3.347 \(\int \frac{\log (c (d+e x^2)^p)}{x^4 (f+g x^2)} \, dx\)

Optimal. Leaf size=651 \[ \frac{i g^{3/2} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{5/2}}+\frac{i g^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{5/2}}-\frac{i g^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{5/2}}+\frac{g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{5/2}}+\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{3 f x^3}-\frac{2 e^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 d^{3/2} f}-\frac{g^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{5/2}}-\frac{g^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{5/2}}-\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f^2}-\frac{2 e p}{3 d f x}+\frac{2 g^{3/2} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{f^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.649823, antiderivative size = 651, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2476, 2455, 325, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac{i g^{3/2} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{5/2}}+\frac{i g^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 f^{5/2}}-\frac{i g^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{f^{5/2}}+\frac{g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{5/2}}+\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{3 f x^3}-\frac{2 e^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 d^{3/2} f}-\frac{g^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{5/2}}-\frac{g^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{f^{5/2}}-\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f^2}-\frac{2 e p}{3 d f x}+\frac{2 g^{3/2} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{f^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*p)/(3*d*f*x) - (2*e^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)*f) - (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/(Sqrt[d]*f^2) + (2*g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)]
)/f^(5/2) - (g^(3/2)*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^(5/2) - (g^(3/2)*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
^(5/2) - Log[c*(d + e*x^2)^p]/(3*f*x^3) + (g*Log[c*(d + e*x^2)^p])/(f^2*x) + (g^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[
f]]*Log[c*(d + e*x^2)^p])/f^(5/2) - (I*g^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2)
+ ((I/2)*g^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sq
rt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) + ((I/2)*g^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqr
t[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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [C]  time = 0.26156, size = 754, normalized size = 1.16 \[ -\frac{2 e g^{3/2} p \left (\frac{i \left (\frac{\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )}{\sqrt{e}}+\frac{\log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )}{\sqrt{e}}\right )}{4 \sqrt{e}}+\frac{i \left (\frac{\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )}{\sqrt{e}}+\frac{\log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )}{\sqrt{e}}\right )}{4 \sqrt{e}}-\frac{i \left (\frac{\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )}{\sqrt{e}}+\frac{\log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )}{\sqrt{e}}\right )}{4 \sqrt{e}}-\frac{i \left (\frac{\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )}{\sqrt{e}}+\frac{\log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )}{\sqrt{e}}\right )}{4 \sqrt{e}}\right )}{f^{5/2}}+\frac{g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{5/2}}+\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{3 f x^3}-\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} f^2}-\frac{2 e p \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e x^2}{d}\right )}{3 d f x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f^2) - (2*e*p*Hypergeometric2F1[-1/2, 1, 1/2, -((e*x^2)/
d)])/(3*d*f*x) - Log[c*(d + e*x^2)^p]/(3*f*x^3) + (g*Log[c*(d + e*x^2)^p])/(f^2*x) + (g^(3/2)*ArcTan[(Sqrt[g]*
x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/f^(5/2) - (2*e*g^(3/2)*p*(((I/4)*((Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*S
qrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[e] + PolyLog[2, (Sqrt[e]*(Sqrt[f] - I
*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])]/Sqrt[e]))/Sqrt[e] + ((I/4)*((Log[-((Sqrt[g]*(Sqrt[-d] + S
qrt[e]*x))/(I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g]))]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[e] + PolyLog[2, (Sqrt
[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])]/Sqrt[e]))/Sqrt[e] - ((I/4)*((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]])/Sqrt[e] + Pol
yLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])]/Sqrt[e]))/Sqrt[e] - ((I/4)*(
(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]
])/Sqrt[e] + PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])]/Sqrt[e]))/Sq
rt[e]))/f^(5/2)

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Maple [C]  time = 0.651, size = 1005, normalized size = 1.5 \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^4/(g*x^2+f),x)

[Out]

(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*g^2/f^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/3*(ln((e*x^2+d)^p)-p*ln(e*x^2+d)
)/f/x^3+(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*g/f^2/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,index=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*_alpha*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))*g/f^2/_alpha,_alpha=RootOf(_Z^2*g+f))-1/3*p/f/x^3*ln(e*x^2+d)-2/3*p/f*e^2/d/(d*e)^
(1/2)*arctan(x*e/(d*e)^(1/2))-2/3*e*p/d/f/x+p*g/f^2/x*ln(e*x^2+d)-2*p*g/f^2*e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/
2))+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*g/f^2/x-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3*g/f^2/x+1/6*I*Pi*csgn(
I*c*(e*x^2+d)^p)^3/f/x^3-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*g/f^2/x+1/2*I*Pi*csgn(I*
c*(e*x^2+d)^p)^2*csgn(I*c)*g^2/f^2/(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*g/f^2/x-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3*g^2/f^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/6*I*Pi*cs
gn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/f/x^3-1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/f/x^3+1/2*I
*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*g^2/f^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/6*I*Pi*csgn(I*(e
*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/f/x^3-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*g^2/f^2/
(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+ln(c)*g^2/f^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/3*ln(c)/f/x^3+ln(c)*g/
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^4/(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^{6} + f x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x^2+d)^p)/x^4/(g*x^2+f),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x^2+d)^p)/x^4/(g*x^2+f),x, algorithm="giac")

[Out]

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