3.4.28 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{x^2 (f+g x^2)^2} \, dx\) [328]

Optimal. Leaf size=919 \[ \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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \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 )}{4 (-f)^{5/2}}+\frac {b e \sqrt {g} 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 (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 \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 )}{4 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 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 )}{2 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}} \]

[Out]

2*b*e*n*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/d/f^2-(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/d/f^2/x+2*b^2*e*n^2*polylog(2,1
+e*x/d)/d/f^2-3/4*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))*g^(1/2)/(-f)^(
5/2)+3/4*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/2)-d*g^(1/2)))*g^(1/2)/(-f)^(5/2)+3/2*
b*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*g^(1/2)/(-f)^(5/2)-3/2*b*n*(a+b
*ln(c*(e*x+d)^n))*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))*g^(1/2)/(-f)^(5/2)-3/2*b^2*n^2*polylog(3
,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*g^(1/2)/(-f)^(5/2)+3/2*b^2*n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^(
1/2)+d*g^(1/2)))*g^(1/2)/(-f)^(5/2)+1/2*b*e*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+
d*g^(1/2)))*g^(1/2)/f^2/(e*(-f)^(1/2)+d*g^(1/2))+1/2*b^2*e*n^2*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/
2)))*g^(1/2)/f^2/(e*(-f)^(1/2)+d*g^(1/2))+1/2*b*e*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^
(1/2)-d*g^(1/2)))*g^(1/2)/f/(e*(-f)^(3/2)+d*f*g^(1/2))+1/2*b^2*e*n^2*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-
d*g^(1/2)))*g^(1/2)/f/(e*(-f)^(3/2)+d*f*g^(1/2))+1/4*g*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f^2/(e*(-f)^(1/2)+d*g^(
1/2))/((-f)^(1/2)-x*g^(1/2))+1/4*g*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f^2/(e*(-f)^(1/2)-d*g^(1/2))/((-f)^(1/2)+x*
g^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 1.26, antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2463, 2444, 2441, 2352, 2456, 2440, 2438, 2443, 2481, 2421, 6724} \begin {gather*} \frac {b^2 e \sqrt {g} \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {b^2 e \sqrt {g} \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {2 b^2 e \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) n^2}{d f^2}-\frac {3 b^2 \sqrt {g} \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{5/2}}+\frac {2 b e \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) n}{d f^2}+\frac {b e \sqrt {g} \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 (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {b e \sqrt {g} \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 \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 b \sqrt {g} \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)^{5/2}}-\frac {3 b \sqrt {g} \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)^{5/2}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {3 \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 )}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{5/2}}+\frac {3 \sqrt {g} \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)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*e*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f^2) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f^2
*x) + (g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])*(Sqrt[-f] - Sqrt[g]*x)) + (g*
(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f^2*(e*Sqrt[-f] - d*Sqrt[g])*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*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*Sqrt[-f] + d
*Sqrt[g])) - (3*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)^(5/2)) + (b*e*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqr
t[g])])/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*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)^(5/2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[
-f] - d*Sqrt[g]))])/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*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]))])/(2*(-f)^(5/2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, (Sqrt[g]*(d +
e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*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])])/(2*(-f)^(5/2)) + (2*b^2*e*n^2*PolyLog[2, 1 + (e*x)/
d])/(d*f^2) - (3*b^2*Sqrt[g]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)) +
(3*b^2*Sqrt[g]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2))

