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

Optimal. Leaf size=571 \[ -\frac{2 b f^{3/2} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac{2 b f^{3/2} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac{2 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}-\frac{2 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}+\frac{2 b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 (-e)^{3/2}}-\frac{2 b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 (-e)^{3/2}}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac{4 b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}+\frac{f^{3/2} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac{f^{3/2} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac{16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac{2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac{2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac{4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 e^{3/2}}-\frac{52 b^2 f m n^2}{27 e x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.931052, antiderivative size = 571, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used = {2305, 2304, 2378, 325, 205, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589} \[ -\frac{2 b f^{3/2} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac{2 b f^{3/2} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac{2 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}-\frac{2 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}+\frac{2 b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 (-e)^{3/2}}-\frac{2 b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{3 (-e)^{3/2}}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac{4 b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}+\frac{f^{3/2} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac{f^{3/2} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac{16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac{2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac{2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac{4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{27 e^{3/2}}-\frac{52 b^2 f m n^2}{27 e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2305

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

Rule 2304

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

Rule 2378

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

Rule 325

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

Rule 205

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2324

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

Rule 12

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

Rule 4848

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2353

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

Rule 2330

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

Rule 2317

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rubi steps

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

Mathematica [A]  time = 0.445522, size = 1083, normalized size = 1.9 \[ \frac{-18 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2(x) x^3-18 b^2 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2\left (c x^n\right ) x^3-4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) x^3-18 a^2 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) x^3-12 a b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) x^3+12 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) x^3+36 a b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) x^3-36 a b f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right ) x^3-12 b^2 f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right ) x^3+36 b^2 f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) \log \left (c x^n\right ) x^3+9 i b^2 f^{3/2} m n^2 \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-6 i b^2 f^{3/2} m n^2 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-18 i a b f^{3/2} m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-18 i b^2 f^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-9 i b^2 f^{3/2} m n^2 \log ^2(x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^3+6 i b^2 f^{3/2} m n^2 \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^3+18 i a b f^{3/2} m n \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^3+18 i b^2 f^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) x^3+6 i b f^{3/2} m n \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-6 i b f^{3/2} m n \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-18 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3+18 i b^2 f^{3/2} m n^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) x^3-52 b^2 \sqrt{e} f m n^2 x^2-18 b^2 \sqrt{e} f m \log ^2\left (c x^n\right ) x^2-18 a^2 \sqrt{e} f m x^2-48 a b \sqrt{e} f m n x^2-36 a b \sqrt{e} f m \log \left (c x^n\right ) x^2-48 b^2 \sqrt{e} f m n \log \left (c x^n\right ) x^2-2 b^2 e^{3/2} n^2 \log \left (d \left (f x^2+e\right )^m\right )-9 b^2 e^{3/2} \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-9 a^2 e^{3/2} \log \left (d \left (f x^2+e\right )^m\right )-6 a b e^{3/2} n \log \left (d \left (f x^2+e\right )^m\right )-18 a b e^{3/2} \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-6 b^2 e^{3/2} n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )}{27 e^{3/2} x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a^2*Sqrt[e]*f*m*x^2 - 48*a*b*Sqrt[e]*f*m*n*x^2 - 52*b^2*Sqrt[e]*f*m*n^2*x^2 - 18*a^2*f^(3/2)*m*x^3*ArcTan
[(Sqrt[f]*x)/Sqrt[e]] - 12*a*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(S
qrt[f]*x)/Sqrt[e]] + 36*a*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 12*b^2*f^(3/2)*m*n^2*x^3*ArcT
an[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 18*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 36*a*b*Sqrt[e
]*f*m*x^2*Log[c*x^n] - 48*b^2*Sqrt[e]*f*m*n*x^2*Log[c*x^n] - 36*a*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*
Log[c*x^n] - 12*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 36*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sq
rt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 18*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 18*b^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[
f]*x)/Sqrt[e]]*Log[c*x^n]^2 - (18*I)*a*b*f^(3/2)*m*n*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^2*f^(
3/2)*m*n^2*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[1 - (I*Sqrt[f]
*x)/Sqrt[e]] - (18*I)*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)*a*b*f^(3/2
)*m*n*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqr
t[e]] - (9*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^2*f^(3/2)*m*n*x^3*Log[x
]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 9*a^2*e^(3/2)*Log[d*(e + f*x^2)^m] - 6*a*b*e^(3/2)*n*Log[d*(e +
f*x^2)^m] - 2*b^2*e^(3/2)*n^2*Log[d*(e + f*x^2)^m] - 18*a*b*e^(3/2)*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 6*b^2*e^
(3/2)*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 9*b^2*e^(3/2)*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + (6*I)*b*f^(3/2)*m*
n*x^3*(3*a + b*n + 3*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b*f^(3/2)*m*n*x^3*(3*a + b*n +
3*b*Log[c*x^n])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, ((-I)*Sqrt[f]*x)/
Sqrt[e]] + (18*I)*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)*x^3)

________________________________________________________________________________________

Maple [F]  time = 9.602, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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))^2*log(d*(f*x^2+e)^m)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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))**2*ln(d*(f*x**2+e)**m)/x**4,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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