3.1.85 \(\int x (a+b \log (c x^n))^3 \log (d (e+f x)^m) \, dx\) [85]

Optimal. Leaf size=603 \[ \frac {21 a b^2 e m n^2 x}{4 f}-\frac {45 b^3 e m n^3 x}{8 f}+\frac {3}{4} b^3 m n^3 x^2+\frac {21 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac {9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac {3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 f^2}+\frac {3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 f^2}-\frac {3 b^3 e^2 m n^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{4 f^2}+\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{2 f^2}-\frac {3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 f^2}-\frac {3 b^3 e^2 m n^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{2 f^2}+\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{f^2}-\frac {3 b^3 e^2 m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{f^2} \]

[Out]

21/4*a*b^2*e*m*n^2*x/f-45/8*b^3*e*m*n^3*x/f+3/4*b^3*m*n^3*x^2+21/4*b^3*e*m*n^2*x*ln(c*x^n)/f-9/8*b^2*m*n^2*x^2
*(a+b*ln(c*x^n))-9/4*b*e*m*n*x*(a+b*ln(c*x^n))^2/f+3/4*b*m*n*x^2*(a+b*ln(c*x^n))^2+1/2*e*m*x*(a+b*ln(c*x^n))^3
/f-1/4*m*x^2*(a+b*ln(c*x^n))^3+3/8*b^3*e^2*m*n^3*ln(f*x+e)/f^2-3/8*b^3*n^3*x^2*ln(d*(f*x+e)^m)+3/4*b^2*n^2*x^2
*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)-3/4*b*n*x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)+1/2*x^2*(a+b*ln(c*x^n))^3*ln(d*
(f*x+e)^m)-3/4*b^2*e^2*m*n^2*(a+b*ln(c*x^n))*ln(1+f*x/e)/f^2+3/4*b*e^2*m*n*(a+b*ln(c*x^n))^2*ln(1+f*x/e)/f^2-1
/2*e^2*m*(a+b*ln(c*x^n))^3*ln(1+f*x/e)/f^2-3/4*b^3*e^2*m*n^3*polylog(2,-f*x/e)/f^2+3/2*b^2*e^2*m*n^2*(a+b*ln(c
*x^n))*polylog(2,-f*x/e)/f^2-3/2*b*e^2*m*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x/e)/f^2-3/2*b^3*e^2*m*n^3*polylog(3
,-f*x/e)/f^2+3*b^2*e^2*m*n^2*(a+b*ln(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 = 0.63, antiderivative size = 603, normalized size of antiderivative = 1.00, 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 = {2342, 2341, 2425, 45, 2393, 2332, 2354, 2438, 2395, 2333, 2421, 6724, 2430} \begin {gather*} \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 \end {gather*}

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 45

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 2332

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

Rule 2333

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 2341

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 2342

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 2354

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 2393

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 2395

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2425

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2438

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 6724

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 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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1431\) vs. \(2(603)=1206\).
time = 0.34, size = 1431, normalized size = 2.37 \begin {gather*} \frac {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 \left (c x^n\right )-36 a b^2 e f m n x \log \left (c x^n\right )+42 b^3 e f m n^2 x \log \left (c x^n\right )-6 a^2 b f^2 m x^2 \log \left (c x^n\right )+12 a b^2 f^2 m n x^2 \log \left (c x^n\right )-9 b^3 f^2 m n^2 x^2 \log \left (c x^n\right )+12 a b^2 e f m x \log ^2\left (c x^n\right )-18 b^3 e f m n x \log ^2\left (c x^n\right )-6 a b^2 f^2 m x^2 \log ^2\left (c x^n\right )+6 b^3 f^2 m n x^2 \log ^2\left (c x^n\right )+4 b^3 e f m x \log ^3\left (c x^n\right )-2 b^3 f^2 m x^2 \log ^3\left (c x^n\right )-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 \log (x) \log (e+f x)-12 a b^2 e^2 m n^2 \log ^2(x) \log (e+f x)+6 b^3 e^2 m n^3 \log ^2(x) \log (e+f x)+4 b^3 e^2 m n^3 \log ^3(x) \log (e+f x)-12 a^2 b e^2 m \log \left (c x^n\right ) \log (e+f x)+12 a b^2 e^2 m n \log \left (c x^n\right ) \log (e+f x)-6 b^3 e^2 m n^2 \log \left (c x^n\right ) \log (e+f x)+24 a b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log (e+f x)-12 b^3 e^2 m n^2 \log (x) \log \left (c x^n\right ) \log (e+f x)-12 b^3 e^2 m n^2 \log ^2(x) \log \left (c x^n\right ) \log (e+f x)-12 a b^2 e^2 m \log ^2\left (c x^n\right ) \log (e+f x)+6 b^3 e^2 m n \log ^2\left (c x^n\right ) \log (e+f x)+12 b^3 e^2 m n \log (x) \log ^2\left (c x^n\right ) \log (e+f x)-4 b^3 e^2 m \log ^3\left (c x^n\right ) \log (e+f x)+4 a^3 f^2 x^2 \log \left (d (e+f x)^m\right )-6 a^2 b f^2 n x^2 \log \left (d (e+f x)^m\right )+6 a b^2 f^2 n^2 x^2 \log \left (d (e+f x)^m\right )-3 b^3 f^2 n^3 x^2 \log \left (d (e+f x)^m\right )+12 a^2 b f^2 x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-12 a b^2 f^2 n x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^3 f^2 n^2 x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+12 a b^2 f^2 x^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 f^2 n x^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+4 b^3 f^2 x^2 \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-12 a^2 b e^2 m n \log (x) \log \left (1+\frac {f x}{e}\right )+12 a b^2 e^2 m n^2 \log (x) \log \left (1+\frac {f x}{e}\right )-6 b^3 e^2 m n^3 \log (x) \log \left (1+\frac {f x}{e}\right )+12 a b^2 e^2 m n^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )-6 b^3 e^2 m n^3 \log ^2(x) \log \left (1+\frac {f x}{e}\right )-4 b^3 e^2 m n^3 \log ^3(x) \log \left (1+\frac {f x}{e}\right )-24 a b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+12 b^3 e^2 m n^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+12 b^3 e^2 m n^2 \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-12 b^3 e^2 m n \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-6 b e^2 m n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+12 b^2 e^2 m n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-24 b^3 e^2 m n^3 \text {Li}_4\left (-\frac {f x}{e}\right )}{8 f^2} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.25, size = 44991, normalized size = 74.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(44991\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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