Rule 2352

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

Rule 2421

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

Rule 2438

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

Rule 2440

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 2441

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 2443

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 2444

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 2456

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 2463

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 2481

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 6724

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 \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^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f \left (f+g x^2\right )^2}-\frac {g \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^2} \, dx}{f^2}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx}{f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 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^2}-\frac {g \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}{f}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f^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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 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)^{5/2}}+\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)^{5/2}}+\frac {g^2 \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^2}+\frac {g^2 \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^2}+\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{-f g-g^2 x^2} \, dx}{2 f^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^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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\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)^{5/2}}+\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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {g^2 \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^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)^{5/2}}-\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)^{5/2}}-\frac {\left (b e g^{3/2} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}+g x} \, dx}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b e g^{3/2} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}-g x} \, dx}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\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)^{5/2}}-\frac {b e \sqrt {g} 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 \sqrt {-f}-d \sqrt {g}\right )}+\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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}+\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{5/2}}+\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{5/2}}+\frac {\left (b \sqrt {g} n\right ) \text {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)^{5/2}}-\frac {\left (b \sqrt {g} n\right ) \text {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)^{5/2}}+\frac {\left (b^2 e^2 \sqrt {g} 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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b^2 e^2 \sqrt {g} 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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} 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 \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \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 )}{4 (-f)^{5/2}}+\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)^{5/2}}-\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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^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}{2 (-f)^{5/2}}-\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}{2 (-f)^{5/2}}-\frac {\left (b^2 \sqrt {g} n^2\right ) \text {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)^{5/2}}+\frac {\left (b^2 \sqrt {g} n^2\right ) \text {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)^{5/2}}+\frac {\left (b^2 e \sqrt {g} n^2\right ) \text {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^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b^2 e \sqrt {g} n^2\right ) \text {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^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} 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 \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\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)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\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)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^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)^{5/2}}+\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)^{5/2}}+\frac {\left (b \sqrt {g} n\right ) \text {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)^{5/2}}-\frac {\left (b \sqrt {g} n\right ) \text {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)^{5/2}}\\ &=\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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} 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 \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 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 )}{2 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^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)^{5/2}}+\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)^{5/2}}-\frac {\left (b^2 \sqrt {g} n^2\right ) \text {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)^{5/2}}+\frac {\left (b^2 \sqrt {g} n^2\right ) \text {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)^{5/2}}\\ &=\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^2}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} 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 \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} 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 \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \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 )}{4 (-f)^{5/2}}-\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 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 )}{2 (-f)^{5/2}}+\frac {b^2 e \sqrt {g} n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 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 )}{2 (-f)^{5/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f^2}-\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.26, size = 1304, normalized size = 1.42 \begin {gather*} \frac {-\frac {4 \sqrt {f} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}-\frac {2 \sqrt {f} g x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}-6 \sqrt {g} \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+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {4 \sqrt {f} (e x \log (x)-(d+e x) \log (d+e x))}{d x}-\frac {\sqrt {f} \sqrt {g} \left (\sqrt {g} (d+e x) \log (d+e x)+i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {\sqrt {f} \sqrt {g} \left (-\sqrt {g} (d+e x) \log (d+e x)+e \left (i \sqrt {f}+\sqrt {g} x\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+3 i \sqrt {g} \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\text {Li}_2\left (-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )-3 i \sqrt {g} \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )\right )+b^2 n^2 \left (\frac {\sqrt {f} \sqrt {g} \left (-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\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 )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {\sqrt {f} \sqrt {g} \left (\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {4 \sqrt {f} \left (2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-(d+e x) \log ^2(d+e x)+2 e x \text {Li}_2\left (1+\frac {e x}{d}\right )\right )}{d x}-3 i \sqrt {g} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+3 i \sqrt {g} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{4 f^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*Sqrt[f]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/x - (2*Sqrt[f]*g*x*(a - b*n*Log[d + e*x] + b*Log
[c*(d + e*x)^n])^2)/(f + g*x^2) - 6*Sqrt[g]*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])*((4*Sqrt[f]*(e*x*Log[x] - (d + e*x)*Log[d + e*
x]))/(d*x) - (Sqrt[f]*Sqrt[g]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + I*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sqrt[f] - Sq
rt[g]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (Sqrt[f]*Sqrt[g]*(-(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)) + (3*I)*Sqrt[g]*(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])]) - (3*I)*Sqrt[g]*(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*((Sqrt
[f]*Sqrt[g]*(-(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*Sqr
t[f] + I*d*Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - (Sqrt[f]*Sqrt[g]*(Log[d + e*x]*(S
qrt[g]*(d + e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt[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)) + (4*Sqrt[f]*(2*e*x*Log[-((e*x)/d)]*Log[d + e*x] - (d + e*x)
*Log[d + e*x]^2 + 2*e*x*PolyLog[2, 1 + (e*x)/d]))/(d*x) - (3*I)*Sqrt[g]*(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])]) + (3*I)*Sqrt[g]*(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])])))/(4*f^(5/2))

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Maple [F]
time = 0.63, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{x^{2} \left (g \,x^{2}+f \right )^{2}}\, dx\]

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)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*((3*g*x^2 + 2*f)/(f^2*g*x^3 + f^3*x) + 3*g*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f^2)) + integrate((b^2*lo
g((x*e + d)^n)^2 + b^2*log(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log((x*e + d)^n))/(g^2*x^6 + 2*f*g*x^4 +
f^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log((x*e + d)^n*c)^2 + 2*a*b*log((x*e + d)^n*c) + a^2)/(g^2*x^6 + 2*f*g*x^4 + f^2*x^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2\,{\left (g\,x^2+f\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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