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

Optimal. Leaf size=1092 \[ \text{result too large to display} \]

[Out]

(52*a*b^2*e*m*n^2*x)/(9*f) - (160*b^3*e*m*n^3*x)/(27*f) + (16*b^3*m*n^3*x^3)/81 + (4*b^3*e^(3/2)*m*n^3*ArcTan[
(Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2)) + (52*b^3*e*m*n^2*x*Log[c*x^n])/(9*f) - (4*b^2*m*n^2*x^3*(a + b*Log[c*x^n])
)/9 - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) - (8*b*e*m*n*x*(a + b*L
og[c*x^n])^2)/(3*f) + (4*b*m*n*x^3*(a + b*Log[c*x^n])^2)/9 + (2*e*m*x*(a + b*Log[c*x^n])^3)/(3*f) - (2*m*x^3*(
a + b*Log[c*x^n])^3)/9 + (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (
(-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*
x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sq
rt[-e]])/(3*f^(3/2)) - (2*b^3*n^3*x^3*Log[d*(e + f*x^2)^m])/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[d*(e +
f*x^2)^m])/9 - (b*n*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/3 + (x^3*(a + b*Log[c*x^n])^3*Log[d*(e + f*
x^2)^m])/3 - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (b*
(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (2*b^2*(-e)^(3/2)*m*n^2*(a
+ b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog
[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (((2*I)/9)*b^3*e^(3/2)*m*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2
) - (((2*I)/9)*b^3*e^(3/2)*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[
3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)
/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2*b^2*(-e)^(3/
2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, -((
Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 1.80723, antiderivative size = 1092, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 18, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {2305, 2304, 2378, 302, 205, 2351, 2295, 2324, 12, 4848, 2391, 2353, 2296, 2330, 2317, 2374, 6589, 2383} \[ \frac{16}{81} m n^3 x^3 b^3-\frac{160 e m n^3 x b^3}{27 f}+\frac{4 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) b^3}{27 f^{3/2}}+\frac{52 e m n^2 x \log \left (c x^n\right ) b^3}{9 f}-\frac{2}{27} n^3 x^3 \log \left (d \left (f x^2+e\right )^m\right ) b^3+\frac{2 i e^{3/2} m n^3 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) b^3}{9 f^{3/2}}-\frac{2 i e^{3/2} m n^3 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) b^3}{9 f^{3/2}}+\frac{2 (-e)^{3/2} m n^3 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{3 f^{3/2}}-\frac{2 (-e)^{3/2} m n^3 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{3 f^{3/2}}+\frac{2 (-e)^{3/2} m n^3 \text{PolyLog}\left (4,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{f^{3/2}}-\frac{2 (-e)^{3/2} m n^3 \text{PolyLog}\left (4,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{f^{3/2}}+\frac{52 a e m n^2 x b^2}{9 f}-\frac{4}{9} m n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) b^2-\frac{4 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{9 f^{3/2}}+\frac{2}{9} n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) b^2-\frac{2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{3 f^{3/2}}+\frac{2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{3 f^{3/2}}-\frac{2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{f^{3/2}}+\frac{2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{f^{3/2}}+\frac{4}{9} m n x^3 \left (a+b \log \left (c x^n\right )\right )^2 b-\frac{8 e m n x \left (a+b \log \left (c x^n\right )\right )^2 b}{3 f}+\frac{(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{3 f^{3/2}}-\frac{(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) b}{3 f^{3/2}}-\frac{1}{3} n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) b+\frac{(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{f^{3/2}}-\frac{(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{f^{3/2}}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(52*a*b^2*e*m*n^2*x)/(9*f) - (160*b^3*e*m*n^3*x)/(27*f) + (16*b^3*m*n^3*x^3)/81 + (4*b^3*e^(3/2)*m*n^3*ArcTan[
(Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2)) + (52*b^3*e*m*n^2*x*Log[c*x^n])/(9*f) - (4*b^2*m*n^2*x^3*(a + b*Log[c*x^n])
)/9 - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) - (8*b*e*m*n*x*(a + b*L
og[c*x^n])^2)/(3*f) + (4*b*m*n*x^3*(a + b*Log[c*x^n])^2)/9 + (2*e*m*x*(a + b*Log[c*x^n])^3)/(3*f) - (2*m*x^3*(
a + b*Log[c*x^n])^3)/9 + (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (
(-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*
x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sq
rt[-e]])/(3*f^(3/2)) - (2*b^3*n^3*x^3*Log[d*(e + f*x^2)^m])/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[d*(e +
f*x^2)^m])/9 - (b*n*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/3 + (x^3*(a + b*Log[c*x^n])^3*Log[d*(e + f*
x^2)^m])/3 - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (b*
(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (2*b^2*(-e)^(3/2)*m*n^2*(a
+ b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog
[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (((2*I)/9)*b^3*e^(3/2)*m*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2
) - (((2*I)/9)*b^3*e^(3/2)*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[
3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)
/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2*b^2*(-e)^(3/
2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, -((
Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2)

