3.114 \(\int \frac{(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^4} \, dx\)

Optimal. Leaf size=1007 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.70208, antiderivative size = 1007, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 16, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {2305, 2304, 2378, 325, 205, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589, 2383} \[ -\frac{4 b^3 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) n^3}{27 e^{3/2}}-\frac{2 b^3 \log \left (d \left (f x^2+e\right )^m\right ) n^3}{27 x^3}+\frac{2 i b^3 f^{3/2} m \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) n^3}{9 e^{3/2}}-\frac{2 i b^3 f^{3/2} m \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) n^3}{9 e^{3/2}}+\frac{2 b^3 f^{3/2} m \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{3 (-e)^{3/2}}-\frac{2 b^3 f^{3/2} m \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{3 (-e)^{3/2}}-\frac{2 b^3 f^{3/2} m \text{PolyLog}\left (4,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{(-e)^{3/2}}+\frac{2 b^3 f^{3/2} m \text{PolyLog}\left (4,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{(-e)^{3/2}}-\frac{160 b^3 f m n^3}{27 e x}-\frac{4 b^2 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e^{3/2}}-\frac{52 b^2 f m \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e x}-\frac{2 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) n^2}{9 x^3}-\frac{2 b^2 f^{3/2} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{3 (-e)^{3/2}}+\frac{2 b^2 f^{3/2} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{3 (-e)^{3/2}}+\frac{2 b^2 f^{3/2} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{(-e)^{3/2}}-\frac{2 b^2 f^{3/2} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{(-e)^{3/2}}-\frac{8 b f m \left (a+b \log \left (c x^n\right )\right )^2 n}{3 e x}+\frac{b f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{3 (-e)^{3/2}}-\frac{b f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) n}{3 (-e)^{3/2}}-\frac{b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) n}{3 x^3}-\frac{b f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{(-e)^{3/2}}+\frac{b f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{(-e)^{3/2}}-\frac{2 f m \left (a+b \log \left (c x^n\right )\right )^3}{3 e x}+\frac{f^{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 (-e)^{3/2}}-\frac{f^{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 (-e)^{3/2}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]

[Out]

(-160*b^3*f*m*n^3)/(27*e*x) - (4*b^3*f^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)) - (52*b^2*f*m*n^2
*(a + b*Log[c*x^n]))/(9*e*x) - (4*b^2*f^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)
) - (8*b*f*m*n*(a + b*Log[c*x^n])^2)/(3*e*x) - (2*f*m*(a + b*Log[c*x^n])^3)/(3*e*x) + (b*f^(3/2)*m*n*(a + b*Lo
g[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) + (f^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x
)/Sqrt[-e]])/(3*(-e)^(3/2)) - (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)
) - (f^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^3*n^3*Log[d*(e + f*x^
2)^m])/(27*x^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(9*x^3) - (b*n*(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*x^3) - (2*b^2*f^(3/2)*m*n^2*(a + b*
Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[
2, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[
-e]])/(3*(-e)^(3/2)) + (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(-e)^(3/2) + (((2
*I)/9)*b^3*f^(3/2)*m*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (((2*I)/9)*b^3*f^(3/2)*m*n^3*PolyLog[
2, (I*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) + (2*b^3*f^(3/2)*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2))
+ (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) - (2*b^3*f^(3/2)*m*n
^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt
[f]*x)/Sqrt[-e]])/(-e)^(3/2) - (2*b^3*f^(3/2)*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (2*b^3*f
^(3/2)*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/(-e)^(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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [B]  time = 0.838992, size = 2488, normalized size = 2.47 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]

[Out]

