\(\int \frac {x^2 \log (c (a+b x^3)^p)}{d+e x} \, dx\) [234]

Optimal result

Integrand size = 23, antiderivative size = 643 \[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\frac {3 d p x}{e^2}-\frac {3 p x^2}{4 e}+\frac {\sqrt {3} \sqrt [3]{a} d p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^2}-\frac {\sqrt {3} a^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3} e}-\frac {\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^2}-\frac {a^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^2}+\frac {a^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e}-\frac {d x \log \left (c \left (a+b x^3\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^3} \]

[Out]

3*d*p*x/e^2-3/4*p*x^2/e-a^(1/3)*d*p*ln(a^(1/3)+b^(1/3)*x)/b^(1/3)/e^2-1/2*a^(2/3)*p*ln(a^(1/3)+b^(1/3)*x)/b^(2
/3)/e-d^2*p*ln(-e*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*d-a^(1/3)*e))*ln(e*x+d)/e^3-d^2*p*ln(-e*((-1)^(2/3)*a^(1/3)+b^(
1/3)*x)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))*ln(e*x+d)/e^3-d^2*p*ln((-1)^(1/3)*e*(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(
b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e))*ln(e*x+d)/e^3+1/2*a^(1/3)*d*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(1/
3)/e^2+1/4*a^(2/3)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(2/3)/e-d*x*ln(c*(b*x^3+a)^p)/e^2+1/2*x^2*ln(
c*(b*x^3+a)^p)/e+d^2*ln(e*x+d)*ln(c*(b*x^3+a)^p)/e^3-d^2*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-a^(1/3)*e))/e^
3-d^2*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e))/e^3-d^2*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/
3)*d-(-1)^(2/3)*a^(1/3)*e))/e^3+a^(1/3)*d*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/b^(1/3)/
e^2-1/2*a^(2/3)*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/b^(2/3)/e

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {2516, 2498, 327, 206, 31, 648, 631, 210, 642, 2505, 298, 2512, 266, 2463, 2441, 2440, 2438} \[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=-\frac {\sqrt {3} a^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3} e}+\frac {\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^2}+\frac {a^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e}-\frac {a^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e}+\frac {\sqrt {3} \sqrt [3]{a} d p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+b x^3\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^3}-\frac {\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^2}+\frac {3 d p x}{e^2}-\frac {3 p x^2}{4 e} \]

[In]

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

[Out]

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

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 210

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

Rule 266

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

Rule 298

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 327

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

Rule 631

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 642

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 648

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 2438

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

Rule 2440

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

Rule 2441

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

Rule 2463

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

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2505

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 2512

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

Rule 2516

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.26 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=-\frac {-12 d e p x+3 e^2 p x^2-\frac {4 \sqrt {3} \sqrt [3]{a} d e p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-3 e^2 p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+\frac {4 \sqrt [3]{a} d e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+4 d^2 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+4 d^2 p \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+4 d^2 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)-\frac {2 \sqrt [3]{a} d e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+4 d e x \log \left (c \left (a+b x^3\right )^p\right )-2 e^2 x^2 \log \left (c \left (a+b x^3\right )^p\right )-4 d^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+4 d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+4 d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+4 d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{4 e^3} \]

[In]

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

[Out]

-1/4*(-12*d*e*p*x + 3*e^2*p*x^2 - (4*Sqrt[3]*a^(1/3)*d*e*p*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3
) - 3*e^2*p*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)] + (4*a^(1/3)*d*e*p*Log[a^(1/3) + b^(1/3)*x])/b^(1
/3) + 4*d^2*p*Log[(e*((-1)^(1/3)*a^(1/3) - b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x] + 4*d^
2*p*Log[(e*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*d) + a^(1/3)*e)]*Log[d + e*x] + 4*d^2*p*Log[(e*((-1)^(2/3)*a^(1/3
) + b^(1/3)*x))/(-(b^(1/3)*d) + (-1)^(2/3)*a^(1/3)*e)]*Log[d + e*x] - (2*a^(1/3)*d*e*p*Log[a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2])/b^(1/3) + 4*d*e*x*Log[c*(a + b*x^3)^p] - 2*e^2*x^2*Log[c*(a + b*x^3)^p] - 4*d^2*Log[d
+ e*x]*Log[c*(a + b*x^3)^p] + 4*d^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)] + 4*d^2*p*PolyL
og[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)] + 4*d^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3
)*d - (-1)^(2/3)*a^(1/3)*e)])/e^3

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.53 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.39

method	result	size
parts	\(\frac {x^{2} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 e}-\frac {d x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e^{3}}-\frac {3 p b \left (\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{3 b}+\frac {\left (e x +d \right )^{2}}{4 b}-\frac {3 \left (e x +d \right ) d}{2 b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\frac {\left (-\textit {\_R} +3 d \right ) \ln \left (e x -\textit {\_R} +d \right )}{\textit {\_R}^{2}-2 \textit {\_R} d +d^{2}}\right ) a \,e^{3}}{6 b^{2}}\right )}{e^{3}}\)	\(250\)
risch	\(\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) x^{2}}{2 e}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) d x}{e^{2}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {p \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{e^{3}}-\frac {3 p \,x^{2}}{4 e}+\frac {3 d p x}{e^{2}}+\frac {15 p \,d^{2}}{4 e^{3}}+\frac {p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\frac {\left (\textit {\_R} -3 d \right ) \ln \left (e x -\textit {\_R} +d \right )}{\textit {\_R}^{2}-2 \textit {\_R} d +d^{2}}\right ) a}{2 b}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{2} e \,x^{2}-d x}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right )}{e^{3}}\right )\)	\(383\)

[In]

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

[Out]

1/2*x^2*ln(c*(b*x^3+a)^p)/e-d*x*ln(c*(b*x^3+a)^p)/e^2+d^2*ln(e*x+d)*ln(c*(b*x^3+a)^p)/e^3-3*p*b/e^3*(1/3*d^2/b
*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*d+3*_Z*b*d^2+a*e^3-b*d^
3))+1/4/b*(e*x+d)^2-3/2/b*(e*x+d)*d+1/6/b^2*sum((-_R+3*d)/(_R^2-2*_R*d+d^2)*ln(e*x-_R+d),_R=RootOf(_Z^3*b-3*_Z
^2*b*d+3*_Z*b*d^2+a*e^3-b*d^3))*a*e^3)

Fricas [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

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

[Out]

integral(x^2*log((b*x^3 + a)^p*c)/(e*x + d), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]

[In]

integrate(x**2*ln(c*(b*x**3+a)**p)/(e*x+d),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int \frac {x^2\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{d+e\,x} \,d x \]

[In]

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

[Out]

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