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

Optimal. Leaf size=473 \[ -\frac {6 b^2 e m n^2 \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {6 b^2 e m n^2 \text {Li}_3\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {3 b e m n \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {3 b e m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {e m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )}{f}-18 b^3 m n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{f}+18 b^3 m n^3 x \]

[Out]

-12*a*b^2*m*n^2*x+18*b^3*m*n^3*x-6*b^2*m*n^2*(-b*n+a)*x-18*b^3*m*n^2*x*ln(c*x^n)+6*b*m*n*x*(a+b*ln(c*x^n))^2-m
*x*(a+b*ln(c*x^n))^3+6*b^2*e*m*n^2*(-b*n+a)*ln(f*x+e)/f+6*a*b^2*n^2*x*ln(d*(f*x+e)^m)-6*b^3*n^3*x*ln(d*(f*x+e)
^m)+6*b^3*n^2*x*ln(c*x^n)*ln(d*(f*x+e)^m)-3*b*n*x*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)+x*(a+b*ln(c*x^n))^3*ln(d*(
f*x+e)^m)+6*b^3*e*m*n^2*ln(c*x^n)*ln(1+f*x/e)/f-3*b*e*m*n*(a+b*ln(c*x^n))^2*ln(1+f*x/e)/f+e*m*(a+b*ln(c*x^n))^
3*ln(1+f*x/e)/f+6*b^3*e*m*n^3*polylog(2,-f*x/e)/f-6*b^2*e*m*n^2*(a+b*ln(c*x^n))*polylog(2,-f*x/e)/f+3*b*e*m*n*
(a+b*ln(c*x^n))^2*polylog(2,-f*x/e)/f+6*b^3*e*m*n^3*polylog(3,-f*x/e)/f-6*b^2*e*m*n^2*(a+b*ln(c*x^n))*polylog(
3,-f*x/e)/f+6*b^3*e*m*n^3*polylog(4,-f*x/e)/f

________________________________________________________________________________________

Rubi [A]  time = 0.65, antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2296, 2295, 2371, 6, 43, 2351, 2317, 2391, 2353, 2374, 6589, 2383} \[ -\frac {6 b^2 e m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {6 b^2 e m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}+\frac {3 b e m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {6 b^3 e m n^3 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {PolyLog}\left (4,-\frac {f x}{e}\right )}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac {3 b e m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {e m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f}+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )}{f}-18 b^3 m n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+18 b^3 m n^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-12*a*b^2*m*n^2*x + 18*b^3*m*n^3*x - 6*b^2*m*n^2*(a - b*n)*x - 18*b^3*m*n^2*x*Log[c*x^n] + 6*b*m*n*x*(a + b*Lo
g[c*x^n])^2 - m*x*(a + b*Log[c*x^n])^3 + (6*b^2*e*m*n^2*(a - b*n)*Log[e + f*x])/f + 6*a*b^2*n^2*x*Log[d*(e + f
*x)^m] - 6*b^3*n^3*x*Log[d*(e + f*x)^m] + 6*b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*b*n*x*(a + b*Log[c*x^n
])^2*Log[d*(e + f*x)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m] + (6*b^3*e*m*n^2*Log[c*x^n]*Log[1 + (f*x)/
e])/f - (3*b*e*m*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f + (e*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/f +
(6*b^3*e*m*n^3*PolyLog[2, -((f*x)/e)])/f - (6*b^2*e*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f + (3*b*
e*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)])/f + (6*b^3*e*m*n^3*PolyLog[3, -((f*x)/e)])/f - (6*b^2*e*m*n
^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/f + (6*b^3*e*m*n^3*PolyLog[4, -((f*x)/e)])/f

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 2295

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

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 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 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 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 2371

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[(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, r, m, n}, x] && IGtQ[p, 0] && IntegerQ[m]

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 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]

