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

Optimal. Leaf size=802 \[ -\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}} \]

[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]
time = 1.19, antiderivative size = 802, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {2526, 2498, 327, 211, 2521, 2513, 815, 649, 266, 2520, 12, 5048, 4966, 2449, 2352, 2497} \begin {gather*} -\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 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g^2 (e f-d g)}+\frac {2 \sqrt {d} p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}-\frac {3 \sqrt {f} p \text {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 \text {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 \text {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} \text {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 \text {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 \text {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 g^{5/2}}-\frac {3 i \sqrt {f} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 g^{5/2}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 2.93, size = 1349, normalized size = 1.68 \begin {gather*} \frac {1}{4} \left (\frac {6 \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (p \log \left (d+e x^2\right )-\log \left (c \left (d+e x^2\right )^p\right )\right )}{g^{5/2}}+\frac {4 x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )}{g^2}+\frac {2 f x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )}{g^2 \left (f+g x^2\right )}+p \left (\frac {4 \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \left (-1+\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )\right )}{g^2}+\frac {4 \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \left (-1+\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )\right )}{g^2}+\frac {i f \left (\frac {\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{\sqrt {f}+i \sqrt {g} x}+\frac {\sqrt {e} \left (-\log \left (i \sqrt {d}-\sqrt {e} x\right )+\log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )}{g^{5/2}}+\frac {i f \left (\frac {\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{\sqrt {f}+i \sqrt {g} x}+\frac {\sqrt {e} \left (-\log \left (i \sqrt {d}+\sqrt {e} x\right )+\log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )}{g^{5/2}}+\frac {f \left (-i \left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+\sqrt {e} \left (i \sqrt {f}+\sqrt {g} x\right ) \left (\log \left (i \sqrt {d}-\sqrt {e} x\right )-\log \left (i \sqrt {f}+\sqrt {g} x\right )\right )\right )}{\left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) g^{5/2} \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {f \left (-\frac {\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{i \sqrt {f}+\sqrt {g} x}-\frac {i \sqrt {e} \left (\log \left (i \sqrt {d}+\sqrt {e} x\right )-\log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )}{g^{5/2}}+4 \left (\frac {x \left (2+\frac {f}{f+g x^2}\right )}{2 g^2}-\frac {3 \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{5/2}}\right ) \left (-\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )-\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+\log \left (d+e x^2\right )\right )-\frac {3 i \sqrt {f} \left (\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{g^{5/2}}+\frac {3 i \sqrt {f} \left (\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{g^{5/2}}+\frac {3 i \sqrt {f} \left (\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{g^{5/2}}-\frac {3 i \sqrt {f} \left (\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{g^{5/2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(p*Log[d + e*x^2] - Log[c*(d + e*x^2)^p]))/g^(5/2) + (4*x*(-(p*Log[d +
e*x^2]) + Log[c*(d + e*x^2)^p]))/g^2 + (2*f*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(g^2*(f + g*x^2))
+ p*((4*(((-I)*Sqrt[d])/Sqrt[e] + x)*(-1 + Log[((-I)*Sqrt[d])/Sqrt[e] + x]))/g^2 + (4*((I*Sqrt[d])/Sqrt[e] +
x)*(-1 + Log[(I*Sqrt[d])/Sqrt[e] + x]))/g^2 + (I*f*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]/(Sqrt[f] + I*Sqrt[g]*x) +
(Sqrt[e]*(-Log[I*Sqrt[d] - Sqrt[e]*x] + Log[I*Sqrt[f] - Sqrt[g]*x]))/(Sqrt[e]*Sqrt[f] - Sqrt[d]*Sqrt[g])))/g^(
5/2) + (I*f*(Log[(I*Sqrt[d])/Sqrt[e] + x]/(Sqrt[f] + I*Sqrt[g]*x) + (Sqrt[e]*(-Log[I*Sqrt[d] + Sqrt[e]*x] + Lo
g[I*Sqrt[f] - Sqrt[g]*x]))/(Sqrt[e]*Sqrt[f] + Sqrt[d]*Sqrt[g])))/g^(5/2) + (f*((-I)*(Sqrt[e]*Sqrt[f] + Sqrt[d]
*Sqrt[g])*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + Sqrt[e]*(I*Sqrt[f] + Sqrt[g]*x)*(Log[I*Sqrt[d] - Sqrt[e]*x] - Log[
I*Sqrt[f] + Sqrt[g]*x])))/((Sqrt[e]*Sqrt[f] + Sqrt[d]*Sqrt[g])*g^(5/2)*(Sqrt[f] - I*Sqrt[g]*x)) - (f*(-(Log[(I
*Sqrt[d])/Sqrt[e] + x]/(I*Sqrt[f] + Sqrt[g]*x)) - (I*Sqrt[e]*(Log[I*Sqrt[d] + Sqrt[e]*x] - Log[I*Sqrt[f] + Sqr
t[g]*x]))/(Sqrt[e]*Sqrt[f] - Sqrt[d]*Sqrt[g])))/g^(5/2) + 4*((x*(2 + f/(f + g*x^2)))/(2*g^2) - (3*Sqrt[f]*ArcT
an[(Sqrt[g]*x)/Sqrt[f]])/(2*g^(5/2)))*(-Log[((-I)*Sqrt[d])/Sqrt[e] + x] - Log[(I*Sqrt[d])/Sqrt[e] + x] + Log[d
+ e*x^2]) - ((3*I)*Sqrt[f]*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[
f] - Sqrt[d]*Sqrt[g])] + PolyLog[2, -((Sqrt[g]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[e]*Sqrt[f] - Sqrt[d]*Sqrt[g]))])
)/g^(5/2) + ((3*I)*Sqrt[f]*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f
] + Sqrt[d]*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[e]*Sqrt[f] + Sqrt[d]*Sqrt[g])]))/g^
(5/2) + ((3*I)*Sqrt[f]*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f]
- Sqrt[d]*Sqrt[g])] + PolyLog[2, -((Sqrt[g]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[e]*Sqrt[f] - Sqrt[d]*Sqrt[g]))]))/
g^(5/2) - ((3*I)*Sqrt[f]*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[
f] + Sqrt[d]*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[e]*Sqrt[f] + Sqrt[d]*Sqrt[g])]))/g
^(5/2)))/4

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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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