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 \[ \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 {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{\sqrt {e}}+\frac {6 i b^3 \sqrt {f} m \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{\sqrt {e}}+\frac {6 b^3 \sqrt {f} m \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e}}-\frac {6 b^3 \sqrt {f} m \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e}}-\frac {6 b^3 \sqrt {f} m \text {Li}_4\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e}}+\frac {6 b^3 \sqrt {f} m \text {Li}_4\left (\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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_3\left (-\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 {Li}_3\left (\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 {Li}_2\left (-\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 {Li}_2\left (\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} \]

[Out]

-6*b^3*n^3*ln(d*(f*x^2+e)^m)/x-6*b^2*n^2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x
^2+e)^m)/x-(a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x+3*b*m*n*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/
(-e)^(1/2)+m*(a+b*ln(c*x^n))^3*ln(1-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-3*b*m*n*(a+b*ln(c*x^n))^2*ln(1+x*
f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-m*(a+b*ln(c*x^n))^3*ln(1+x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-6*b^2
*m*n^2*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-3*b*m*n*(a+b*ln(c*x^n))^2*polylog(2
,-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)+6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))*f^(1/2)
/(-e)^(1/2)+3*b*m*n*(a+b*ln(c*x^n))^2*polylog(2,x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)+6*b^3*m*n^3*polylog(3
,-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)+6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(3,-x*f^(1/2)/(-e)^(1/2))*f^(1/2
)/(-e)^(1/2)-6*b^3*m*n^3*polylog(3,x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-6*b^2*m*n^2*(a+b*ln(c*x^n))*polylo
g(3,x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)-6*b^3*m*n^3*polylog(4,-x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)+6
*b^3*m*n^3*polylog(4,x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/2)+12*b^3*m*n^3*arctan(x*f^(1/2)/e^(1/2))*f^(1/2)/e
^(1/2)+12*b^2*m*n^2*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))*f^(1/2)/e^(1/2)-6*I*b^3*m*n^3*polylog(2,-I*x*f^(
1/2)/e^(1/2))*f^(1/2)/e^(1/2)+6*I*b^3*m*n^3*polylog(2,I*x*f^(1/2)/e^(1/2))*f^(1/2)/e^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.11, antiderivative size = 879, normalized size of antiderivative = 1.00, 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 12

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

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 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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \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*}

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Mathematica [B]  time = 0.71, 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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fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}\,{d x} \]

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)

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maple [F]  time = 112.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b^{3} m \log \left (x^{n}\right )^{3} + 3 \, {\left (m n + m \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, m n^{2} + 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} a b^{2} + {\left (6 \, m n^{3} + 6 \, m n^{2} \log \relax (c) + 3 \, m n \log \relax (c)^{2} + m \log \relax (c)^{3}\right )} b^{3} + a^{3} m + 3 \, {\left ({\left (m n + m \log \relax (c)\right )} b^{3} + a b^{2} m\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (2 \, {\left (m n + m \log \relax (c)\right )} a b^{2} + {\left (2 \, m n^{2} + 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} b^{3} + a^{2} b m\right )} \log \left (x^{n}\right )\right )} \log \left (f x^{2} + e\right )}{x} + \int \frac {b^{3} e \log \relax (c)^{3} \log \relax (d) + 3 \, a b^{2} e \log \relax (c)^{2} \log \relax (d) + 3 \, a^{2} b e \log \relax (c) \log \relax (d) + a^{3} e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} b^{3} x^{2} + b^{3} e \log \relax (d)\right )} \log \left (x^{n}\right )^{3} + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a^{3} + 3 \, {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} a^{2} b + 3 \, {\left (4 \, f m n^{2} + 4 \, f m n \log \relax (c) + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)^{2}\right )} a b^{2} + {\left (12 \, f m n^{3} + 12 \, f m n^{2} \log \relax (c) + 6 \, f m n \log \relax (c)^{2} + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)^{3}\right )} b^{3}\right )} x^{2} + 3 \, {\left (b^{3} e \log \relax (c) \log \relax (d) + a b^{2} e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a b^{2} + {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b^{2} e \log \relax (c) \log \relax (d) + a^{2} b e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a^{2} b + 2 \, {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} a b^{2} + {\left (4 \, f m n^{2} + 4 \, f m n \log \relax (c) + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )}{f x^{4} + e x^{2}}\,{d x} \]

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]

-(b^3*m*log(x^n)^3 + 3*(m*n + m*log(c))*a^2*b + 3*(2*m*n^2 + 2*m*n*log(c) + m*log(c)^2)*a*b^2 + (6*m*n^3 + 6*m
*n^2*log(c) + 3*m*n*log(c)^2 + m*log(c)^3)*b^3 + a^3*m + 3*((m*n + m*log(c))*b^3 + a*b^2*m)*log(x^n)^2 + 3*(2*
(m*n + m*log(c))*a*b^2 + (2*m*n^2 + 2*m*n*log(c) + m*log(c)^2)*b^3 + a^2*b*m)*log(x^n))*log(f*x^2 + e)/x + int
egrate((b^3*e*log(c)^3*log(d) + 3*a*b^2*e*log(c)^2*log(d) + 3*a^2*b*e*log(c)*log(d) + a^3*e*log(d) + ((2*f*m +
f*log(d))*b^3*x^2 + b^3*e*log(d))*log(x^n)^3 + ((2*f*m + f*log(d))*a^3 + 3*(2*f*m*n + (2*f*m + f*log(d))*log(
c))*a^2*b + 3*(4*f*m*n^2 + 4*f*m*n*log(c) + (2*f*m + f*log(d))*log(c)^2)*a*b^2 + (12*f*m*n^3 + 12*f*m*n^2*log(
c) + 6*f*m*n*log(c)^2 + (2*f*m + f*log(d))*log(c)^3)*b^3)*x^2 + 3*(b^3*e*log(c)*log(d) + a*b^2*e*log(d) + ((2*
f*m + f*log(d))*a*b^2 + (2*f*m*n + (2*f*m + f*log(d))*log(c))*b^3)*x^2)*log(x^n)^2 + 3*(b^3*e*log(c)^2*log(d)
+ 2*a*b^2*e*log(c)*log(d) + a^2*b*e*log(d) + ((2*f*m + f*log(d))*a^2*b + 2*(2*f*m*n + (2*f*m + f*log(d))*log(c
))*a*b^2 + (4*f*m*n^2 + 4*f*m*n*log(c) + (2*f*m + f*log(d))*log(c)^2)*b^3)*x^2)*log(x^n))/(f*x^4 + e*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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