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

Optimal. Leaf size=879 \[ \text{result too large to display} \]

[Out]

(12*b^3*Sqrt[f]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (12*b^2*Sqrt[f]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]
*(a + b*Log[c*x^n]))/Sqrt[e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] +
(Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])
^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sq
rt[-e] - (6*b^3*n^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - (3*b*n*(
a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*Sqrt[f]*m
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*
PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)
/Sqrt[-e]])/Sqrt[-e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - ((6*
I)*b^3*Sqrt[f]*m*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + ((6*I)*b^3*Sqrt[f]*m*n^3*PolyLog[2, (I*Sq
rt[f]*x)/Sqrt[e]])/Sqrt[e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[
f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, (S
qrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt
[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, (S
qrt[f]*x)/Sqrt[-e]])/Sqrt[-e]

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Rubi [A]  time = 1.10587, antiderivative size = 879, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {2305, 2304, 2378, 205, 2324, 12, 4848, 2391, 2330, 2317, 2374, 6589, 2383} \[ \frac{12 b^3 \sqrt{f} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) n^3}{\sqrt{e}}-\frac{6 b^3 \log \left (d \left (f x^2+e\right )^m\right ) n^3}{x}-\frac{6 i b^3 \sqrt{f} m \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) n^3}{\sqrt{e}}+\frac{6 i b^3 \sqrt{f} m \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) n^3}{\sqrt{e}}+\frac{6 b^3 \sqrt{f} m \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{\sqrt{-e}}-\frac{6 b^3 \sqrt{f} m \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{\sqrt{-e}}-\frac{6 b^3 \sqrt{f} m \text{PolyLog}\left (4,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{\sqrt{-e}}+\frac{6 b^3 \sqrt{f} m \text{PolyLog}\left (4,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^3}{\sqrt{-e}}+\frac{12 b^2 \sqrt{f} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{\sqrt{e}}-\frac{6 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}{x}-\frac{6 b^2 \sqrt{f} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{\sqrt{-e}}-\frac{6 b^2 \sqrt{f} m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n^2}{\sqrt{-e}}+\frac{3 b \sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{\sqrt{-e}}-\frac{3 b \sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) n}{\sqrt{-e}}-\frac{3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) n}{x}-\frac{3 b \sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{\sqrt{-e}}+\frac{3 b \sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) n}{\sqrt{-e}}+\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right )}{\sqrt{-e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*b^3*Sqrt[f]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (12*b^2*Sqrt[f]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]
*(a + b*Log[c*x^n]))/Sqrt[e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] +
(Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])
^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sq
rt[-e] - (6*b^3*n^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - (3*b*n*(
a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*Sqrt[f]*m
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*
PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)
/Sqrt[-e]])/Sqrt[-e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - ((6*
I)*b^3*Sqrt[f]*m*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + ((6*I)*b^3*Sqrt[f]*m*n^3*PolyLog[2, (I*Sq
rt[f]*x)/Sqrt[e]])/Sqrt[e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[
f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, (S
qrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt
[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, (S
qrt[f]*x)/Sqrt[-e]])/Sqrt[-e]

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 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 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 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^2} \, dx &=-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-(2 f m) \int \left (-\frac{6 b^3 n^3}{e+f x^2}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{e+f x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2}\right ) \, dx\\ &=-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}+(2 f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+(6 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+\left (12 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x^2} \, dx+\left (12 b^3 f m n^3\right ) \int \frac{1}{e+f x^2} \, dx\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}+(2 f m) \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+(6 b f m n) \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-\left (12 b^3 f m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} \sqrt{f} x} \, dx\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}-\sqrt{f} x} \, dx}{\sqrt{-e}}-\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}+\sqrt{f} x} \, dx}{\sqrt{-e}}-\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}-\sqrt{f} x} \, dx}{\sqrt{-e}}-\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}+\sqrt{f} x} \, dx}{\sqrt{-e}}-\frac{\left (12 b^3 \sqrt{f} m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{e}}\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (3 b \sqrt{f} 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}{\sqrt{-e}}+\frac{\left (3 b \sqrt{f} 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}{\sqrt{-e}}-\frac{\left (6 b^2 \sqrt{f} 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}{\sqrt{-e}}+\frac{\left (6 b^2 \sqrt{f} 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}{\sqrt{-e}}-\frac{\left (6 i b^3 \sqrt{f} m n^3\right ) \int \frac{\log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{e}}+\frac{\left (6 i b^3 \sqrt{f} m n^3\right ) \int \frac{\log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{e}}\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{\left (6 b^2 \sqrt{f} 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}{\sqrt{-e}}-\frac{\left (6 b^2 \sqrt{f} 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}{\sqrt{-e}}+\frac{\left (6 b^3 \sqrt{f} m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{-e}}-\frac{\left (6 b^3 \sqrt{f} m n^3\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{-e}}\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\left (6 b^3 \sqrt{f} m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{-e}}+\frac{\left (6 b^3 \sqrt{f} m n^3\right ) \int \frac{\text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{-e}}\\ &=\frac{12 b^3 \sqrt{f} m n^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{12 b^2 \sqrt{f} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{\sqrt{f} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 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 )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{3 b \sqrt{f} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{6 i b^3 \sqrt{f} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e}}+\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^2 \sqrt{f} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}-\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_4\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}+\frac{6 b^3 \sqrt{f} m n^3 \text{Li}_4\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{-e}}\\ \end{align*}