Rule 2305

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

Rule 2304

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

Rule 2378

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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 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^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \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{2 b^3 n^3 x^4}{27 \left (e+f x^2\right )}+\frac{2 b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{9 \left (e+f x^2\right )}-\frac{b n x^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 \left (e+f x^2\right )}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )^3}{3 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 f m) \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+\frac{1}{3} (2 b f m n) \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\frac{1}{9} \left (4 b^2 f m n^2\right ) \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx+\frac{1}{27} \left (4 b^3 f m n^3\right ) \int \frac{x^4}{e+f x^2} \, dx\\ &=-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 f m) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{x^2 \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 \left (e+f x^2\right )}\right ) \, dx+\frac{1}{3} (2 b f m n) \int \left (-\frac{e \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^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 \left (e+f x^2\right )}\right ) \, dx-\frac{1}{9} \left (4 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^2 \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 \left (e+f x^2\right )}\right ) \, dx+\frac{1}{27} \left (4 b^3 f m n^3\right ) \int \left (-\frac{e}{f^2}+\frac{x^2}{f}+\frac{e^2}{f^2 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac{4 b^3 e m n^3 x}{27 f}+\frac{4}{81} b^3 m n^3 x^3-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (2 m) \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx+\frac{(2 e m) \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx}{3 f}-\frac{\left (2 e^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx}{3 f}+\frac{1}{3} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{(2 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}+\frac{\left (2 b e^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{3 f}-\frac{1}{9} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}-\frac{\left (4 b^2 e^2 m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x^2} \, dx}{9 f}+\frac{\left (4 b^3 e^2 m n^3\right ) \int \frac{1}{e+f x^2} \, dx}{27 f}\\ &=\frac{4 a b^2 e m n^2 x}{9 f}-\frac{4 b^3 e m n^3 x}{27 f}+\frac{8}{81} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}-\frac{4}{27} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{2 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{2}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{\left (2 e^2 m\right ) \int \left (\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^3}{2 e \left (\sqrt{-e}-\sqrt{f} x\right )}+\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^3}{2 e \left (\sqrt{-e}+\sqrt{f} x\right )}\right ) \, dx}{3 f}+\frac{1}{3} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{(2 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{f}+\frac{\left (2 b e^2 m n\right ) \int \left (\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}-\sqrt{f} x\right )}+\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}+\sqrt{f} x\right )}\right ) \, dx}{3 f}-\frac{1}{9} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}+\frac{\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{9 f}+\frac{\left (4 b^3 e^2 m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} \sqrt{f} x} \, dx}{9 f}\\ &=\frac{16 a b^2 e m n^2 x}{9 f}-\frac{16 b^3 e m n^3 x}{27 f}+\frac{4}{27} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{4 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac{8}{27} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left ((-e)^{3/2} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}-\sqrt{f} x} \, dx}{3 f}+\frac{\left ((-e)^{3/2} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}+\sqrt{f} x} \, dx}{3 f}-\frac{\left (b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}-\sqrt{f} x} \, dx}{3 f}-\frac{\left (b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}+\sqrt{f} x} \, dx}{3 f}-\frac{1}{9} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{f}+\frac{\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{3 f}+\frac{\left (4 b^3 e^{3/2} m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=\frac{52 a b^2 e m n^2 x}{9 f}-\frac{52 b^3 e m n^3 x}{27 f}+\frac{16}{81} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{16 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac{4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left (b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}-\frac{\left (b (-e)^{3/2} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}-\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac{\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{f}+\frac{\left (2 i b^3 e^{3/2} m n^3\right ) \int \frac{\log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}-\frac{\left (2 i b^3 e^{3/2} m n^3\right ) \int \frac{\log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=\frac{52 a b^2 e m n^2 x}{9 f}-\frac{160 b^3 e m n^3 x}{27 f}+\frac{16}{81} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac{4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}+\frac{\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}+\frac{\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac{\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=\frac{52 a b^2 e m n^2 x}{9 f}-\frac{160 b^3 e m n^3 x}{27 f}+\frac{16}{81} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac{4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}-\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}-\frac{\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac{\text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{f^{3/2}}\\ &=\frac{52 a b^2 e m n^2 x}{9 f}-\frac{160 b^3 e m n^3 x}{27 f}+\frac{16}{81} b^3 m n^3 x^3+\frac{4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 f^{3/2}}+\frac{52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac{4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac{8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac{4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{2 i b^3 e^{3/2} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}-\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}-\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 f^{3/2}}+\frac{2 b^2 (-e)^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}+\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_4\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}-\frac{2 b^3 (-e)^{3/2} m n^3 \text{Li}_4\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{f^{3/2}}\\ \end{align*}

