3.3.91 \(\int \frac {\log (c (d+e x^2)^p)}{f+g x^3} \, dx\) [291]

Optimal. Leaf size=1165 \[ -\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.15, antiderivative size = 1165, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2521, 2512, 266, 2463, 2441, 2440, 2438} \begin {gather*} -\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (c \left (e x^2+d\right )^p\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt [3]{-1} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{(-1)^{2/3} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt [3]{-1} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}{(-1)^{2/3} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^2)^p]/(f + g*x^3),x]

[Out]

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

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 2438

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

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[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]))

Rubi steps

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

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Mathematica [A]
time = 0.59, size = 990, normalized size = 0.85 \begin {gather*} \frac {-p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )-p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )-(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )-(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )+\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )+\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )+\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )+(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )-\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right )-(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )-(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )+\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )+\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^2)^p]/(f + g*x^3),x]

[Out]

(-(p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) - g^(1/3)*x]) - p
*Log[(g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(-(Sqrt[e]*f^(1/3)) + Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) - g^(1/3)*x] - (-1
)^(2/3)*p*Log[((-1)^(1/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(-(Sqrt[e]*f^(1/3)) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))]*
Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x] - (-1)^(2/3)*p*Log[((-1)^(1/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f
^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x] + (-1)^(1/3)*p*Log[((-1)^(2/3)*g^(
1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3
)*x] + (-1)^(1/3)*p*Log[((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(-(Sqrt[e]*f^(1/3)) + (-1)^(2/3)*Sqrt[-d]*
g^(1/3))]*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x] + Log[-f^(1/3) - g^(1/3)*x]*Log[c*(d + e*x^2)^p] + (-1)^(2/3)*L
og[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p] - (-1)^(1/3)*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x]*Log
[c*(d + e*x^2)^p] - p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))] - p*Pol
yLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))] - (-1)^(2/3)*p*PolyLog[2, (Sqrt[
e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1/3))] - (-1)^(2/3)*p*PolyLog[2
, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))] + (-1)^(1/3)*p*P
olyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3))] + (-1)^(
1/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))])
/(3*f^(2/3)*g^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.07, size = 1180, normalized size = 1.01

method	result	size
risch	\(\text {Expression too large to display}\)	\(1180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x^2+d)^p)/(g*x^3+f),x,method=_RETURNVERBOSE)

[Out]

1/3*(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/g/(f/g)^(2/3)*ln(x+(f/g)^(1/3))-1/6*(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/g/(f/g
)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/3*(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3
*3^(1/2)*(2/(f/g)^(1/3)*x-1))+1/3*p/g*sum(1/_alpha^2*(ln(x-_alpha)*ln(e*x^2+d)-ln(x-_alpha)*(ln((RootOf(_Z^2*e
+2*_Z*_alpha*e+_alpha^2*e+d,index=1)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=1))+ln((RootOf(_
Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2)))-dilog(
(RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=1)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=1)
)-dilog((RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,
index=2))),_alpha=RootOf(_Z^3*g+f))-1/12*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/
3)*x+(f/g)^(2/3))-1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(f/g)^(1/3)*x-1
))+1/6*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(f/g)^(1/3
)*x-1))+1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g/(f/g)^(2/3)*ln(x+(f/g)^(1/3))+1/6*I*Pi*csgn(I*c*(e*x^2+d)
^p)^2*csgn(I*c)/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(f/g)^(1/3)*x-1))-1/12*I*Pi*csgn(I*(e*x^2+d)^p)*cs
gn(I*c*(e*x^2+d)^p)^2/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))-1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g/(f/g)
^(2/3)*ln(x+(f/g)^(1/3))-1/6*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/g/(f/g)^(2/3)*ln(x+(f/g)
^(1/3))+1/12*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/12*I*Pi*csgn(I*(e*
x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/6*I*Pi*csgn(I*(e*x
^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g/(f/g)^(2/3)*ln(x+(f/g)^(1/3))-1/6*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+
d)^p)*csgn(I*c)/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(f/g)^(1/3)*x-1))+1/3*ln(c)/g/(f/g)^(2/3)*ln(x+(f/
g)^(1/3))-1/6*ln(c)/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/3*ln(c)/g/(f/g)^(2/3)*3^(1/2)*arctan(1/3
*3^(1/2)*(2/(f/g)^(1/3)*x-1))

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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(log(c*(e*x^2+d)^p)/(g*x^3+f),x, algorithm="maxima")

[Out]

integrate(log((x^2*e + d)^p*c)/(g*x^3 + f), 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(log(c*(e*x^2+d)^p)/(g*x^3+f),x, algorithm="fricas")

[Out]

integral(log((x^2*e + d)^p*c)/(g*x^3 + f), 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(ln(c*(e*x**2+d)**p)/(g*x**3+f),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(log(c*(e*x^2+d)^p)/(g*x^3+f),x, algorithm="giac")

[Out]

integrate(log((x^2*e + d)^p*c)/(g*x^3 + f), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^3+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^2)^p)/(f + g*x^3),x)

[Out]

int(log(c*(d + e*x^2)^p)/(f + g*x^3), x)

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