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

Optimal. Leaf size=1861 \[ \text{result too large to display} \]

[Out]

(2*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(e*f^(2/3) + d*g^(2/3))) + (2*(-1)^(2/3)*Sqrt[d]*
Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((1 + (-1)^(1/3))^4*f^(4/3)*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))) + (4*Sq
rt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))) - (2*e*p*Lo
g[f^(1/3) + g^(1/3)*x])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - (2*p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqr
t[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) - (2*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])/(9*f^(5/3)*g^(1/3)) + (2*(-1)
^(1/3)*e*p*Log[f^(1/3) - (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/
3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + (4*(-1)^(1/3)*e*p*Log[f^(1/3) + (-1)^(2/3)*g^(1/3
)*x])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - (2*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)^(1/3))
^4*f^(5/3)*g^(1/3)) - (2*p*Log[-(((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqr
t[-d]*g^(1/3)))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) + (e*p*Log[d + e*x^
2])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - ((-1)^(1/3)*e*p*Log[d + e*x^2])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) +
(-1)^(2/3)*d*g^(2/3))*g^(1/3)) - (2*(-1)^(1/3)*e*p*Log[d + e*x^2])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2
/3))*g^(1/3)) - Log[c*(d + e*x^2)^p]/(9*f^(4/3)*g^(1/3)*(f^(1/3) + g^(1/3)*x)) - Log[c*(d + e*x^2)^p]/((1 + (-
1)^(1/3))^4*f^(4/3)*g^(1/3)*((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + ((-1)^(1/3)*Log[c*(d + e*x^2)^p])/(9*f^(4/3)*g
^(1/3)*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x)) + (2*Log[f^(1/3) + g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*f^(5/3)*g^(1/3
)) - ((2*I)*Sqrt[3]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(
1/3)) + (2*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2
*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) - (2*
p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) + ((2*
I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) - (2*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))])/((1 + (-1)^(1/3
))^4*f^(5/3)*g^(1/3)) - (2*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))])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 2.88701, antiderivative size = 1863, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2471, 2463, 801, 635, 205, 260, 2462, 2416, 2394, 2393, 2391} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(e*f^(2/3) + d*g^(2/3))) + (2*(-1)^(2/3)*Sqrt[d]*
Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((1 + (-1)^(1/3))^4*f^(4/3)*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))) + (4*Sq
rt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) + I*(I - Sqrt[3])*d*g^(2/3))) - (2*e*p*Lo
g[f^(1/3) + g^(1/3)*x])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - (2*p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqr
t[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) - (2*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])/(9*f^(5/3)*g^(1/3)) + (2*(-1)
^(1/3)*e*p*Log[f^(1/3) - (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/
3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + (4*(-1)^(1/3)*e*p*Log[f^(1/3) + (-1)^(2/3)*g^(1/3
)*x])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - (2*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)^(1/3))
^4*f^(5/3)*g^(1/3)) - (2*p*Log[-(((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqr
t[-d]*g^(1/3)))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) + (e*p*Log[d + e*x^
2])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - ((-1)^(1/3)*e*p*Log[d + e*x^2])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) +
(-1)^(2/3)*d*g^(2/3))*g^(1/3)) - (2*(-1)^(1/3)*e*p*Log[d + e*x^2])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2
/3))*g^(1/3)) - Log[c*(d + e*x^2)^p]/(9*f^(4/3)*g^(1/3)*(f^(1/3) + g^(1/3)*x)) - Log[c*(d + e*x^2)^p]/((1 + (-
1)^(1/3))^4*f^(4/3)*g^(1/3)*((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + ((-1)^(1/3)*Log[c*(d + e*x^2)^p])/(9*f^(4/3)*g
^(1/3)*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x)) + (2*Log[f^(1/3) + g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*f^(5/3)*g^(1/3
)) - ((2*I)*Sqrt[3]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(
1/3)) + (2*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2
*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) - (2*
p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) + ((2*
I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[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)^(1/3))^5*f^(5/3)*g^(1/3)) - (2*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))])/((1 + (-1)^(1/3
))^4*f^(5/3)*g^(1/3)) - (2*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))])/((1 + (-1)^(1/3))^4*f^(5/3)*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 2463

