3.2.14 \(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^4} \, dx\) [114]

Optimal. Leaf size=1007 \[ -\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}} \]

[Out]

-2/3*f*m*(a+b*ln(c*x^n))^3/e/x-2/27*b^3*n^3*ln(d*(f*x^2+e)^m)/x^3-160/27*b^3*f*m*n^3/e/x-1/3*(a+b*ln(c*x^n))^3
*ln(d*(f*x^2+e)^m)/x^3-4/27*b^3*f^(3/2)*m*n^3*arctan(x*f^(1/2)/e^(1/2))/e^(3/2)+2/3*b^3*f^(3/2)*m*n^3*polylog(
3,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b^3*f^(3/2)*m*n^3*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2*b^3*f^(
3/2)*m*n^3*polylog(4,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2*b^3*f^(3/2)*m*n^3*polylog(4,x*f^(1/2)/(-e)^(1/2))/(-e
)^(3/2)-2/9*b^2*n^2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^3-1/3*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^3+1/3*
f^(3/2)*m*(a+b*ln(c*x^n))^3*ln(1-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-1/3*f^(3/2)*m*(a+b*ln(c*x^n))^3*ln(1+x*f^(1/
2)/(-e)^(1/2))/(-e)^(3/2)-52/9*b^2*f*m*n^2*(a+b*ln(c*x^n))/e/x-8/3*b*f*m*n*(a+b*ln(c*x^n))^2/e/x-4/9*b^2*f^(3/
2)*m*n^2*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))/e^(3/2)+1/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/
(-e)^(1/2))/(-e)^(3/2)-1/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b^2*f^(3/
2)*m*n^2*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/3*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^n))*pol
ylog(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,-x*f^(1/2)/(-e)^(1/2))/(
-e)^(3/2)-2*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/9*I*b^3*f^(3/2)*m*n
^3*polylog(2,I*x*f^(1/2)/e^(1/2))/e^(3/2)-b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*polylog(2,-x*f^(1/2)/(-e)^(1/2))/(-e
)^(3/2)+b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*polylog(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/9*I*b^3*f^(3/2)*m*n^3*pol
ylog(2,-I*x*f^(1/2)/e^(1/2))/e^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 1007, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {2342, 2341, 2425, 331, 211, 2380, 2361, 12, 4940, 2438, 2367, 2354, 2421, 6724, 2430} \begin {gather*} -\frac {4 b^3 f^{3/2} m \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}

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 12

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

Rule 211

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

Rule 331

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 2341

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 2342

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 2354

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 2361

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 2367

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 2380

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2425

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2438

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 4940

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 6724

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^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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2488\) vs. \(2(1007)=2014\).
time = 0.55, size = 2488, normalized size = 2.47 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{4}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

-1/27*(9*b^3*m*log(x^n)^3 + 9*(m*n + 3*m*log(c))*a^2*b + 3*(2*m*n^2 + 6*m*n*log(c) + 9*m*log(c)^2)*a*b^2 + (2*
m*n^3 + 6*m*n^2*log(c) + 9*m*n*log(c)^2 + 9*m*log(c)^3)*b^3 + 9*a^3*m + 9*((m*n + 3*m*log(c))*b^3 + 3*a*b^2*m)
*log(x^n)^2 + 3*(6*(m*n + 3*m*log(c))*a*b^2 + (2*m*n^2 + 6*m*n*log(c) + 9*m*log(c)^2)*b^3 + 9*a^2*b*m)*log(x^n
))*log(f*x^2 + e)/x^3 + integrate(1/27*(9*((2*f*m + 3*f*log(d))*b^3*x^2 + 3*b^3*e*log(d))*log(x^n)^3 + (9*(2*f
*m + 3*f*log(d))*a^3 + 9*(2*f*m*n + 3*(2*f*m + 3*f*log(d))*log(c))*a^2*b + 3*(4*f*m*n^2 + 12*f*m*n*log(c) + 9*
(2*f*m + 3*f*log(d))*log(c)^2)*a*b^2 + (4*f*m*n^3 + 12*f*m*n^2*log(c) + 18*f*m*n*log(c)^2 + 9*(2*f*m + 3*f*log
(d))*log(c)^3)*b^3)*x^2 + 9*((3*(2*f*m + 3*f*log(d))*a*b^2 + (2*f*m*n + 3*(2*f*m + 3*f*log(d))*log(c))*b^3)*x^
2 + 9*(b^3*log(c)*log(d) + a*b^2*log(d))*e)*log(x^n)^2 + 27*(b^3*log(c)^3*log(d) + 3*a*b^2*log(c)^2*log(d) + 3
*a^2*b*log(c)*log(d) + a^3*log(d))*e + 3*((9*(2*f*m + 3*f*log(d))*a^2*b + 6*(2*f*m*n + 3*(2*f*m + 3*f*log(d))*
log(c))*a*b^2 + (4*f*m*n^2 + 12*f*m*n*log(c) + 9*(2*f*m + 3*f*log(d))*log(c)^2)*b^3)*x^2 + 27*(b^3*log(c)^2*lo
g(d) + 2*a*b^2*log(c)*log(d) + a^2*b*log(d))*e)*log(x^n))/(f*x^6 + x^4*e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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^4} \,d x \end {gather*}

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^4,x)

[Out]

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

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