3.356 \(\int \frac{\log (c (d+e x^2)^p)}{x^2 (f+g x^2)^2} \, dx\)

Optimal. Leaf size=803 \[ -\frac{\sqrt{d} \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{f^2 (e f-d g)}+\frac{2 \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f^2}-\frac{e \sqrt{g} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 (-f)^{3/2} (e f-d g)}-\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{5/2}}+\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 f^{5/2}}+\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 f^{5/2}}+\frac{e \sqrt{g} p \log \left (\sqrt{g} x+\sqrt{-f}\right )}{2 (-f)^{3/2} (e f-d g)}-\frac{3 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{2 f^{5/2}}-\frac{\log \left (c \left (e x^2+d\right )^p\right )}{f^2 x}+\frac{\sqrt{g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt{g} x+\sqrt{-f}\right )}+\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 f^{5/2}}-\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}+1\right )}{4 f^{5/2}}-\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 f^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.54166, antiderivative size = 803, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2476, 2455, 205, 2471, 2463, 801, 635, 260, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ -\frac{\sqrt{d} \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{f^2 (e f-d g)}+\frac{2 \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f^2}-\frac{e \sqrt{g} p \log \left (\sqrt{-f}-\sqrt{g} x\right )}{2 (-f)^{3/2} (e f-d g)}-\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{5/2}}+\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 f^{5/2}}+\frac{3 \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 f^{5/2}}+\frac{e \sqrt{g} p \log \left (\sqrt{g} x+\sqrt{-f}\right )}{2 (-f)^{3/2} (e f-d g)}-\frac{3 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{2 f^{5/2}}-\frac{\log \left (c \left (e x^2+d\right )^p\right )}{f^2 x}+\frac{\sqrt{g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt{g} x+\sqrt{-f}\right )}+\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{2 f^{5/2}}-\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}+1\right )}{4 f^{5/2}}-\frac{3 i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{4 f^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2476

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

Rule 2455

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2471

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

Rule 2463

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2470

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 2402

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

Rule 2315

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

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [A]  time = 4.50928, size = 1438, normalized size = 1.79 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.51, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) }{{x}^{2} \left ( g{x}^{2}+f \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(c*(e*x^2+d)^p)/x^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(log(c*(e*x^2+d)^p)/x^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{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g^{2} x^{6} + 2 \, f g x^{4} + f^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x**2+d)**p)/x**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{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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