c*x^n] + 4*b^3*f^2*m*n^2*x^2*Log[x]^3*Log[c*x^n] + 12*a*b^2*e*f*m*x*Log[c*x^n]^2 + 18*b^3*e*f*m
*n*x*Log[c*x^n]^2 + 12*a*b^2*f^2*m*x^2*Log[x]*Log[c*x^n]^2 + 6*b^3*f^2*m*n*x^2*Log[x]*Log[c*x^n]^2 - 6*b^3*f^2
*m*n*x^2*Log[x]^2*Log[c*x^n]^2 + 4*b^3*e*f*m*x*Log[c*x^n]^3 + 4*b^3*f^2*m*x^2*Log[x]*Log[c*x^n]^3 - 4*a^3*f^2*
m*x^2*Log[e + f*x] - 6*a^2*b*f^2*m*n*x^2*Log[e + f*x] - 6*a*b^2*f^2*m*n^2*x^2*Log[e + f*x] - 3*b^3*f^2*m*n^3*x
^2*Log[e + f*x] + 12*a^2*b*f^2*m*n*x^2*Log[x]*Log[e + f*x] + 12*a*b^2*f^2*m*n^2*x^2*Log[x]*Log[e + f*x] + 6*b^
3*f^2*m*n^3*x^2*Log[x]*Log[e + f*x] - 12*a*b^2*f^2*m*n^2*x^2*Log[x]^2*Log[e + f*x] - 6*b^3*f^2*m*n^3*x^2*Log[x
]^2*Log[e + f*x] + 4*b^3*f^2*m*n^3*x^2*Log[x]^3*Log[e + f*x] - 12*a^2*b*f^2*m*x^2*Log[c*x^n]*Log[e + f*x] - 12
*a*b^2*f^2*m*n*x^2*Log[c*x^n]*Log[e + f*x] - 6*b^3*f^2*m*n^2*x^2*Log[c*x^n]*Log[e + f*x] + 24*a*b^2*f^2*m*n*x^
2*Log[x]*Log[c*x^n]*Log[e + f*x] + 12*b^3*f^2*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[e + f*x] - 12*b^3*f^2*m*n^2*x^2*
Log[x]^2*Log[c*x^n]*Log[e + f*x] - 12*a*b^2*f^2*m*x^2*Log[c*x^n]^2*Log[e + f*x] - 6*b^3*f^2*m*n*x^2*Log[c*x^n]
^2*Log[e + f*x] + 12*b^3*f^2*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[e + f*x] - 4*b^3*f^2*m*x^2*Log[c*x^n]^3*Log[e + f
*x] + 4*a^3*e^2*Log[d*(e + f*x)^m] + 6*a^2*b*e^2*n*Log[d*(e + f*x)^m] + 6*a*b^2*e^2*n^2*Log[d*(e + f*x)^m] + 3
*b^3*e^2*n^3*Log[d*(e + f*x)^m] + 12*a^2*b*e^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 12*a*b^2*e^2*n*Log[c*x^n]*Log[d
*(e + f*x)^m] + 6*b^3*e^2*n^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 12*a*b^2*e^2*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 6
*b^3*e^2*n*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 4*b^3*e^2*Log[c*x^n]^3*Log[d*(e + f*x)^m] - 12*a^2*b*f^2*m*n*x^2*
Log[x]*Log[1 + (f*x)/e] - 12*a*b^2*f^2*m*n^2*x^2*Log[x]*Log[1 + (f*x)/e] - 6*b^3*f^2*m*n^3*x^2*Log[x]*Log[1 +
(f*x)/e] + 12*a*b^2*f^2*m*n^2*x^2*Log[x]^2*Log[1 + (f*x)/e] + 6*b^3*f^2*m*n^3*x^2*Log[x]^2*Log[1 + (f*x)/e] -
4*b^3*f^2*m*n^3*x^2*Log[x]^3*Log[1 + (f*x)/e] - 24*a*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] - 12*b
^3*f^2*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 12*b^3*f^2*m*n^2*x^2*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e
] - 12*b^3*f^2*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] - 6*b*f^2*m*n*x^2*(2*a^2 + 2*a*b*n + b^2*n^2 + 2*b
*(2*a + b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, -((f*x)/e)] + 12*b^2*f^2*m*n^2*x^2*(2*a + b*n + 2*b*L
og[c*x^n])*PolyLog[3, -((f*x)/e)] - 24*b^3*f^2*m*n^3*x^2*PolyLog[4, -((f*x)/e)])/(8*e^2*x^2)

