3.78 \(\int x^2 (a+b \log (c x^n))^2 \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=452 \[ \frac{2 b e^3 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{2 b^2 e^3 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{9 f^3}-\frac{2 b^2 e^3 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{e^3 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^3}-\frac{2 b e^3 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{8 a b e^2 m n x}{9 f^2}+\frac{8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{26 b^2 e^2 m n^2 x}{27 f^2}+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{19 b^2 e m n^2 x^2}{108 f}-\frac{2}{27} b^2 m n^2 x^3 \]

[Out]

(8*a*b*e^2*m*n*x)/(9*f^2) - (26*b^2*e^2*m*n^2*x)/(27*f^2) + (19*b^2*e*m*n^2*x^2)/(108*f) - (2*b^2*m*n^2*x^3)/2
7 + (8*b^2*e^2*m*n*x*Log[c*x^n])/(9*f^2) - (5*b*e*m*n*x^2*(a + b*Log[c*x^n]))/(18*f) + (4*b*m*n*x^3*(a + b*Log
[c*x^n]))/27 - (e^2*m*x*(a + b*Log[c*x^n])^2)/(3*f^2) + (e*m*x^2*(a + b*Log[c*x^n])^2)/(6*f) - (m*x^3*(a + b*L
og[c*x^n])^2)/9 + (2*b^2*e^3*m*n^2*Log[e + f*x])/(27*f^3) + (2*b^2*n^2*x^3*Log[d*(e + f*x)^m])/27 - (2*b*n*x^3
*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/3 - (2*b*e^3*m*n*(a
+ b*Log[c*x^n])*Log[1 + (f*x)/e])/(9*f^3) + (e^3*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(3*f^3) - (2*b^2*e^3
*m*n^2*PolyLog[2, -((f*x)/e)])/(9*f^3) + (2*b*e^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(3*f^3) - (2*
b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(3*f^3)

________________________________________________________________________________________

Rubi [A]  time = 0.683452, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2305, 2304, 2378, 43, 2351, 2295, 2317, 2391, 2353, 2296, 2374, 6589} \[ \frac{2 b e^3 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{2 b^2 e^3 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{9 f^3}-\frac{2 b^2 e^3 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{e^3 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^3}-\frac{2 b e^3 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{8 a b e^2 m n x}{9 f^2}+\frac{8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{26 b^2 e^2 m n^2 x}{27 f^2}+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{19 b^2 e m n^2 x^2}{108 f}-\frac{2}{27} b^2 m n^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*b*e^2*m*n*x)/(9*f^2) - (26*b^2*e^2*m*n^2*x)/(27*f^2) + (19*b^2*e*m*n^2*x^2)/(108*f) - (2*b^2*m*n^2*x^3)/2
7 + (8*b^2*e^2*m*n*x*Log[c*x^n])/(9*f^2) - (5*b*e*m*n*x^2*(a + b*Log[c*x^n]))/(18*f) + (4*b*m*n*x^3*(a + b*Log
[c*x^n]))/27 - (e^2*m*x*(a + b*Log[c*x^n])^2)/(3*f^2) + (e*m*x^2*(a + b*Log[c*x^n])^2)/(6*f) - (m*x^3*(a + b*L
og[c*x^n])^2)/9 + (2*b^2*e^3*m*n^2*Log[e + f*x])/(27*f^3) + (2*b^2*n^2*x^3*Log[d*(e + f*x)^m])/27 - (2*b*n*x^3
*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/3 - (2*b*e^3*m*n*(a
+ b*Log[c*x^n])*Log[1 + (f*x)/e])/(9*f^3) + (e^3*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(3*f^3) - (2*b^2*e^3
*m*n^2*PolyLog[2, -((f*x)/e)])/(9*f^3) + (2*b*e^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(3*f^3) - (2*
b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(3*f^3)

