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

Optimal. Leaf size=814 \[ -\frac{b^2 e \left (\sqrt{g} d+e \sqrt{-f}\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 (-f)^{3/2} \left (g d^2+e^2 f\right )}+\frac{b^2 e \left (\sqrt{-f} \sqrt{g} d+e f\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 f^2 \left (g d^2+e^2 f\right )}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{f^2}+\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{f^2}-\frac{2 b^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right ) n^2}{f^2}+\frac{b e \left (\sqrt{-f} \sqrt{g} d+e f\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 f^2 \left (g d^2+e^2 f\right )}+\frac{b e \left (e f-d \sqrt{-f} \sqrt{g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 f^2 \left (g d^2+e^2 f\right )}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{f^2}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{f^2}+\frac{2 b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) n}{f^2}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (g d^2+e^2 f\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (g x^2+f\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{2 f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2} \]

[Out]

-(e^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*f*(e^2*f + d^2*g)) + (a + b*Log[c*(d + e*x)^n])^2/(2*f*(f + g*x^2)) + (
Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^2 + (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n
])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) - ((a + b*Log[c*(d + e*x)
^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*n*(
a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) -
((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) - (b^2*e*(e*Sq
rt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*(e^2*f + d^
2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 + (b^
2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d
^2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 + (2*b*
n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f^2 + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[
-f] - d*Sqrt[g]))])/f^2 + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 - (2*b^2*n^2*
PolyLog[3, 1 + (e*x)/d])/f^2

________________________________________________________________________________________

Rubi [A]  time = 1.30039, antiderivative size = 814, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2416, 2396, 2433, 2374, 6589, 2413, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{b^2 e \left (\sqrt{g} d+e \sqrt{-f}\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 (-f)^{3/2} \left (g d^2+e^2 f\right )}+\frac{b^2 e \left (\sqrt{-f} \sqrt{g} d+e f\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 f^2 \left (g d^2+e^2 f\right )}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{f^2}+\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{f^2}-\frac{2 b^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right ) n^2}{f^2}+\frac{b e \left (\sqrt{-f} \sqrt{g} d+e f\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 f^2 \left (g d^2+e^2 f\right )}+\frac{b e \left (e f-d \sqrt{-f} \sqrt{g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 f^2 \left (g d^2+e^2 f\right )}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{f^2}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{f^2}+\frac{2 b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) n}{f^2}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (g d^2+e^2 f\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (g x^2+f\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{2 f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*f*(e^2*f + d^2*g)) + (a + b*Log[c*(d + e*x)^n])^2/(2*f*(f + g*x^2)) + (
Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^2 + (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n
])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) - ((a + b*Log[c*(d + e*x)
^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*n*(
a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) -
((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) - (b^2*e*(e*Sq
rt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*(e^2*f + d^
2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 + (b^
2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d
^2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 + (2*b*
n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f^2 + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[
-f] - d*Sqrt[g]))])/f^2 + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 - (2*b^2*n^2*
PolyLog[3, 1 + (e*x)/d])/f^2

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

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

Rule 6589

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

Rule 2413

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x \left (f+g x^2\right )^2} \, dx &=\int \left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x}-\frac{g x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f \left (f+g x^2\right )^2}-\frac{g x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{f^2}-\frac{g \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2}-\frac{g \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx}{f}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{g \int \left (-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f^2}-\frac{(2 b e n) \int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{f^2}-\frac{(b e n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f+g x^2\right )} \, dx}{f}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac{\sqrt{g} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f^2}-\frac{\sqrt{g} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f^2}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x\right )}{f^2}-\frac{(b e n) \int \left (\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) (d+e x)}-\frac{g (-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}\right ) \, dx}{f}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}+\frac{(b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{f^2}+\frac{(b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{f^2}-\frac{\left (b e^3 n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{f \left (e^2 f+d^2 g\right )}+\frac{(b e g n) \int \frac{(-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{f \left (e^2 f+d^2 g\right )}-\frac{\left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{f^2}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}-\frac{2 b^2 n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^2}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}+d \sqrt{g}}{e}-\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}-d \sqrt{g}}{e}+\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{f \left (e^2 f+d^2 g\right )}+\frac{(b e g n) \int \left (\frac{\left (-d \sqrt{-f}-\frac{e f}{\sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (-d \sqrt{-f}+\frac{e f}{\sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f \left (e^2 f+d^2 g\right )}\\ &=-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (e^2 f+d^2 g\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}-\frac{2 b^2 n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^2}+\frac{\left (b e \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) g n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f \left (e^2 f+d^2 g\right )}-\frac{\left (b e \left (\frac{d f}{(-f)^{3/2}}+\frac{e}{\sqrt{g}}\right ) g n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f \left (e^2 f+d^2 g\right )}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}\\ &=-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (e^2 f+d^2 g\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{b e \left (e f-d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}-\frac{2 b^2 n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^2}-\frac{\left (b^2 e^2 \left (e f+d \sqrt{-f} \sqrt{g}\right ) n^2\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (b^2 e^2 \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) \sqrt{g} n^2\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{2 f \left (e^2 f+d^2 g\right )}\\ &=-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (e^2 f+d^2 g\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{b e \left (e f-d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}-\frac{2 b^2 n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^2}-\frac{\left (b^2 e \left (e f+d \sqrt{-f} \sqrt{g}\right ) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (b^2 e \left (\frac{d}{\sqrt{-f}}+\frac{e}{\sqrt{g}}\right ) \sqrt{g} n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 f \left (e^2 f+d^2 g\right )}\\ &=-\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (e^2 f+d^2 g\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (f+g x^2\right )}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac{b e \left (e f+d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{b e \left (e f-d \sqrt{-f} \sqrt{g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{b^2 e \left (e f-d \sqrt{-f} \sqrt{g}\right ) n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}+\frac{b^2 e \left (e f+d \sqrt{-f} \sqrt{g}\right ) n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac{b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}+\frac{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^2}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{f^2}-\frac{2 b^2 n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{f^2}\\ \end{align*}

