\(\int \frac {x^4 \log (c (d+e x^2)^p)}{(f+g x^2)^2} \, dx\) [353]

Optimal result

Integrand size = 25, antiderivative size = 802 \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {2 p x}{g^2}+\frac {2 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}+\frac {\sqrt {d} \sqrt {e} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g^2 (e f-d g)}-\frac {e (-f)^{3/2} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{5/2} (e f-d g)}-\frac {3 \sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}+\frac {3 \sqrt {f} 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 g^{5/2}}+\frac {3 \sqrt {f} 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 g^{5/2}}+\frac {e (-f)^{3/2} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{5/2} (e f-d g)}+\frac {x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {3 \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 g^{5/2}}+\frac {3 i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 g^{5/2}}-\frac {3 i \sqrt {f} 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 g^{5/2}}-\frac {3 i \sqrt {f} 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 g^{5/2}} \]

[Out]

-2*p*x/g^2+x*ln(c*(e*x^2+d)^p)/g^2-1/2*e*(-f)^(3/2)*p*ln((-f)^(1/2)-x*g^(1/2))/g^(5/2)/(-d*g+e*f)+1/2*e*(-f)^(
3/2)*p*ln((-f)^(1/2)+x*g^(1/2))/g^(5/2)/(-d*g+e*f)+2*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/g^2/e^(1/2)+f*p*arcta
n(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/g^2/(-d*g+e*f)-3/2*arctan(x*g^(1/2)/f^(1/2))*ln(c*(e*x^2+d)^p)*f^(1/2)/g^
(5/2)-3*p*arctan(x*g^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))*f^(1/2)/g^(5/2)+3/2*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^(1/2)/g^(5/2)+3/2*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^(1/2)/g^(5/2)+3/2*I*p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I*x*
g^(1/2)))*f^(1/2)/g^(5/2)-3/4*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^(1/2)/g^(5/2)-3/4*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^(1/2)/g^(5/2)-1/4*f*ln(c*(e*x^2+d)^p)/g^
(5/2)/((-f)^(1/2)-x*g^(1/2))+1/4*f*ln(c*(e*x^2+d)^p)/g^(5/2)/((-f)^(1/2)+x*g^(1/2))

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {2526, 2498, 327, 211, 2521, 2513, 815, 649, 266, 2520, 12, 5048, 4966, 2449, 2352, 2497} \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right ) (-f)^{3/2}}{2 g^{5/2} (e f-d g)}+\frac {e p \log \left (\sqrt {g} x+\sqrt {-f}\right ) (-f)^{3/2}}{2 g^{5/2} (e f-d g)}-\frac {2 p x}{g^2}+\frac {\sqrt {d} \sqrt {e} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g^2 (e f-d g)}+\frac {2 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}-\frac {3 \sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}+\frac {3 \sqrt {f} 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 g^{5/2}}+\frac {3 \sqrt {f} p \arctan \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 g^{5/2}}+\frac {x \log \left (c \left (e x^2+d\right )^p\right )}{g^2}-\frac {3 \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{2 g^{5/2}}-\frac {f \log \left (c \left (e x^2+d\right )^p\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \log \left (c \left (e x^2+d\right )^p\right )}{4 g^{5/2} \left (\sqrt {g} x+\sqrt {-f}\right )}+\frac {3 i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 g^{5/2}}-\frac {3 i \sqrt {f} 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 g^{5/2}}-\frac {3 i \sqrt {f} 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 g^{5/2}} \]

[In]

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

[Out]

(-2*p*x)/g^2 + (2*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2) + (Sqrt[d]*Sqrt[e]*f*p*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/(g^2*(e*f - d*g)) - (e*(-f)^(3/2)*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*g^(5/2)*(e*f - d*g)) - (3*Sqrt[
f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + (3*Sqrt[f]*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*g^(5/2)) + (3*Sqrt[f]*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*g^(5/2)) + (e*(-f)^(3/2)*p*
Log[Sqrt[-f] + Sqrt[g]*x])/(2*g^(5/2)*(e*f - d*g)) + (x*Log[c*(d + e*x^2)^p])/g^2 - (f*Log[c*(d + e*x^2)^p])/(
4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*Log[c*(d + e*x^2)^p])/(4*g^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (3*Sqrt[f]*A
rcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*g^(5/2)) + (((3*I)/2)*Sqrt[f]*p*PolyLog[2, 1 - (2*Sqrt[f])
/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) - (((3*I)/4)*Sqrt[f]*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))])/g^(5/2) - (((3*I)/4)*Sqrt[f]*p*PolyLo
g[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))])/g^(5/2)

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 327

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

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 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, 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 2526

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

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.14 \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {-8 \sqrt {g} p x+\frac {8 \sqrt {d} \sqrt {g} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {2 \sqrt {-d} \sqrt {e} f \sqrt {g} p \log \left (\sqrt {-d}-\sqrt {e} x\right )}{-e f+d g}+\frac {2 \sqrt {-d} \sqrt {e} f \sqrt {g} p \log \left (\sqrt {-d}+\sqrt {e} x\right )}{e f-d g}+\frac {2 e \sqrt {-f} f p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{e f-d g}+\frac {2 e (-f)^{3/2} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{e f-d g}+3 i \sqrt {f} 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 )+3 i \sqrt {f} 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 )-3 i \sqrt {f} 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 )-3 i \sqrt {f} 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 )+4 \sqrt {g} x \log \left (c \left (d+e x^2\right )^p\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}-\sqrt {g} x}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}+\sqrt {g} x}-6 \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+3 i \sqrt {f} 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 )+3 i \sqrt {f} 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 )-3 i \sqrt {f} 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 )-3 i \sqrt {f} 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 )}{4 g^{5/2}} \]

[In]

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

[Out]

(-8*Sqrt[g]*p*x + (8*Sqrt[d]*Sqrt[g]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (2*Sqrt[-d]*Sqrt[e]*f*Sqrt[g]*p*
Log[Sqrt[-d] - Sqrt[e]*x])/(-(e*f) + d*g) + (2*Sqrt[-d]*Sqrt[e]*f*Sqrt[g]*p*Log[Sqrt[-d] + Sqrt[e]*x])/(e*f -
d*g) + (2*e*Sqrt[-f]*f*p*Log[Sqrt[-f] - Sqrt[g]*x])/(e*f - d*g) + (2*e*(-f)^(3/2)*p*Log[Sqrt[-f] + Sqrt[g]*x])
/(e*f - d*g) + (3*I)*Sqrt[f]*p*Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Lo
g[1 - (I*Sqrt[g]*x)/Sqrt[f]] + (3*I)*Sqrt[f]*p*Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sq
rt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] - (3*I)*Sqrt[f]*p*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]] - (3*I)*Sqrt[f]*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]] + 4*Sqrt[g]*x*Log[c*(d + e*
x^2)^p] - (f*Log[c*(d + e*x^2)^p])/(Sqrt[-f] - Sqrt[g]*x) + (f*Log[c*(d + e*x^2)^p])/(Sqrt[-f] + Sqrt[g]*x) -
6*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p] + (3*I)*Sqrt[f]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*
Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] + (3*I)*Sqrt[f]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]
*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])] - (3*I)*Sqrt[f]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(S
qrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - (3*I)*Sqrt[f]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*
Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/(4*g^(5/2))

Maple [F]

\[\int \frac {x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4*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]

\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]

[In]

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

[Out]

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