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2378

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

Rule 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]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{2 b^2 n^2 x^3}{27 (e+f x)}-\frac{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 (e+f x)}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 (e+f x)}\right ) \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{3} (f m) \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\frac{1}{9} (2 b f m n) \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx-\frac{1}{27} \left (2 b^2 f m n^2\right ) \int \frac{x^3}{e+f x} \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{3} (f m) \int \left (\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )^2}{f^3 (e+f x)}\right ) \, dx+\frac{1}{9} (2 b f m n) \int \left (\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{f^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{f^3 (e+f x)}\right ) \, dx-\frac{1}{27} \left (2 b^2 f m n^2\right ) \int \left (\frac{e^2}{f^3}-\frac{e x}{f^2}+\frac{x^2}{f}-\frac{e^3}{f^3 (e+f x)}\right ) \, dx\\ &=-\frac{2 b^2 e^2 m n^2 x}{27 f^2}+\frac{b^2 e m n^2 x^2}{27 f}-\frac{2}{81} b^2 m n^2 x^3+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{3} m \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{\left (e^2 m\right ) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f^2}+\frac{\left (e^3 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{3 f^2}+\frac{(e m) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}+\frac{1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f^2}-\frac{\left (2 b e^3 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{9 f^2}-\frac{(2 b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}\\ &=\frac{2 a b e^2 m n x}{9 f^2}-\frac{2 b^2 e^2 m n^2 x}{27 f^2}+\frac{5 b^2 e m n^2 x^2}{54 f}-\frac{4}{81} b^2 m n^2 x^3-\frac{b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{9 f}+\frac{2}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 f^3}+\frac{1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{\left (2 b e^3 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{3 f^3}+\frac{\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f^2}+\frac{\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{9 f^2}-\frac{(b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}+\frac{\left (2 b^2 e^3 m n^2\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{9 f^3}\\ &=\frac{8 a b e^2 m n x}{9 f^2}-\frac{8 b^2 e^2 m n^2 x}{27 f^2}+\frac{19 b^2 e m n^2 x^2}{108 f}-\frac{2}{27} b^2 m n^2 x^3+\frac{2 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac{5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 f^3}-\frac{2 b^2 e^3 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{9 f^3}+\frac{2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{3 f^3}+\frac{\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{3 f^2}-\frac{\left (2 b^2 e^3 m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{3 f^3}\\ &=\frac{8 a b e^2 m n x}{9 f^2}-\frac{26 b^2 e^2 m n^2 x}{27 f^2}+\frac{19 b^2 e m n^2 x^2}{108 f}-\frac{2}{27} b^2 m n^2 x^3+\frac{8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac{5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac{4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac{2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{9 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{3 f^3}-\frac{2 b^2 e^3 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{9 f^3}+\frac{2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{3 f^3}-\frac{2 b^2 e^3 m n^2 \text{Li}_3\left (-\frac{f x}{e}\right )}{3 f^3}\\ \end{align*}

