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

Optimal. Leaf size=821 \[ -\frac{b^2 e \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}-\frac{b^2 e \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \sqrt{g}}-\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 (-f)^{3/2} \sqrt{g}}+\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt{g}}-\frac{b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \sqrt{g}}-\frac{b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 (-f)^{3/2} \sqrt{g}}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 (-f)^{3/2} \sqrt{g}}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{g} x+\sqrt{-f}\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{4 (-f)^{3/2} \sqrt{g}}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 (-f)^{3/2} \sqrt{g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.846161, antiderivative size = 821, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2409, 2397, 2394, 2393, 2391, 2396, 2433, 2374, 6589} \[ -\frac{b^2 e \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}-\frac{b^2 e \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \sqrt{g}}-\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n^2}{2 (-f)^{3/2} \sqrt{g}}+\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt{g}}-\frac{b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \sqrt{g}}-\frac{b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) n}{2 (-f)^{3/2} \sqrt{g}}-\frac{b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right ) n}{2 (-f)^{3/2} \sqrt{g}}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt{g} d+e \sqrt{-f}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{g} x+\sqrt{-f}\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{4 (-f)^{3/2} \sqrt{g}}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 (-f)^{3/2} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

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}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt{-f} \sqrt{g}-g x\right )^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt{-f} \sqrt{g}+g x\right )^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx\\ &=-\frac{g \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (\sqrt{-f} \sqrt{g}-g x\right )^2} \, dx}{4 f}-\frac{g \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (\sqrt{-f} \sqrt{g}+g x\right )^2} \, dx}{4 f}-\frac{g \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{-f g-g^2 x^2} \, dx}{2 f}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{g \int \left (-\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f g \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 g \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{2 f}+\frac{\left (b e \sqrt{g} n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f} \sqrt{g}+g x} \, dx}{2 f \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{\left (b e \sqrt{g} n\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f} \sqrt{g}-g x} \, dx}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right )}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{b e 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 \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\frac{b e 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 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\frac{\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{4 (-f)^{3/2}}+\frac{\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{4 (-f)^{3/2}}+\frac{\left (b^2 e^2 n^2\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f} \sqrt{g}-g x\right )}{e \sqrt{-f} \sqrt{g}+d g}\right )}{d+e x} \, dx}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}+\frac{\left (b^2 e^2 n^2\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-f} \sqrt{g}+g x\right )}{e \sqrt{-f} \sqrt{g}-d g}\right )}{d+e x} \, dx}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{b e 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 \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b e 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 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{4 (-f)^{3/2} \sqrt{g}}+\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}{2 (-f)^{3/2} \sqrt{g}}-\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}{2 (-f)^{3/2} \sqrt{g}}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{g x}{e \sqrt{-f} \sqrt{g}+d g}\right )}{x} \, dx,x,d+e x\right )}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}+\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e \sqrt{-f} \sqrt{g}-d g}\right )}{x} \, dx,x,d+e x\right )}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{b e 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 \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b e 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 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}+\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 )}{2 (-f)^{3/2} \sqrt{g}}-\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 )}{2 (-f)^{3/2} \sqrt{g}}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{b e 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 \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b e 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 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{2 (-f)^{3/2} \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{2 (-f)^{3/2} \sqrt{g}}-\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 )}{2 (-f)^{3/2} \sqrt{g}}+\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 )}{2 (-f)^{3/2} \sqrt{g}}\\ &=-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt{-f}-d \sqrt{g}\right ) \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{b e 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 \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b e 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 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{4 (-f)^{3/2} \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt{g}\right ) \sqrt{g}}+\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 )}{2 (-f)^{3/2} \sqrt{g}}-\frac{b^2 e n^2 \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f \left (e \sqrt{-f}+d \sqrt{g}\right ) \sqrt{g}}-\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 )}{2 (-f)^{3/2} \sqrt{g}}-\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{3/2} \sqrt{g}}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{3/2} \sqrt{g}}\\ \end{align*}

Mathematica [C]  time = 2.62324, size = 1143, normalized size = 1.39 \[ \frac{\frac{b^2 \left (-\frac{\sqrt{f} \left (-\sqrt{g} (d+e x) \log ^2(d+e x)+2 e \left (\sqrt{g} x+i \sqrt{f}\right ) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{i \sqrt{g} d+e \sqrt{f}}\right ) \log (d+e x)+2 e \left (\sqrt{g} x+i \sqrt{f}\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 (\sqrt{g} (d+e x) \log (d+e x)+2 i 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 )\right )+2 i 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 )}+i \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right )-i \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right )\right ) n^2}{\sqrt{g}}+\frac{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 (\sqrt{g} (d+e x) \log (d+e x)+i e \left (i \sqrt{g} x+\sqrt{f}\right ) \log \left (i \sqrt{f}-\sqrt{g} x\right )\right )}{\left (e \sqrt{f}-i d \sqrt{g}\right ) \left (i \sqrt{g} x+\sqrt{f}\right )}+\frac{\sqrt{f} \left (\sqrt{g} (d+e x) \log (d+e x)+e \left (-\sqrt{g} x-i \sqrt{f}\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 )}-i \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 )+i \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 )\right ) n}{\sqrt{g}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{g}}+\frac{2 \sqrt{f} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{g x^2+f}}{4 f^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[f]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a
- b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/Sqrt[g] + (2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*
((Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + I*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)) + (Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + e*((-I)*Sqrt[f] - Sqrt
[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - I*(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*(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])])))/Sqrt[g] + (b^2*n^2*(-((Sqrt[f]*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2)
+ 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e*(
I*Sqrt[f] + 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))) + (Sqrt[f]*(Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sq
rt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*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)) + I*(
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
*(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])])))/Sqrt[g])
/(4*f^(3/2))

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Maple [F]  time = 11.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{ \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/(g*x^2+f)^2,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,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/(g*x^2+f)^2,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^{2} x^{4} + 2 \, f g x^{2} + f^{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/(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^4 + 2*f*g*x^2 + f^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/(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}}\,{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/(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)