Mathematica [B]  time = 0.925172, size = 2544, normalized size = 2.33 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54*a^3*e*Sqrt[f]*m*x - 216*a^2*b*e*Sqrt[f]*m*n*x + 468*a*b^2*e*Sqrt[f]*m*n^2*x - 480*b^3*e*Sqrt[f]*m*n^3*x -
18*a^3*f^(3/2)*m*x^3 + 36*a^2*b*f^(3/2)*m*n*x^3 - 36*a*b^2*f^(3/2)*m*n^2*x^3 + 16*b^3*f^(3/2)*m*n^3*x^3 - 54*a
^3*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 54*a^2*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 36*a*b^2*e^(3/2)
*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 162*a^2*b*e^(3/2)*m*n*
ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 108*a*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 36*b^3*e^(3/
2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 162*a*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 5
4*b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 54*b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Lo
g[x]^3 + 162*a^2*b*e*Sqrt[f]*m*x*Log[c*x^n] - 432*a*b^2*e*Sqrt[f]*m*n*x*Log[c*x^n] + 468*b^3*e*Sqrt[f]*m*n^2*x
*Log[c*x^n] - 54*a^2*b*f^(3/2)*m*x^3*Log[c*x^n] + 72*a*b^2*f^(3/2)*m*n*x^3*Log[c*x^n] - 36*b^3*f^(3/2)*m*n^2*x
^3*Log[c*x^n] - 162*a^2*b*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 108*a*b^2*e^(3/2)*m*n*ArcTan[(Sqr
t[f]*x)/Sqrt[e]]*Log[c*x^n] - 36*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 324*a*b^2*e^(3/2)*
m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 108*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*L
og[c*x^n] - 162*b^3*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] + 162*a*b^2*e*Sqrt[f]*m*x*Lo
g[c*x^n]^2 - 216*b^3*e*Sqrt[f]*m*n*x*Log[c*x^n]^2 - 54*a*b^2*f^(3/2)*m*x^3*Log[c*x^n]^2 + 36*b^3*f^(3/2)*m*n*x
^3*Log[c*x^n]^2 - 162*a*b^2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 54*b^3*e^(3/2)*m*n*ArcTan[(Sq
rt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 162*b^3*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^2 + 54*b^3*
e*Sqrt[f]*m*x*Log[c*x^n]^3 - 18*b^3*f^(3/2)*m*x^3*Log[c*x^n]^3 - 54*b^3*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*
Log[c*x^n]^3 - (81*I)*a^2*b*e^(3/2)*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (54*I)*a*b^2*e^(3/2)*m*n^2*Log
[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^3*e^(3/2)*m*n^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (81*I)*a
*b^2*e^(3/2)*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (27*I)*b^3*e^(3/2)*m*n^3*Log[x]^2*Log[1 - (I*Sqrt
[f]*x)/Sqrt[e]] - (27*I)*b^3*e^(3/2)*m*n^3*Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (162*I)*a*b^2*e^(3/2)*m*n
*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (54*I)*b^3*e^(3/2)*m*n^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt