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 801

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 635

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 205

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

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 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 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 )}{\left (f+g x^3\right )^2} \, dx &=\int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )^2}+\frac{2 \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{(-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )^2}-\frac{2 (-1)^{5/6} \sqrt{3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}+\frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )^2}+\frac{2 (-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}\right ) \, dx\\ &=\frac{2 \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt [3]{f}+\sqrt [3]{g} x} \, dx}{9 f^{5/3}}+\frac{2 \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x} \, dx}{9 f^{5/3}}-\frac{\left (2 (-1)^{5/6} \sqrt{3}\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3}}+\frac{\int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt [3]{f}+\sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}}+\frac{\int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}}+\frac{\int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}}\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{(4 e p) \int \frac{x \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{d+e x^2} \, dx}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (4 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (4 i \sqrt{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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{(2 e p) \int \frac{x}{\left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac{\left (2 \sqrt [3]{-1} e p\right ) \int \frac{x}{\left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac{\left (2 (-1)^{2/3} e p\right ) \int \frac{x}{\left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}}\\ &=-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{(4 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}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (4 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (4 i \sqrt{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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{(2 e p) \int \left (-\frac{\sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}+d g^{2/3}\right ) \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{d \sqrt [3]{g}+e \sqrt [3]{f} x}{\left (e f^{2/3}+d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac{\left (2 \sqrt [3]{-1} e p\right ) \int \left (-\frac{(-1)^{2/3} \sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{(-1)^{2/3} d \sqrt [3]{g}+e \sqrt [3]{f} x}{\left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac{\left (2 (-1)^{2/3} e p\right ) \int \left (-\frac{\sqrt [3]{-1} \sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} d \sqrt [3]{g}-e \sqrt [3]{f} x}{\left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}}\\ &=-\frac{2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{2 (-1)^{2/3} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (2 \sqrt{e} p\right ) \int \frac{\log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 \sqrt{e} p\right ) \int \frac{\log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac{\left (2 \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}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 i \sqrt{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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{\left (2 i \sqrt{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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{(2 e p) \int \frac{d \sqrt [3]{g}+e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\left (2 \sqrt [3]{-1} e p\right ) \int \frac{(-1)^{2/3} d \sqrt [3]{g}+e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\left (2 (-1)^{2/3} e p\right ) \int \frac{\sqrt [3]{-1} d \sqrt [3]{g}-e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}\\ &=-\frac{2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 (-1)^{2/3} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{(2 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}{9 f^{5/3}}+\frac{(2 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}{9 f^{5/3}}+\frac{(2 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}{9 f^{5/3}}+\frac{(2 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}{9 f^{5/3}}-\frac{\left (2 (-1)^{5/6} \sqrt{3} p\right ) \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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3}}-\frac{\left (2 (-1)^{5/6} \sqrt{3} p\right ) \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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3}}+\frac{(2 d e p) \int \frac{1}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right )}+\frac{(2 d e p) \int \frac{1}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right )}+\frac{(2 d e p) \int \frac{1}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right )}+\frac{\left (2 e^2 p\right ) \int \frac{x}{d+e x^2} \, dx}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\left (2 \sqrt [3]{-1} e^2 p\right ) \int \frac{x}{d+e x^2} \, dx}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{\left (2 (-1)^{2/3} e^2 p\right ) \int \frac{x}{d+e x^2} \, dx}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}\\ &=\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right )}+\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right )}+\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right )}-\frac{2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 (-1)^{2/3} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\sqrt [3]{-1} e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{(-1)^{2/3} e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{(2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{(2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{\left (2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}\\ &=\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right )}+\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right )}+\frac{2 \sqrt{d} \sqrt{e} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right )}-\frac{2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 (-1)^{2/3} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\sqrt [3]{-1} e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}+\frac{(-1)^{2/3} e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}-\frac{\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac{2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac{2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 i \sqrt{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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac{2 \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 )}{9 f^{5/3} \sqrt [3]{g}}\\ \end{align*}