(-18*a^3*Sqrt[e]*f*m*x^2 - 72*a^2*b*Sqrt[e]*f*m*n*x^2 - 156*a*b^2*Sqrt[e]*f*m*n^2*x^2 - 160*b^3*Sqrt[e]*f*m*n^
3*x^2 - 18*a^3*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 18*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]
] - 12*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e
]] + 54*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 36*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*
x)/Sqrt[e]]*Log[x] + 12*b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 54*a*b^2*f^(3/2)*m*n^2*x^3*
ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 18*b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 18*b^3*
f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 54*a^2*b*Sqrt[e]*f*m*x^2*Log[c*x^n] - 144*a*b^2*Sqrt[
e]*f*m*n*x^2*Log[c*x^n] - 156*b^3*Sqrt[e]*f*m*n^2*x^2*Log[c*x^n] - 54*a^2*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/S
qrt[e]]*Log[c*x^n] - 36*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*b^3*f^(3/2)*m*n^2*x^
3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 108*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*
x^n] + 36*b^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 54*b^3*f^(3/2)*m*n^2*x^3*ArcTa
n[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 54*a*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 72*b^3*Sqrt[e]*f*m*n*x^2*
Log[c*x^n]^2 - 54*a*b^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - 18*b^3*f^(3/2)*m*n*x^3*ArcTan
[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 54*b^3*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^2 -
18*b^3*Sqrt[e]*f*m*x^2*Log[c*x^n]^3 - 18*b^3*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^3 - (27*I)*a
^2*b*f^(3/2)*m*n*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*a*b^2*f^(3/2)*m*n^2*x^3*Log[x]*Log[1 - (I*
Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*f^(3/2)*m*n^3*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (27*I)*a*b^2*f^(3/2)
*m*n^2*x^3*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^3*f^(3/2)*m*n^3*x^3*Log[x]^2*Log[1 - (I*Sqrt[f]*x
)/Sqrt[e]] - (9*I)*b^3*f^(3/2)*m*n^3*x^3*Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (54*I)*a*b^2*f^(3/2)*m*n*x^
3*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^3*f^(3/2)*m*n^2*x^3*Log[x]*Log[c*x^n]*Log[1 - (I
*Sqrt[f]*x)/Sqrt[e]] + (27*I)*b^3*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (27*I
)*b^3*f^(3/2)*m*n*x^3*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (27*I)*a^2*b*f^(3/2)*m*n*x^3*Log[x]
*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*a*b^2*f^(3/2)*m*n^2*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)
*b^3*f^(3/2)*m*n^3*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (27*I)*a*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[1 +
(I*Sqrt[f]*x)/Sqrt[e]] - (9*I)*b^3*f^(3/2)*m*n^3*x^3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^3*f^(3
/2)*m*n^3*x^3*Log[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (54*I)*a*b^2*f^(3/2)*m*n*x^3*Log[x]*Log[c*x^n]*Log[1 +
(I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^3*f^(3/2)*m*n^2*x^3*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (27*
I)*b^3*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (27*I)*b^3*f^(3/2)*m*n*x^3*Log[x
]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 9*a^3*e^(3/2)*Log[d*(e + f*x^2)^m] - 9*a^2*b*e^(3/2)*n*Log[d*(
e + f*x^2)^m] - 6*a*b^2*e^(3/2)*n^2*Log[d*(e + f*x^2)^m] - 2*b^3*e^(3/2)*n^3*Log[d*(e + f*x^2)^m] - 27*a^2*b*e
^(3/2)*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 18*a*b^2*e^(3/2)*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 6*b^3*e^(3/2)*n^
2*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 27*a*b^2*e^(3/2)*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - 9*b^3*e^(3/2)*n*Log[c
*x^n]^2*Log[d*(e + f*x^2)^m] - 9*b^3*e^(3/2)*Log[c*x^n]^3*Log[d*(e + f*x^2)^m] + (3*I)*b*f^(3/2)*m*n*x^3*(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]
] - (3*I)*b*f^(3/2)*m*n*x^3*(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)*Po
lyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - (54*I)*a*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (18*I)
*b^3*f^(3/2)*m*n^3*x^3*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (54*I)*b^3*f^(3/2)*m*n^2*x^3*Log[c*x^n]*PolyLog[
3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (54*I)*a*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^3*f
^(3/2)*m*n^3*x^3*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + (54*I)*b^3*f^(3/2)*m*n^2*x^3*Log[c*x^n]*PolyLog[3, (I*Sqr
t[f]*x)/Sqrt[e]] + (54*I)*b^3*f^(3/2)*m*n^3*x^3*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (54*I)*b^3*f^(3/2)*m*n^
3*x^3*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)*x^3)

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Maple [F]  time = 44.451, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,x)

[Out]

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,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((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,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 (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="fricas")

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log((f*x^2 + e)^m*d)/x^4, 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((a+b*ln(c*x**n))**3*ln(d*(f*x**2+e)**m)/x**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="giac")

[Out]

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