[f]*x)/Sqrt[e]] + (81*I)*b^3*e^(3/2)*m*n^2*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (81*I)*b^3*e^(
3/2)*m*n*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (81*I)*a^2*b*e^(3/2)*m*n*Log[x]*Log[1 + (I*Sqrt[
f]*x)/Sqrt[e]] - (54*I)*a*b^2*e^(3/2)*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^3*e^(3/2)*m*n^3*L
og[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (81*I)*a*b^2*e^(3/2)*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (2
7*I)*b^3*e^(3/2)*m*n^3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (27*I)*b^3*e^(3/2)*m*n^3*Log[x]^3*Log[1 + (I*
Sqrt[f]*x)/Sqrt[e]] + (162*I)*a*b^2*e^(3/2)*m*n*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (54*I)*b^3*
e^(3/2)*m*n^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (81*I)*b^3*e^(3/2)*m*n^2*Log[x]^2*Log[c*x^n]*
Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (81*I)*b^3*e^(3/2)*m*n*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 2
7*a^3*f^(3/2)*x^3*Log[d*(e + f*x^2)^m] - 27*a^2*b*f^(3/2)*n*x^3*Log[d*(e + f*x^2)^m] + 18*a*b^2*f^(3/2)*n^2*x^
3*Log[d*(e + f*x^2)^m] - 6*b^3*f^(3/2)*n^3*x^3*Log[d*(e + f*x^2)^m] + 81*a^2*b*f^(3/2)*x^3*Log[c*x^n]*Log[d*(e
+ f*x^2)^m] - 54*a*b^2*f^(3/2)*n*x^3*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 18*b^3*f^(3/2)*n^2*x^3*Log[c*x^n]*Log[
d*(e + f*x^2)^m] + 81*a*b^2*f^(3/2)*x^3*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - 27*b^3*f^(3/2)*n*x^3*Log[c*x^n]^2*
Log[d*(e + f*x^2)^m] + 27*b^3*f^(3/2)*x^3*Log[c*x^n]^3*Log[d*(e + f*x^2)^m] + (9*I)*b*e^(3/2)*m*n*(9*a^2 - 6*a
*b*n + 2*b^2*n^2 - 6*b*(-3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2)*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (9
*I)*b*e^(3/2)*m*n*(9*a^2 - 6*a*b*n + 2*b^2*n^2 - 6*b*(-3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2)*PolyLog[2,
(I*Sqrt[f]*x)/Sqrt[e]] - (162*I)*a*b^2*e^(3/2)*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (54*I)*b^3*e^(3/2)
*m*n^3*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (162*I)*b^3*e^(3/2)*m*n^2*Log[c*x^n]*PolyLog[3, ((-I)*Sqrt[f]*x)
/Sqrt[e]] + (162*I)*a*b^2*e^(3/2)*m*n^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (54*I)*b^3*e^(3/2)*m*n^3*PolyLog[3
, (I*Sqrt[f]*x)/Sqrt[e]] + (162*I)*b^3*e^(3/2)*m*n^2*Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + (162*I)*b^
3*e^(3/2)*m*n^3*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (162*I)*b^3*e^(3/2)*m*n^3*PolyLog[4, (I*Sqrt[f]*x)/Sqrt
[e]])/(81*f^(3/2))

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Maple [F]  time = 103.929, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

[Out]

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