Mathematica [A]  time = 7.04227, size = 2168, normalized size = 1.16 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(3*f*(f + g*x^3)) + (2*ArcTan[(-f^(1/3) + 2*g^(1/3)*x)/(Sqrt[
3]*f^(1/3))]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(3*Sqrt[3]*f^(5/3)*g^(1/3)) + (2*Log[f^(1/3) + g^(1
/3)*x]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(9*f^(5/3)*g^(1/3)) - ((-(p*Log[d + e*x^2]) + Log[c*(d +
e*x^2)^p])*Log[f^(2/3) - f^(1/3)*g^(1/3)*x + g^(2/3)*x^2])/(9*f^(5/3)*g^(1/3)) + p*(-((-1 + (-1)^(1/3))*(-(Log
[((-I)*Sqrt[d])/Sqrt[e] + x]/((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] - Sqrt[e]*x] - Log[-(
(-1)^(2/3)*f^(1/3)) - g^(1/3)*x]))/((-1)^(2/3)*Sqrt[e]*f^(1/3) + I*Sqrt[d]*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^
(4/3)*g^(1/3)) - ((-1 + (-1)^(1/3))*(-(Log[(I*Sqrt[d])/Sqrt[e] + x]/((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + (Sqrt[
e]*(Log[I*Sqrt[d] + Sqrt[e]*x] - Log[-((-1)^(2/3)*f^(1/3)) - g^(1/3)*x]))/((-1)^(2/3)*Sqrt[e]*f^(1/3) - I*Sqrt
[d]*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-1)^(1/3)*(-(Log[((-I)*Sqrt[d])/Sqrt[e] + x]/(f^(1/3
) + g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] - Sqrt[e]*x] - Log[f^(1/3) + g^(1/3)*x]))/(Sqrt[e]*f^(1/3) + I*Sqrt[
d]*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-1)^(1/3)*(-(Log[(I*Sqrt[d])/Sqrt[e] + x]/(f^(1/3) +
g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] + Sqrt[e]*x] - Log[f^(1/3) + g^(1/3)*x]))/(Sqrt[e]*f^(1/3) - I*Sqrt[d]*g
^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) - (Log[((-I)*Sqrt[d])/Sqrt[e] + x]/((-1)^(1/3)*f^(1/3) - g^(1
/3)*x) + (Sqrt[e]*(-Log[I*Sqrt[d] - Sqrt[e]*x] + Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]))/((-1)^(1/3)*Sqrt[e]*f^(
1/3) - I*Sqrt[d]*g^(1/3)))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) - (Log[(I*Sqrt[d])/Sqrt[e] + x]/((-1)^(1/3)*
f^(1/3) - g^(1/3)*x) + (Sqrt[e]*(-Log[I*Sqrt[d] + Sqrt[e]*x] + Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]))/((-1)^(1/
3)*Sqrt[e]*f^(1/3) + I*Sqrt[d]*g^(1/3)))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-Log[((-I)*Sqrt[d])/Sqrt[e
] + x] - Log[(I*Sqrt[d])/Sqrt[e] + x] + Log[d + e*x^2])*((3*f^(2/3)*x)/(f + g*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2
*g^(1/3)*x)/f^(1/3))/Sqrt[3]])/g^(1/3) + (2*Log[f^(1/3) + g^(1/3)*x])/g^(1/3) - Log[f^(2/3) - f^(1/3)*g^(1/3)*
x + g^(2/3)*x^2]/g^(1/3)))/(9*f^(5/3)) - (2*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[((-1)^(2/3)*f^(1/3) + g^(1/3)*x)
/((-1)^(2/3)*f^(1/3) - (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, -((g^(1/3)*(Sqrt[d] - I*Sqrt[e]*x))/((-1)^(1
/6)*Sqrt[e]*f^(1/3) - Sqrt[d]*g^(1/3)))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) - (2*(-1 + (-1)^(1/3))*(Log[
(I*Sqrt[d])/Sqrt[e] + x]*Log[-((-((-1)^(1/3)*f^(1/3)) + g^(1/3)*x)/((-1)^(1/3)*f^(1/3) + (I*Sqrt[d]*g^(1/3))/S
qrt[e]))] + PolyLog[2, -((g^(1/3)*(Sqrt[d] - I*Sqrt[e]*x))/((-1)^(5/6)*Sqrt[e]*f^(1/3) - Sqrt[d]*g^(1/3)))]))/
(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(f^(1/3) + g^(1/3)
*x)/(f^(1/3) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (I*g^(1/3)*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[e]*f^(1/3)
+ I*Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) - (2*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[((-1
)^(2/3)*f^(1/3) + g^(1/3)*x)/((-1)^(2/3)*f^(1/3) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (g^(1/3)*(Sqrt[d
] + I*Sqrt[e]*x))/((-1)^(1/6)*Sqrt[e]*f^(1/3) + Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) - (
2*(-1 + (-1)^(1/3))*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(-((-1)^(1/3)*f^(1/3)) + g^(1/3)*x)/(-((-1)^(1/3)*f^(
1/3)) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (g^(1/3)*(Sqrt[d] + I*Sqrt[e]*x))/((-1)^(5/6)*Sqrt[e]*f^(1/
3) + Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*(Log[(I*Sqrt[d])/Sqrt[e] + x]*
Log[(f^(1/3) + g^(1/3)*x)/(f^(1/3) - (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, -((g^(1/3)*(I*Sqrt[d] + Sqrt[e
]*x))/(Sqrt[e]*f^(1/3) - I*Sqrt[d]*g^(1/3)))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)))

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Maple [F]  time = 1.281, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) }{ \left ( g{x}^{3}+f \right ) ^{2}}}\, dx \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)^2,x)

[Out]

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

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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)^2,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^{2} x^{6} + 2 \, f g x^{3} + f^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(log((e*x^2 + d)^p*c)/(g^2*x^6 + 2*f*g*x^3 + f^2), 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)**2,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 )}{{\left (g x^{3} + f\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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