3.133 \(\int x \log ^2(c (d+e x^3)^p) \, dx\)

Optimal. Leaf size=1294 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.91987, antiderivative size = 1300, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 19, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.187, Rules used = {2457, 2476, 2455, 321, 292, 31, 634, 617, 204, 628, 2462, 260, 2416, 2390, 2301, 2394, 2393, 2391, 12} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[x*Log[c*(d + e*x^3)^p]^2,x]

[Out]

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

Rule 2457

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

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

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

Rule 321

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

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx &=\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \frac{x^4 \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \left (\frac{x \log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac{d x \log \left (c \left (d+e x^3\right )^p\right )}{e \left (d+e x^3\right )}\right ) \, dx\\ &=\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 p) \int x \log \left (c \left (d+e x^3\right )^p\right ) \, dx+(3 d p) \int \frac{x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+(3 d p) \int \left (-\frac{\log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac{(-1)^{2/3} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx+\frac{1}{2} \left (9 e p^2\right ) \int \frac{x^4}{d+e x^3} \, dx\\ &=\frac{9 p^2 x^2}{4}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac{\left (d^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left (\sqrt [3]{-1} d^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left ((-1)^{2/3} d^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{1}{2} \left (9 d p^2\right ) \int \frac{x}{d+e x^3} \, dx\\ &=\frac{9 p^2 x^2}{4}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac{\left (3 d^{2/3} p^2\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 \sqrt [3]{e}}-\frac{\left (3 d^{2/3} p^2\right ) \int \frac{\sqrt [3]{d}+\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx\\ &=\frac{9 p^2 x^2}{4}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac{\left (3 d^{2/3} p^2\right ) \int \frac{-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 e^{2/3}}-\frac{\left (9 d p^2\right ) \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx\\ &=\frac{9 p^2 x^2}{4}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac{3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac{\left (9 d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac{\left (d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left (d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left (d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}\\ &=\frac{9 p^2 x^2}{4}+\frac{3 \sqrt{3} d^{2/3} p^2 \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac{\left (d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\left (d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{-1} \log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{\left (d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{(-1)^{2/3} \log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\left (d^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left (d^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac{\log \left (\frac{(-1)^{2/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \int \frac{\log \left (\frac{(-1)^{2/3} \sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (-\frac{\sqrt [3]{-1} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}-\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (-\frac{\sqrt [3]{-1} \sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}\\ &=\frac{9 p^2 x^2}{4}+\frac{3 \sqrt{3} d^{2/3} p^2 \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac{\left (d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\left (d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}\\ &=\frac{9 p^2 x^2}{4}+\frac{3 \sqrt{3} d^{2/3} p^2 \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac{d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \text{Li}_2\left (\frac{2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt{3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \text{Li}_2\left (-\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{\left ((-1)^{2/3} d^{2/3} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{e^{2/3}}\\ &=\frac{9 p^2 x^2}{4}+\frac{3 \sqrt{3} d^{2/3} p^2 \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac{3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac{3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac{d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{\sqrt [3]{-1} d^{2/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac{1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac{d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{d^{2/3} p^2 \text{Li}_2\left (\frac{2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt{3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \text{Li}_2\left (-\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{\sqrt [3]{-1} d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}-\frac{(-1)^{2/3} d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac{(-1)^{2/3} d^{2/3} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}\\ \end{align*}

Mathematica [C]  time = 1.30223, size = 823, normalized size = 0.64 \[ \frac{1}{4} \left (2 x^2 \log ^2\left (c \left (e x^3+d\right )^p\right )+\frac{p \left (-9 e^{2/3} p \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{e x^3}{d}\right )-1\right ) x^2-6 e^{2/3} \log \left (c \left (e x^3+d\right )^p\right ) x^2-4 d^{2/3} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )+4 \sqrt [3]{-1} d^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )-4 (-1)^{2/3} d^{2/3} \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )-2 \sqrt [3]{-1} d^{2/3} p \left (\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \log \left (\frac{(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 (-1)^{2/3} d^{2/3} p \left (\log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (2 \log \left (\frac{(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )\right )+2 \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 d^{2/3} p \left (\log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (\log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \left (\log \left (\frac{\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\frac{-\frac{2 i \sqrt [3]{e} x}{\sqrt [3]{d}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{2 i \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{3 i+\sqrt{3}}\right )\right )\right )}{e^{2/3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[c*(d + e*x^3)^p]^2,x]

[Out]

(2*x^2*Log[c*(d + e*x^3)^p]^2 + (p*(-9*e^(2/3)*p*x^2*(-1 + Hypergeometric2F1[2/3, 1, 5/3, -((e*x^3)/d)]) - 6*e
^(2/3)*x^2*Log[c*(d + e*x^3)^p] - 4*d^(2/3)*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p] + 4*(-1)^(1/3)*d^(2
/3)*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p] - 4*(-1)^(2/3)*d^(2/3)*Log[-d^(1/3) - (-1)^(2/3)
*e^(1/3)*x]*Log[c*(d + e*x^3)^p] - 2*(-1)^(1/3)*d^(2/3)*p*(Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*(2*Log[((-1)^(
1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))] + Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x] + 2*Log[((-1)^(
2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((-1 + (-1)^(2/3))*d^(1/3))]) + 2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1
/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*PolyLog[2, (-d^(1/3) + (-1)^(1/3)*e^(1/3)*x)/((-1 + (-1)^(2/3))*d^(1/3)
)]) + 2*(-1)^(2/3)*d^(2/3)*p*(Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*(2*Log[((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/(
(-1 + (-1)^(2/3))*d^(1/3))] + 2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]
+ Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]) + 2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1
/3))] + 2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))]) + 2*d^(2/3)*p*(Log[-d^(1/3)
- e^(1/3)*x]*(Log[-d^(1/3) - e^(1/3)*x] + 2*(Log[((-1)^(1/3)*d^(1/3) - e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))]
+ Log[(I + Sqrt[3] - ((2*I)*e^(1/3)*x)/d^(1/3))/(3*I + Sqrt[3])])) + 2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 +
(-1)^(1/3))*d^(1/3))] + 2*PolyLog[2, ((2*I)*(1 + (e^(1/3)*x)/d^(1/3)))/(3*I + Sqrt[3])])))/e^(2/3))/4

________________________________________________________________________________________

Maple [F]  time = 0.672, size = 0, normalized size = 0. \begin{align*} \int x \left ( \ln \left ( c \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*log((e*x^3 + d)^p*c)^2, x)

________________________________________________________________________________________

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(x*ln(c*(e*x**3+d)**p)**2,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*(e*x^3+d)^p)^2,x, algorithm="giac")

[Out]

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