3.108 \(\int x (a+b \log (c x^n))^3 \log (d (e+f x^2)^m) \, dx\)

Optimal. Leaf size=514 \[ -\frac{3 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{3 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}+\frac{3 b e m n \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right )}{8 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )}{8 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right )}{8 f}+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{3 b^2 e m n^2 \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{3 b e m n \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{e m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f}+\frac{3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}+\frac{3}{2} b^3 m n^3 x^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.924466, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2305, 2304, 2378, 266, 43, 2351, 2337, 2391, 2353, 2374, 6589, 2383} \[ -\frac{3 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{3 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}+\frac{3 b e m n \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right )}{8 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )}{8 f}+\frac{3 b^3 e m n^3 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right )}{8 f}+\frac{3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{3 b^2 e m n^2 \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac{9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{3 b e m n \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac{e m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f}+\frac{3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}+\frac{3}{2} b^3 m n^3 x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m],x]

[Out]

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

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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

Mathematica [C]  time = 0.514654, size = 1911, normalized size = 3.72 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m],x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 4.701, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( f{x}^{2}+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^2+e)^m),x)

[Out]

int(x*(a+b*ln(c*x^n))^3*ln(d*(f*x^2+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^2+e)^m),x, algorithm="maxima")

[Out]

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

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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^{2} + 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^2+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^2 + 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*(a+b*ln(c*x**n))**3*ln(d*(f*x**2+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 )}^{3} x \log \left ({\left (f x^{2} + 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^2+e)^m),x, algorithm="giac")

[Out]

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