Mathematica [B]  time = 0.682536, size = 2166, normalized size = 2.46 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 6*a^2*b*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*
Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a^2*b*Sqr
t[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*a*b^2*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] -
12*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 6*a*b^2*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]
]*Log[x]^2 + 6*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 2*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f
]*x)/Sqrt[e]]*Log[x]^3 + 6*a^2*b*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*a*b^2*Sqrt[f]*m*n*x*A
rcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b
^2*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 12*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqr
t[e]]*Log[x]*Log[c*x^n] + 6*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] + 6*a*b^2*Sqrt
[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 6*b^3*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^
2 - 6*b^3*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^2 + 2*b^3*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)
/Sqrt[e]]*Log[c*x^n]^3 + (3*I)*a^2*b*Sqrt[f]*m*n*x*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*Sqrt[f]
*m*n^2*x*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[f]*m*n^3*x*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e
]] - (3*I)*a*b^2*Sqrt[f]*m*n^2*x*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[f]*m*n^3*x*Log[x]^2*
Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + I*b^3*Sqrt[f]*m*n^3*x*Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*S
qrt[f]*m*n*x*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[f]*m*n^2*x*Log[x]*Log[c*x^n]*Lo
g[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[f]*m*n^2*x*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] +
(3*I)*b^3*Sqrt[f]*m*n*x*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*a^2*b*Sqrt[f]*m*n*x*Log[x]*
Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*Sqrt[f]*m*n^2*x*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3
*Sqrt[f]*m*n^3*x*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*a*b^2*Sqrt[f]*m*n^2*x*Log[x]^2*Log[1 + (I*Sqrt[
f]*x)/Sqrt[e]] + (3*I)*b^3*Sqrt[f]*m*n^3*x*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - I*b^3*Sqrt[f]*m*n^3*x*Log
[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*Sqrt[f]*m*n*x*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[
e]] - (6*I)*b^3*Sqrt[f]*m*n^2*x*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b^3*Sqrt[f]*m*n^2*x*L
og[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[f]*m*n*x*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqr
t[f]*x)/Sqrt[e]] - a^3*Sqrt[e]*Log[d*(e + f*x^2)^m] - 3*a^2*b*Sqrt[e]*n*Log[d*(e + f*x^2)^m] - 6*a*b^2*Sqrt[e]
*n^2*Log[d*(e + f*x^2)^m] - 6*b^3*Sqrt[e]*n^3*Log[d*(e + f*x^2)^m] - 3*a^2*b*Sqrt[e]*Log[c*x^n]*Log[d*(e + f*x
^2)^m] - 6*a*b^2*Sqrt[e]*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 6*b^3*Sqrt[e]*n^2*Log[c*x^n]*Log[d*(e + f*x^2)^m]
- 3*a*b^2*Sqrt[e]*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - 3*b^3*Sqrt[e]*n*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - b^3
*Sqrt[e]*Log[c*x^n]^3*Log[d*(e + f*x^2)^m] - (3*I)*b*Sqrt[f]*m*n*x*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*b*(a + b*n)*
Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b*Sqrt[f]*m*n*x*(a^2 + 2*a*b*n + 2
*b^2*n^2 + 2*b*(a + b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*Sqrt[f
]*m*n^2*x*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[f]*m*n^3*x*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]
] + (6*I)*b^3*Sqrt[f]*m*n^2*x*Log[c*x^n]*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*Sqrt[f]*m*n^2*x*Po
lyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqrt[f]*m*n^3*x*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqrt
[f]*m*n^2*x*Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqrt[f]*m*n^3*x*PolyLog[4, ((-I)*Sqrt[f]*
x)/Sqrt[e]] + (6*I)*b^3*Sqrt[f]*m*n^3*x*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*x)

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Maple [F]  time = 27.737, 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}^{2}}}\, 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^2,x)

[Out]

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

[Out]

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