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 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 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {6 a b^2 n^2 x}{e+f x}-\frac {6 b^3 n^3 x}{e+f x}+\frac {6 b^3 n^2 x \log \left (c x^n\right )}{e+f x}-\frac {3 b n x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {\left (6 a b^2 n^2-6 b^3 n^3\right ) x}{e+f x}+\frac {6 b^3 n^2 x \log \left (c x^n\right )}{e+f x}-\frac {3 b n x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx+(3 b f m n) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx-\left (6 b^3 f m n^2\right ) \int \frac {x \log \left (c x^n\right )}{e+f x} \, dx-\left (6 b^2 f m n^2 (a-b n)\right ) \int \frac {x}{e+f x} \, dx\\ &=6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{f (e+f x)}\right ) \, dx+(3 b f m n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{f (e+f x)}\right ) \, dx-\left (6 b^3 f m n^2\right ) \int \left (\frac {\log \left (c x^n\right )}{f}-\frac {e \log \left (c x^n\right )}{f (e+f x)}\right ) \, dx-\left (6 b^2 f m n^2 (a-b n)\right ) \int \left (\frac {1}{f}-\frac {e}{f (e+f x)}\right ) \, dx\\ &=-6 b^2 m n^2 (a-b n) x+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-m \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx+(e m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx+(3 b m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(3 b e m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx+\left (6 b^3 e m n^2\right ) \int \frac {\log \left (c x^n\right )}{e+f x} \, dx\\ &=6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-6 b^3 m n^2 x \log \left (c x^n\right )+3 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{f}+(3 b m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {(3 b e m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{f}-\left (6 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (6 b^2 e 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}{f}-\frac {\left (6 b^3 e m n^3\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{f}\\ &=-6 a b^2 m n^2 x+6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-6 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}-\left (6 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac {\left (6 b^2 e 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}+\frac {\left (6 b^3 e m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{f}\\ &=-12 a b^2 m n^2 x+12 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-12 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{f}-\left (6 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx+\frac {\left (6 b^3 e m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx}{f}\\ &=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x-18 b^3 m n^2 x \log \left (c x^n\right )+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {6 b^2 e m n^2 (a-b n) \log (e+f x)}{f}+6 a b^2 n^2 x \log \left (d (e+f x)^m\right )-6 b^3 n^3 x \log \left (d (e+f x)^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )+\frac {6 b^3 e m n^2 \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{f}-\frac {6 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{f}+\frac {6 b^3 e m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{f}\\ \end {align*}

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Mathematica [B]  time = 0.44, size = 1122, normalized size = 2.37 \[ \frac {-f m x a^3+e m \log (e+f x) a^3+f x \log \left (d (e+f x)^m\right ) a^3+6 b f m n x a^2-3 b f m x \log \left (c x^n\right ) a^2-3 b e m n \log (e+f x) a^2-3 b e m n \log (x) \log (e+f x) a^2+3 b e m \log \left (c x^n\right ) \log (e+f x) a^2-3 b f n x \log \left (d (e+f x)^m\right ) a^2+3 b f x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) a^2+3 b e m n \log (x) \log \left (\frac {f x}{e}+1\right ) a^2-3 b^2 f m x \log ^2\left (c x^n\right ) a-18 b^2 f m n^2 x a+12 b^2 f m n x \log \left (c x^n\right ) a+6 b^2 e m n^2 \log (e+f x) a+3 b^2 e m n^2 \log ^2(x) \log (e+f x) a+3 b^2 e m \log ^2\left (c x^n\right ) \log (e+f x) a+6 b^2 e m n^2 \log (x) \log (e+f x) a-6 b^2 e m n \log \left (c x^n\right ) \log (e+f x) a-6 b^2 e m n \log (x) \log \left (c x^n\right ) \log (e+f x) a+3 b^2 f x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right ) a+6 b^2 f n^2 x \log \left (d (e+f x)^m\right ) a-6 b^2 f n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right ) a-3 b^2 e m n^2 \log ^2(x) \log \left (\frac {f x}{e}+1\right ) a-6 b^2 e m n^2 \log (x) \log \left (\frac {f x}{e}+1\right ) a+6 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right ) a-b^3 f m x \log ^3\left (c x^n\right )+6 b^3 f m n x \log ^2\left (c x^n\right )+24 b^3 f m n^3 x-18 b^3 f m n^2 x \log \left (c x^n\right )-6 b^3 e m n^3 \log (e+f x)-b^3 e m n^3 \log ^3(x) \log (e+f x)+b^3 e m \log ^3\left (c x^n\right ) \log (e+f x)-3 b^3 e m n^3 \log ^2(x) \log (e+f x)-3 b^3 e m n \log ^2\left (c x^n\right ) \log (e+f x)-3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log (e+f x)-6 b^3 e m n^3 \log (x) \log (e+f x)+6 b^3 e m n^2 \log \left (c x^n\right ) \log (e+f x)+3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log (e+f x)+6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log (e+f x)+b^3 f x \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 b^3 f n x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 f n^3 x \log \left (d (e+f x)^m\right )+6 b^3 f n^2 x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 e m n^3 \log ^3(x) \log \left (\frac {f x}{e}+1\right )+3 b^3 e m n^3 \log ^2(x) \log \left (\frac {f x}{e}+1\right )+3 b^3 e m n \log (x) \log ^2\left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )+6 b^3 e m n^3 \log (x) \log \left (\frac {f x}{e}+1\right )-3 b^3 e m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-6 b^3 e m n^2 \log (x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )+3 b e m n \left (a^2-2 b n a+2 b^2 n^2+b^2 \log ^2\left (c x^n\right )+2 b (a-b n) \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )-6 b^2 e m n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )+6 b^3 e m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \]

