3.4.25 \(\int \frac {x^4 (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [325]

Optimal. Leaf size=897 \[ -\frac {2 a b n x}{g^2}+\frac {2 b^2 n^2 x}{g^2}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e f 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 \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {3 \sqrt {-f} \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 g^{5/2}}+\frac {b e f 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 \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 \sqrt {-f} \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 g^{5/2}}+\frac {b^2 e f n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 b \sqrt {-f} 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 g^{5/2}}-\frac {b^2 e f n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {3 b \sqrt {-f} 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 g^{5/2}}+\frac {3 b^2 \sqrt {-f} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {3 b^2 \sqrt {-f} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.36, antiderivative size = 897, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2463, 2436, 2333, 2332, 2456, 2444, 2441, 2440, 2438, 2443, 2481, 2421, 6724} \begin {gather*} \frac {2 n^2 x b^2}{g^2}-\frac {2 n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e g^2}+\frac {e f n^2 \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) b^2}{2 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {e f n^2 \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) b^2}{2 \left (\sqrt {g} d+e \sqrt {-f}\right ) g^{5/2}}+\frac {3 \sqrt {-f} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) b^2}{2 g^{5/2}}-\frac {3 \sqrt {-f} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) b^2}{2 g^{5/2}}-\frac {2 a n x b}{g^2}-\frac {e f n \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 ) b}{2 \left (\sqrt {g} d+e \sqrt {-f}\right ) g^{5/2}}+\frac {e f n \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 ) b}{2 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 \sqrt {-f} n \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 ) b}{2 g^{5/2}}+\frac {3 \sqrt {-f} n \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 ) b}{2 g^{5/2}}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (\sqrt {g} d+e \sqrt {-f}\right ) g^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^2 \left (\sqrt {g} x+\sqrt {-f}\right )}+\frac {3 \sqrt {-f} \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 g^{5/2}}-\frac {3 \sqrt {-f} \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 g^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2436

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 2.30, size = 1237, normalized size = 1.38 \begin {gather*} \frac {4 \sqrt {g} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\frac {2 f \sqrt {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 {f} \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 {g} (d+e x) (-1+\log (d+e x))}{e}+\frac {f \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 {f \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 {f} \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 {f} \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 {4 \sqrt {g} \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )}{e}-\frac {f \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 {f \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 )}-3 i \sqrt {f} \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 {f} \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 g^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.71, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{\left (g \,x^{2}+f \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*(a+b*ln(c*(e*x+d)^n))^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(x^4*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="maxima")

[Out]

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

[Out]

integral((b^2*x^4*log((x*e + d)^n*c)^2 + 2*a*b*x^4*log((x*e + d)^n*c) + a^2*x^4)/(g^2*x^4 + 2*f*g*x^2 + f^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(x**4*(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 = 0.00, 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(x^4*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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