Mathematica [A]  time = 0.342029, size = 788, normalized size = 1.74 \[ \frac{24 b e^3 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )-72 b^2 e^3 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )+36 a^2 f^3 x^3 \log \left (d (e+f x)^m\right )-36 a^2 e^2 f m x+36 a^2 e^3 m \log (e+f x)+18 a^2 e f^2 m x^2-12 a^2 f^3 m x^3+72 a b f^3 x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+72 a b e^3 m \log \left (c x^n\right ) \log (e+f x)-72 a b e^2 f m x \log \left (c x^n\right )+36 a b e f^2 m x^2 \log \left (c x^n\right )-24 a b f^3 m x^3 \log \left (c x^n\right )-24 a b f^3 n x^3 \log \left (d (e+f x)^m\right )+96 a b e^2 f m n x-24 a b e^3 m n \log (e+f x)-72 a b e^3 m n \log (x) \log (e+f x)+72 a b e^3 m n \log (x) \log \left (\frac{f x}{e}+1\right )-30 a b e f^2 m n x^2+16 a b f^3 m n x^3+36 b^2 f^3 x^3 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-24 b^2 f^3 n x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+36 b^2 e^3 m \log ^2\left (c x^n\right ) \log (e+f x)-36 b^2 e^2 f m x \log ^2\left (c x^n\right )-24 b^2 e^3 m n \log \left (c x^n\right ) \log (e+f x)-72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log (e+f x)+72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+96 b^2 e^2 f m n x \log \left (c x^n\right )+18 b^2 e f^2 m x^2 \log ^2\left (c x^n\right )-30 b^2 e f^2 m n x^2 \log \left (c x^n\right )-12 b^2 f^3 m x^3 \log ^2\left (c x^n\right )+16 b^2 f^3 m n x^3 \log \left (c x^n\right )+8 b^2 f^3 n^2 x^3 \log \left (d (e+f x)^m\right )-104 b^2 e^2 f m n^2 x+36 b^2 e^3 m n^2 \log ^2(x) \log (e+f x)-36 b^2 e^3 m n^2 \log ^2(x) \log \left (\frac{f x}{e}+1\right )+8 b^2 e^3 m n^2 \log (e+f x)+24 b^2 e^3 m n^2 \log (x) \log (e+f x)-24 b^2 e^3 m n^2 \log (x) \log \left (\frac{f x}{e}+1\right )+19 b^2 e f^2 m n^2 x^2-8 b^2 f^3 m n^2 x^3}{108 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*a^2*e^2*f*m*x + 96*a*b*e^2*f*m*n*x - 104*b^2*e^2*f*m*n^2*x + 18*a^2*e*f^2*m*x^2 - 30*a*b*e*f^2*m*n*x^2 +
19*b^2*e*f^2*m*n^2*x^2 - 12*a^2*f^3*m*x^3 + 16*a*b*f^3*m*n*x^3 - 8*b^2*f^3*m*n^2*x^3 - 72*a*b*e^2*f*m*x*Log[c*
x^n] + 96*b^2*e^2*f*m*n*x*Log[c*x^n] + 36*a*b*e*f^2*m*x^2*Log[c*x^n] - 30*b^2*e*f^2*m*n*x^2*Log[c*x^n] - 24*a*
b*f^3*m*x^3*Log[c*x^n] + 16*b^2*f^3*m*n*x^3*Log[c*x^n] - 36*b^2*e^2*f*m*x*Log[c*x^n]^2 + 18*b^2*e*f^2*m*x^2*Lo
g[c*x^n]^2 - 12*b^2*f^3*m*x^3*Log[c*x^n]^2 + 36*a^2*e^3*m*Log[e + f*x] - 24*a*b*e^3*m*n*Log[e + f*x] + 8*b^2*e
^3*m*n^2*Log[e + f*x] - 72*a*b*e^3*m*n*Log[x]*Log[e + f*x] + 24*b^2*e^3*m*n^2*Log[x]*Log[e + f*x] + 36*b^2*e^3
*m*n^2*Log[x]^2*Log[e + f*x] + 72*a*b*e^3*m*Log[c*x^n]*Log[e + f*x] - 24*b^2*e^3*m*n*Log[c*x^n]*Log[e + f*x] -
72*b^2*e^3*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] + 36*b^2*e^3*m*Log[c*x^n]^2*Log[e + f*x] + 36*a^2*f^3*x^3*Log[d
*(e + f*x)^m] - 24*a*b*f^3*n*x^3*Log[d*(e + f*x)^m] + 8*b^2*f^3*n^2*x^3*Log[d*(e + f*x)^m] + 72*a*b*f^3*x^3*Lo
g[c*x^n]*Log[d*(e + f*x)^m] - 24*b^2*f^3*n*x^3*Log[c*x^n]*Log[d*(e + f*x)^m] + 36*b^2*f^3*x^3*Log[c*x^n]^2*Log
[d*(e + f*x)^m] + 72*a*b*e^3*m*n*Log[x]*Log[1 + (f*x)/e] - 24*b^2*e^3*m*n^2*Log[x]*Log[1 + (f*x)/e] - 36*b^2*e
^3*m*n^2*Log[x]^2*Log[1 + (f*x)/e] + 72*b^2*e^3*m*n*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 24*b*e^3*m*n*(3*a - b
*n + 3*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] - 72*b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(108*f^3)

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Maple [F]  time = 1.819, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\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^2*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \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^2*(a+b*log(c*x^n))^2*log(d*(f*x+e)^m),x, algorithm="giac")

[Out]

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