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, 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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \]

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, algorithm="giac")

[Out]

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

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maple [F]  time = 11.85, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f x +e \right )^{m}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{3} e m \log \left (f x + e\right ) - {\left (f m - f \log \relax (d)\right )} b^{3} x\right )} \log \left (x^{n}\right )^{3} + {\left (b^{3} f x \log \left (x^{n}\right )^{3} - 3 \, {\left ({\left (f n - f \log \relax (c)\right )} b^{3} - a b^{2} f\right )} x \log \left (x^{n}\right )^{2} - 3 \, {\left (2 \, {\left (f n - f \log \relax (c)\right )} a b^{2} - {\left (2 \, f n^{2} - 2 \, f n \log \relax (c) + f \log \relax (c)^{2}\right )} b^{3} - a^{2} b f\right )} x \log \left (x^{n}\right ) - {\left (3 \, {\left (f n - f \log \relax (c)\right )} a^{2} b - 3 \, {\left (2 \, f n^{2} - 2 \, f n \log \relax (c) + f \log \relax (c)^{2}\right )} a b^{2} + {\left (6 \, f n^{3} - 6 \, f n^{2} \log \relax (c) + 3 \, f n \log \relax (c)^{2} - f \log \relax (c)^{3}\right )} b^{3} - a^{3} f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{f} - \int \frac {{\left ({\left (f^{2} m - f^{2} \log \relax (d)\right )} a^{3} - 3 \, {\left (f^{2} m n - {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, f^{2} m n^{2} - 2 \, f^{2} m n \log \relax (c) + {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)^{2}\right )} a b^{2} - {\left (6 \, f^{2} m n^{3} - 6 \, f^{2} m n^{2} \log \relax (c) + 3 \, f^{2} m n \log \relax (c)^{2} - {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)^{3}\right )} b^{3}\right )} x^{2} + 3 \, {\left ({\left ({\left (f^{2} m - f^{2} \log \relax (d)\right )} a b^{2} - {\left (2 \, f^{2} m n - f^{2} n \log \relax (d) - {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)\right )} b^{3}\right )} x^{2} - {\left (a b^{2} e f \log \relax (d) + {\left (e f m n - e f n \log \relax (d) + e f \log \relax (c) \log \relax (d)\right )} b^{3}\right )} x + {\left (b^{3} e f m n x + b^{3} e^{2} m n\right )} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )^{2} - {\left (b^{3} e f \log \relax (c)^{3} \log \relax (d) + 3 \, a b^{2} e f \log \relax (c)^{2} \log \relax (d) + 3 \, a^{2} b e f \log \relax (c) \log \relax (d) + a^{3} e f \log \relax (d)\right )} x + 3 \, {\left ({\left ({\left (f^{2} m - f^{2} \log \relax (d)\right )} a^{2} b - 2 \, {\left (f^{2} m n - {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)\right )} a b^{2} + {\left (2 \, f^{2} m n^{2} - 2 \, f^{2} m n \log \relax (c) + {\left (f^{2} m - f^{2} \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{3}\right )} x^{2} - {\left (b^{3} e f \log \relax (c)^{2} \log \relax (d) + 2 \, a b^{2} e f \log \relax (c) \log \relax (d) + a^{2} b e f \log \relax (d)\right )} x\right )} \log \left (x^{n}\right )}{f^{2} x^{2} + e f x}\,{d x} \]

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, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)

[Out]

Timed out

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