\(\int \frac {\log (c (d+e x^2)^p)}{(f+g x^2)^2} \, dx\) [272]

Optimal result

Integrand size = 22, antiderivative size = 751 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2} \sqrt {g}}-\frac {p \arctan \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^{3/2} \sqrt {g}}-\frac {p \arctan \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^{3/2} \sqrt {g}}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\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^{3/2} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\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^{3/2} \sqrt {g}} \]

[Out]

p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/f/(-d*g+e*f)+1/2*arctan(x*g^(1/2)/f^(1/2))*ln(c*(e*x^2+d)^p)/f^(3/
2)/g^(1/2)+p*arctan(x*g^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))/f^(3/2)/g^(1/2)-1/2*p*arctan(x*g^(1
/2)/f^(1/2))*ln(-2*((-d)^(1/2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)-(-d)^(1/2)*
g^(1/2)))/f^(3/2)/g^(1/2)-1/2*p*arctan(x*g^(1/2)/f^(1/2))*ln(2*((-d)^(1/2)+x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)
-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2)))/f^(3/2)/g^(1/2)-1/2*I*p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I
*x*g^(1/2)))/f^(3/2)/g^(1/2)+1/4*I*p*polylog(2,1+2*((-d)^(1/2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2)
)/(I*e^(1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2)))/f^(3/2)/g^(1/2)+1/4*I*p*polylog(2,1-2*((-d)^(1/2)+x*e^(1/2))*f^(1/2)
*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2)))/f^(3/2)/g^(1/2)-1/2*e*p*ln((-f)^(1/2)-x
*g^(1/2))/(-d*g+e*f)/(-f)^(1/2)/g^(1/2)+1/2*e*p*ln((-f)^(1/2)+x*g^(1/2))/(-d*g+e*f)/(-f)^(1/2)/g^(1/2)-1/4*ln(
c*(e*x^2+d)^p)/f/g^(1/2)/((-f)^(1/2)-x*g^(1/2))+1/4*ln(c*(e*x^2+d)^p)/f/g^(1/2)/((-f)^(1/2)+x*g^(1/2))

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {2521, 2513, 815, 649, 211, 266, 2520, 12, 5048, 4966, 2449, 2352, 2497} \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \arctan \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 )}{2 f^{3/2} \sqrt {g}}-\frac {p \arctan \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 )}{2 f^{3/2} \sqrt {g}}+\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2} \sqrt {g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {i p \operatorname {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^{3/2} \sqrt {g}}+\frac {i p \operatorname {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^{3/2} \sqrt {g}}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}} \]

[In]

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

[Out]

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

Rule 12

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

Rule 211

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

Rule 266

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 649

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 815

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 2352

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

Rule 2449

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 2497

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]]

Rule 2513

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 2520

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 2521

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 4966

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 5048

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])

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

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.17 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} p \log \left (\sqrt {-d}-\sqrt {e} x\right )}{-e f+d g}+\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} p \log \left (\sqrt {-d}+\sqrt {e} x\right )}{e f-d g}+\frac {2 e \sqrt {-f^2} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} (e f-d g)}+\frac {2 e \sqrt {-f^2} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {g} (-e f+d g)}-\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {\sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt {-f} \sqrt {g}+g x}+\frac {\sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f} \sqrt {g}+g x}+\frac {2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {g}}-\frac {i p \operatorname {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 {g}}-\frac {i p \operatorname {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 {g}}+\frac {i p \operatorname {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 {g}}+\frac {i p \operatorname {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 {g}}}{4 f^{3/2}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]

[In]

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

[Out]

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