3.85 \(\int x (a+b \log (c x^n))^3 \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=603 \[ \frac{3 b^2 e^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}+\frac{3 b^2 e^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{3 b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{4 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{f^2}+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{21 a b^2 e m n^2 x}{4 f}-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{3 b e^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}-\frac{e^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^2}-\frac{9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}+\frac{3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{21 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{45 b^3 e m n^3 x}{8 f}+\frac{3}{4} b^3 m n^3 x^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.974442, antiderivative size = 603, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {2305, 2304, 2378, 43, 2351, 2295, 2317, 2391, 2353, 2296, 2374, 6589, 2383} \[ \frac{3 b^2 e^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}+\frac{3 b^2 e^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{3 b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{4 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{f^2}+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac{9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{21 a b^2 e m n^2 x}{4 f}-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac{3 b e^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}-\frac{e^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^2}-\frac{9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}+\frac{3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{21 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{45 b^3 e m n^3 x}{8 f}+\frac{3}{4} b^3 m n^3 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*a*b^2*e*m*n^2*x)/(4*f) - (45*b^3*e*m*n^3*x)/(8*f) + (3*b^3*m*n^3*x^2)/4 + (21*b^3*e*m*n^2*x*Log[c*x^n])/(4
*f) - (9*b^2*m*n^2*x^2*(a + b*Log[c*x^n]))/8 - (9*b*e*m*n*x*(a + b*Log[c*x^n])^2)/(4*f) + (3*b*m*n*x^2*(a + b*
Log[c*x^n])^2)/4 + (e*m*x*(a + b*Log[c*x^n])^3)/(2*f) - (m*x^2*(a + b*Log[c*x^n])^3)/4 + (3*b^3*e^2*m*n^3*Log[
e + f*x])/(8*f^2) - (3*b^3*n^3*x^2*Log[d*(e + f*x)^m])/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m
])/4 - (3*b*n*x^2*(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
- (3*b^2*e^2*m*n^2*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(4*f^2) + (3*b*e^2*m*n*(a + b*Log[c*x^n])^2*Log[1 + (
f*x)/e])/(4*f^2) - (e^2*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/(2*f^2) - (3*b^3*e^2*m*n^3*PolyLog[2, -((f*x)
/e)])/(4*f^2) + (3*b^2*e^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(2*f^2) - (3*b*e^2*m*n*(a + b*Log[
c*x^n])^2*PolyLog[2, -((f*x)/e)])/(2*f^2) - (3*b^3*e^2*m*n^3*PolyLog[3, -((f*x)/e)])/(2*f^2) + (3*b^2*e^2*m*n^
2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/f^2 - (3*b^3*e^2*m*n^3*PolyLog[4, -((f*x)/e)])/f^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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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 2296

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

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 x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx &=-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (-\frac{3 b^3 n^3 x^2}{8 (e+f x)}+\frac{3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{4 (e+f x)}-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 (e+f x)}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e+f x)}\right ) \, dx\\ &=-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{1}{2} (f m) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx+\frac{1}{4} (3 b f m n) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx-\frac{1}{4} \left (3 b^2 f m n^2\right ) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx+\frac{1}{8} \left (3 b^3 f m n^3\right ) \int \frac{x^2}{e+f x} \, dx\\ &=-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{1}{2} (f m) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{f^2 (e+f x)}\right ) \, dx+\frac{1}{4} (3 b f m n) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 (e+f x)}\right ) \, dx-\frac{1}{4} \left (3 b^2 f m n^2\right ) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{f^2 (e+f x)}\right ) \, dx+\frac{1}{8} \left (3 b^3 f m n^3\right ) \int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx\\ &=-\frac{3 b^3 e m n^3 x}{8 f}+\frac{3}{16} b^3 m n^3 x^2+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{1}{2} m \int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx+\frac{(e m) \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx}{2 f}-\frac{\left (e^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{2 f}+\frac{1}{4} (3 b m n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{(3 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 f}+\frac{\left (3 b e^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{4 f}-\frac{1}{4} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (3 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 f}-\frac{\left (3 b^2 e^2 m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{4 f}\\ &=\frac{3 a b^2 e m n^2 x}{4 f}-\frac{3 b^3 e m n^3 x}{8 f}+\frac{3}{8} b^3 m n^3 x^2-\frac{3}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3}{8} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 f^2}+\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 f^2}+\frac{1}{4} (3 b m n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac{\left (3 b e^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 f^2}-\frac{(3 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 f}-\frac{1}{4} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{\left (3 b^2 e^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 f^2}+\frac{\left (3 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f}+\frac{\left (3 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{4 f}+\frac{\left (3 b^3 e^2 m n^3\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{4 f^2}\\ &=\frac{9 a b^2 e m n^2 x}{4 f}-\frac{9 b^3 e m n^3 x}{8 f}+\frac{9}{16} b^3 m n^3 x^2+\frac{3 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac{3}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 f^2}+\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 f^2}+\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{1}{4} \left (3 b^2 m n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (3 b^2 e^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}{f^2}+\frac{\left (3 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{f}+\frac{\left (3 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 f}-\frac{\left (3 b^3 e^2 m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{2 f^2}\\ &=\frac{21 a b^2 e m n^2 x}{4 f}-\frac{21 b^3 e m n^3 x}{8 f}+\frac{3}{4} b^3 m n^3 x^2+\frac{9 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac{9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 f^2}+\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 f^2}+\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{2 f^2}+\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{f^2}+\frac{\left (3 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{f}-\frac{\left (3 b^3 e^2 m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx}{f^2}\\ &=\frac{21 a b^2 e m n^2 x}{4 f}-\frac{45 b^3 e m n^3 x}{8 f}+\frac{3}{4} b^3 m n^3 x^2+\frac{21 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac{9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac{3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{4 f^2}+\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{4 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{4 f^2}+\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{2 f^2}+\frac{3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{f^2}-\frac{3 b^3 e^2 m n^3 \text{Li}_4\left (-\frac{f x}{e}\right )}{f^2}\\ \end{align*}