Mathematica [C]  time = 2.03173, size = 1209, normalized size = 1.49 \[ \frac{b^2 \left (4 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)-2 \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log ^2(d+e x)-2 \log \left (1-\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log ^2(d+e x)-4 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log (d+e x)-4 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log (d+e x)+8 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \log (d+e x)+\frac{\sqrt{f} \left (\log (d+e x) \left (i \sqrt{g} (d+e x) \log (d+e x)+2 e \left (\sqrt{f}-i \sqrt{g} x\right ) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{i \sqrt{g} d+e \sqrt{f}}\right )\right )+2 e \left (\sqrt{f}-i \sqrt{g} x\right ) \text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{i \sqrt{g} d+e \sqrt{f}}\right )\right )}{\left (i \sqrt{g} d+e \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}+\frac{\sqrt{f} \left (\log (d+e x) \left (2 e \left (i \sqrt{g} x+\sqrt{f}\right ) \log \left (\frac{e \left (i \sqrt{g} x+\sqrt{f}\right )}{e \sqrt{f}-i d \sqrt{g}}\right )-i \sqrt{g} (d+e x) \log (d+e x)\right )+2 e \left (i \sqrt{g} x+\sqrt{f}\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right )}{\left (e \sqrt{f}-i d \sqrt{g}\right ) \left (i \sqrt{g} x+\sqrt{f}\right )}+4 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+4 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )-8 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )\right ) n^2+2 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac{\sqrt{f} \left (e \left (i \sqrt{g} x+\sqrt{f}\right ) \log \left (i \sqrt{f}-\sqrt{g} x\right )-i \sqrt{g} (d+e x) \log (d+e x)\right )}{\left (e \sqrt{f}-i d \sqrt{g}\right ) \left (i \sqrt{g} x+\sqrt{f}\right )}+\frac{\sqrt{f} \left (i \sqrt{g} (d+e x) \log (d+e x)+e \left (\sqrt{f}-i \sqrt{g} x\right ) \log \left (\sqrt{g} x+i \sqrt{f}\right )\right )}{\left (i \sqrt{g} d+e \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}-2 \left (\log (d+e x) \log \left (\frac{e \left (i \sqrt{g} x+\sqrt{f}\right )}{e \sqrt{f}-i d \sqrt{g}}\right )+\text{PolyLog}\left (2,-\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}-i d \sqrt{g}}\right )\right )-2 \left (\log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{i \sqrt{g} d+e \sqrt{f}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{i \sqrt{g} d+e \sqrt{f}}\right )\right )+4 \left (\log \left (-\frac{e x}{d}\right ) \log (d+e x)+\text{PolyLog}\left (2,\frac{e x}{d}+1\right )\right )\right ) n+4 \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\frac{2 f \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{g x^2+f}-2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (g x^2+f\right )}{4 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*f*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + 4*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*
(d + e*x)^n])^2 - 2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] + 2*b*n*(a - b*n*Log[d + e*
x] + b*Log[c*(d + e*x)^n])*((Sqrt[f]*((-I)*Sqrt[g]*(d + e*x)*Log[d + e*x] + e*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sq
rt[f] - Sqrt[g]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (Sqrt[f]*(I*Sqrt[g]*(d + e*x)*Log[d
+ e*x] + e*(Sqrt[f] - I*Sqrt[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[
g]*x)) - 2*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]
*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) - 2*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqr
t[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]) + 4*(Log[-((e*x)/d)]*Log[d + e*x] + Poly
Log[2, 1 + (e*x)/d])) + b^2*n^2*(4*Log[-((e*x)/d)]*Log[d + e*x]^2 - 2*Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x
))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + (
Sqrt[f]*(Log[d + e*x]*(I*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2*e*(Sqrt[f] - I*Sqrt[g]*x)*Log[(e*(Sqrt[f] - I*Sqrt
[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])]) + 2*e*(Sqrt[f] - I*Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f]
+ I*d*Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - 4*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d
+ e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 4*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g]
)] + (Sqrt[f]*(Log[d + e*x]*((-I)*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f]
+ I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + 2*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e
*Sqrt[f] + d*Sqrt[g])]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + 8*Log[d + e*x]*PolyLog[2, 1 + (
e*x)/d] + 4*PolyLog[3, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + 4*PolyLog[3, (Sqrt[g]*(d + e*x))/(I
*e*Sqrt[f] + d*Sqrt[g])] - 8*PolyLog[3, 1 + (e*x)/d]))/(4*f^2)

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Maple [F]  time = 1.363, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{x \left ( g{x}^{2}+f \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{1}{f g x^{2} + f^{2}} - \frac{\log \left (g x^{2} + f\right )}{f^{2}} + \frac{2 \, \log \left (x\right )}{f^{2}}\right )} + \int \frac{b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g^{2} x^{5} + 2 \, f g x^{3} + f^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*(1/(f*g*x^2 + f^2) - log(g*x^2 + f)/f^2 + 2*log(x)/f^2) + integrate((b^2*log((e*x + d)^n)^2 + b^2*log(
c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log((e*x + d)^n))/(g^2*x^5 + 2*f*g*x^3 + f^2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g^{2} x^{5} + 2 \, f g x^{3} + f^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log((e*x + d)^n*c)^2 + 2*a*b*log((e*x + d)^n*c) + a^2)/(g^2*x^5 + 2*f*g*x^3 + f^2*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**2/x/(g*x**2+f)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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