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

Optimal. Leaf size=1165 \[ \text{result too large to display} \]

[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])/(3*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*Po
lyLog[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*PolyL
og[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, (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))

________________________________________________________________________________________

Rubi [A]  time = 1.60361, antiderivative size = 1165, normalized size of antiderivative = 1., 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 = {2471, 2462, 260, 2416, 2394, 2393, 2391} \[ -\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}} \]

Antiderivative was successfully verified.

[In]

Int[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])/(3*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*Po
lyLog[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*PolyL
og[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, (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))

Rule 2471

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 2462

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 260

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 2416

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 2394

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 2393

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 2391

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]

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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 0.812465, size = 990, normalized size = 0.85 \[ \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 )-p \log \left (\frac{\sqrt [3]{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{\sqrt{-d} \sqrt [3]{g}-\sqrt{e} \sqrt [3]{f}}\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )+\log \left (c \left (e x^2+d\right )^p\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )-(-1)^{2/3} p \log \left (\frac{\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt [3]{-1} \sqrt{-d} \sqrt [3]{g}-\sqrt{e} \sqrt [3]{f}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )-(-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 )+\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 )+\sqrt [3]{-1} p \log \left (\frac{(-1)^{2/3} \sqrt [3]{g} \left (\sqrt{e} x+\sqrt{-d}\right )}{(-1)^{2/3} \sqrt{-d} \sqrt [3]{g}-\sqrt{e} \sqrt [3]{f}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right )+(-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 )-\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 )-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 )-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 )-(-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 )-(-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 )+\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 )+\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}} \]

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]  time = 0.609, size = 1180, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))-1/6*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))-1/6*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))-1/12*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^2-(f/
g)^(1/3)*x+(f/g)^(2/3))+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))-1/6*
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)*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)*ln(x+(f/g)^(1/3))-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))+1/12*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*c*(e*x^2+d)^p)^3/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))+1/6*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))+1/3*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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f}, x\right ) \end{align*}

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((e*x^2 + d)^p*c)/(g*x^3 + f), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f}\,{d x} \end{align*}

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((e*x^2 + d)^p*c)/(g*x^3 + f), x)