Mathematica [B]  time = 0.554275, size = 1431, normalized size = 2.37 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m],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 - 2*a^3*f^2*m*x^2 + 6*a^2*b*f^
2*m*n*x^2 - 9*a*b^2*f^2*m*n^2*x^2 + 6*b^3*f^2*m*n^3*x^2 + 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] - 6*a^2*b*f^2*m*x^2*Log[c*x^n] + 12*a*b^2*f^2*m*n*x^2*Log[c*x^n] - 9*b
^3*f^2*m*n^2*x^2*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 - 6*a*b^2*f^2*m*x^
2*Log[c*x^n]^2 + 6*b^3*f^2*m*n*x^2*Log[c*x^n]^2 + 4*b^3*e*f*m*x*Log[c*x^n]^3 - 2*b^3*f^2*m*x^2*Log[c*x^n]^3 -
4*a^3*e^2*m*Log[e + f*x] + 6*a^2*b*e^2*m*n*Log[e + f*x] - 6*a*b^2*e^2*m*n^2*Log[e + f*x] + 3*b^3*e^2*m*n^3*Log
[e + f*x] + 12*a^2*b*e^2*m*n*Log[x]*Log[e + f*x] - 12*a*b^2*e^2*m*n^2*Log[x]*Log[e + f*x] + 6*b^3*e^2*m*n^3*Lo
g[x]*Log[e + f*x] - 12*a*b^2*e^2*m*n^2*Log[x]^2*Log[e + f*x] + 6*b^3*e^2*m*n^3*Log[x]^2*Log[e + f*x] + 4*b^3*e
^2*m*n^3*Log[x]^3*Log[e + f*x] - 12*a^2*b*e^2*m*Log[c*x^n]*Log[e + f*x] + 12*a*b^2*e^2*m*n*Log[c*x^n]*Log[e +
f*x] - 6*b^3*e^2*m*n^2*Log[c*x^n]*Log[e + f*x] + 24*a*b^2*e^2*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] - 12*b^3*e^2*
m*n^2*Log[x]*Log[c*x^n]*Log[e + f*x] - 12*b^3*e^2*m*n^2*Log[x]^2*Log[c*x^n]*Log[e + f*x] - 12*a*b^2*e^2*m*Log[
c*x^n]^2*Log[e + f*x] + 6*b^3*e^2*m*n*Log[c*x^n]^2*Log[e + f*x] + 12*b^3*e^2*m*n*Log[x]*Log[c*x^n]^2*Log[e + f
*x] - 4*b^3*e^2*m*Log[c*x^n]^3*Log[e + f*x] + 4*a^3*f^2*x^2*Log[d*(e + f*x)^m] - 6*a^2*b*f^2*n*x^2*Log[d*(e +
f*x)^m] + 6*a*b^2*f^2*n^2*x^2*Log[d*(e + f*x)^m] - 3*b^3*f^2*n^3*x^2*Log[d*(e + f*x)^m] + 12*a^2*b*f^2*x^2*Log
[c*x^n]*Log[d*(e + f*x)^m] - 12*a*b^2*f^2*n*x^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 6*b^3*f^2*n^2*x^2*Log[c*x^n]*L
og[d*(e + f*x)^m] + 12*a*b^2*f^2*x^2*Log[c*x^n]^2*Log[d*(e + f*x)^m] - 6*b^3*f^2*n*x^2*Log[c*x^n]^2*Log[d*(e +
f*x)^m] + 4*b^3*f^2*x^2*Log[c*x^n]^3*Log[d*(e + f*x)^m] - 12*a^2*b*e^2*m*n*Log[x]*Log[1 + (f*x)/e] + 12*a*b^2
*e^2*m*n^2*Log[x]*Log[1 + (f*x)/e] - 6*b^3*e^2*m*n^3*Log[x]*Log[1 + (f*x)/e] + 12*a*b^2*e^2*m*n^2*Log[x]^2*Log
[1 + (f*x)/e] - 6*b^3*e^2*m*n^3*Log[x]^2*Log[1 + (f*x)/e] - 4*b^3*e^2*m*n^3*Log[x]^3*Log[1 + (f*x)/e] - 24*a*b
^2*e^2*m*n*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 12*b^3*e^2*m*n^2*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 12*b^3*e
^2*m*n^2*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e] - 12*b^3*e^2*m*n*Log[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] - 6*b*e^2*
m*n*(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*e^2*m*n^2*(2*a - b*n + 2*b*Log[c*x^n])*PolyLog[3, -((f*x)/e)] - 24*b^3*e^2*m*n^3*PolyLog[4, -((f*x)/e)])/
(8*f^2)

________________________________________________________________________________________

Maple [F]  time = 36.276, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*x*log(c*x^n)^3 + 3*a*b^2*x*log(c*x^n)^2 + 3*a^2*b*x*log(c*x^n) + a^3*x)*log((f*x + e)^m*d), 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(x*(a+b*ln(c*x**n))**3*ln(d*(f*x+e)**m),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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