________________________________________________________________________________________

Maple [F]  time = 6.108, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/x^3,x)

[Out]

int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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(c*x^n))^3*log(d*(f*x+e)^m)/x^3,x, algorithm="maxima")

[Out]

1/8*(4*(b^3*f^2*m*x^2*log(f*x + e) - b^3*f^2*m*x^2*log(x) - b^3*e*f*m*x - b^3*e^2*log(d))*log(x^n)^3 - (4*b^3*
e^2*log(x^n)^3 + 4*a^3*e^2 + 6*(e^2*n + 2*e^2*log(c))*a^2*b + 6*(e^2*n^2 + 2*e^2*n*log(c) + 2*e^2*log(c)^2)*a*
b^2 + (3*e^2*n^3 + 6*e^2*n^2*log(c) + 6*e^2*n*log(c)^2 + 4*e^2*log(c)^3)*b^3 + 6*(2*a*b^2*e^2 + (e^2*n + 2*e^2
*log(c))*b^3)*log(x^n)^2 + 6*(2*a^2*b*e^2 + 2*(e^2*n + 2*e^2*log(c))*a*b^2 + (e^2*n^2 + 2*e^2*n*log(c) + 2*e^2
*log(c)^2)*b^3)*log(x^n))*log((f*x + e)^m))/(e^2*x^2) - integrate(-1/8*(8*b^3*e^3*log(c)^3*log(d) + 24*a*b^2*e
^3*log(c)^2*log(d) + 24*a^2*b*e^3*log(c)*log(d) + 8*a^3*e^3*log(d) + 6*(2*b^3*e*f^2*m*n*x^2 + 4*a*b^2*e^3*log(
d) + 2*(e^3*n*log(d) + 2*e^3*log(c)*log(d))*b^3 + (2*(e^2*f*m + 2*e^2*f*log(d))*a*b^2 + (3*e^2*f*m*n + 2*e^2*f
*n*log(d) + 2*(e^2*f*m + 2*e^2*f*log(d))*log(c))*b^3)*x - 2*(b^3*f^3*m*n*x^3 + b^3*e*f^2*m*n*x^2)*log(f*x + e)
+ 2*(b^3*f^3*m*n*x^3 + b^3*e*f^2*m*n*x^2)*log(x))*log(x^n)^2 + (4*(e^2*f*m + 2*e^2*f*log(d))*a^3 + 6*(e^2*f*m
*n + 2*(e^2*f*m + 2*e^2*f*log(d))*log(c))*a^2*b + 6*(e^2*f*m*n^2 + 2*e^2*f*m*n*log(c) + 2*(e^2*f*m + 2*e^2*f*l
og(d))*log(c)^2)*a*b^2 + (3*e^2*f*m*n^3 + 6*e^2*f*m*n^2*log(c) + 6*e^2*f*m*n*log(c)^2 + 4*(e^2*f*m + 2*e^2*f*l
og(d))*log(c)^3)*b^3)*x + 6*(4*b^3*e^3*log(c)^2*log(d) + 8*a*b^2*e^3*log(c)*log(d) + 4*a^2*b*e^3*log(d) + (2*(
e^2*f*m + 2*e^2*f*log(d))*a^2*b + 2*(e^2*f*m*n + 2*(e^2*f*m + 2*e^2*f*log(d))*log(c))*a*b^2 + (e^2*f*m*n^2 + 2
*e^2*f*m*n*log(c) + 2*(e^2*f*m + 2*e^2*f*log(d))*log(c)^2)*b^3)*x)*log(x^n))/(e^2*f*x^4 + e^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x+e)^m)/x^3,x, algorithm="fricas")

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log((f*x + e)^m*d)/x^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(c*x**n))**3*ln(d*(f*x+e)**m)/x**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 (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x+e)^m)/x^3,x, algorithm="giac")

[Out]

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