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

Optimal. Leaf size=461 \[ -\frac{b \sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac{b \sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac{b^2 \sqrt{g} n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{3/2}}-\frac{b^2 \sqrt{g} n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{(-f)^{3/2}}+\frac{2 b^2 e n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d f}+\frac{\sqrt{g} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}-\frac{\sqrt{g} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}+\frac{2 b e n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.633778, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2416, 2397, 2394, 2315, 2409, 2396, 2433, 2374, 6589} \[ -\frac{b \sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac{b \sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac{b^2 \sqrt{g} n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{3/2}}-\frac{b^2 \sqrt{g} n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{(-f)^{3/2}}+\frac{2 b^2 e n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d f}+\frac{\sqrt{g} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}-\frac{\sqrt{g} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}+\frac{2 b e n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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]

Rubi steps

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

Mathematica [C]  time = 0.494154, size = 668, normalized size = 1.45 \[ \frac{2 b n \left (i d \sqrt{g} x \left (\text{PolyLog}\left (2,-\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}-i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}+i \sqrt{g} x\right )}{e \sqrt{f}-i d \sqrt{g}}\right )\right )-i d \sqrt{g} x \left (\text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}+i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{e \sqrt{f}+i d \sqrt{g}}\right )\right )+2 \sqrt{f} (e x \log (x)-(d+e x) \log (d+e x))\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+b^2 n^2 \left (-i d \sqrt{g} x \left (-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+\log ^2(d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right )+i d \sqrt{g} x \left (-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+\log ^2(d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )\right )+2 \sqrt{f} \left (2 e x \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-(d+e x) \log ^2(d+e x)+2 e x \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )\right )-2 d \sqrt{g} x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-2 d \sqrt{f} \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{2 d f^{3/2} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 20.142, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{{x}^{2} \left ( g{x}^{2}+f \right ) }}\, 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^2/(g*x^2+f),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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 x^{4} + f x^{2}}, 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^2/(g*x^2+f),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*x^4 + f*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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