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

Optimal. Leaf size=581 \[ -\frac{i \sqrt{g} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{3/2}}-\frac{i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{3/2}}+\frac{i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{3/2}}-\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{f x}+\frac{\sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{3/2}}+\frac{\sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{3/2}}+\frac{2 \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f}-\frac{2 \sqrt{g} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{f^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.596958, antiderivative size = 581, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2476, 2455, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ -\frac{i \sqrt{g} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{3/2}}-\frac{i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{3/2}}+\frac{i \sqrt{g} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{3/2}}-\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{f x}+\frac{\sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{3/2}}+\frac{\sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{3/2}}+\frac{2 \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f}-\frac{2 \sqrt{g} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{f^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.298604, size = 673, normalized size = 1.16 \[ \frac{i \sqrt{g} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )+i \sqrt{g} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-i \sqrt{g} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )-i \sqrt{g} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 \sqrt{f} \log \left (c \left (d+e x^2\right )^p\right )}{x}+i \sqrt{g} p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+i \sqrt{g} p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-i \sqrt{g} p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-i \sqrt{g} p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+\frac{4 \sqrt{e} \sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}}{2 f^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((4*Sqrt[e]*Sqrt[f]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + I*Sqrt[g]*p*Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/
(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + I*Sqrt[g]*p*Log[(Sqrt[g]*(Sqrt[-d] +
Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] - I*Sqrt[g]*p*Log[(Sqrt[
g]*(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] - I*Sqrt[
g]*p*Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f
]] - (2*Sqrt[f]*Log[c*(d + e*x^2)^p])/x - 2*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p] + I*Sqrt[
g]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] + I*Sqrt[g]*p*PolyLo
g[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])] - I*Sqrt[g]*p*PolyLog[2, (Sqrt[
e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - I*Sqrt[g]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f]
+ I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/(2*f^(3/2))

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Maple [C]  time = 0.652, size = 755, normalized size = 1.3 \begin{align*} \text{result too large to display} \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),x)

[Out]

-(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/f*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/f/x+p
*Sum(-1/2*(ln(x-_alpha)*ln(e*x^2+d)-2*e*(1/2*ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1
)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,inde
x=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e+1/2*(dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*
g+d*g-e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+dilog((RootOf(_Z^2*e*g+2*_Z*_al
pha*e*g+d*g-e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e))/f/_alpha,_alpha=Root
Of(_Z^2*g+f))-p/f/x*ln(e*x^2+d)+2*p/f*e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(
I*c*(e*x^2+d)^p)^2/f*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^
2/f/x+1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/f*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/2
*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/f/x+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/f*g/(f*g)^(1/2)
*arctan(x*g/(f*g)^(1/2))+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/f/x-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/f*g/(
f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/f/x-ln(c)/f*g/(f*g)^(1/2)*arctan
(x*g/(f*g)^(1/2))-ln(c)/f/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{4} + f 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),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} 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),x, algorithm="giac")

[Out]

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