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

Optimal. Leaf size=692 \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{\sqrt{3} a^{2/3} d p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac{\sqrt{3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.891271, antiderivative size = 692, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 20, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.87, Rules used = {2466, 2448, 321, 200, 31, 634, 617, 204, 628, 2455, 292, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391} \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{\sqrt{3} a^{2/3} d p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac{\sqrt{3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2466

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]

Rule 2448

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 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 200

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 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 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 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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2462

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

Rule 260

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

Rule 2416

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

Rule 2394

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

Rule 2393

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

Rule 2391

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

Rubi steps

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

Mathematica [C]  time = 0.601297, size = 497, normalized size = 0.72 \[ -\frac{-12 d^3 p \left (\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )+\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} e-\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{a} e-\sqrt [3]{b} d}\right )\right )+\frac{6 d^2 e p \left (\sqrt [3]{a} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 \sqrt [3]{b} x\right )}{\sqrt [3]{b}}+12 d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )-12 d^2 e x \log \left (c \left (a+b x^3\right )^p\right )+6 d e^2 x^2 \log \left (c \left (a+b x^3\right )^p\right )+\frac{4 e^3 \left (b p x^3-\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )\right )}{b}+9 d e^2 p x^2 \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )-1\right )}{12 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(9*d*e^2*p*x^2*(-1 + Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)]) + (6*d^2*e*p*(6*b^(1/3)*x - 2*a^(1/3)*Log[
a^(1/3) + b^(1/3)*x] + a^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])))/b^(1/3) - 12*d^2*e*x*Log[c*(a + b*x^3)^p] + 6*d*e^2*x^2*Log[c*(a + b*x^3)^p] + 12*
d^3*Log[d + e*x]*Log[c*(a + b*x^3)^p] + (4*e^3*(b*p*x^3 - (a + b*x^3)*Log[c*(a + b*x^3)^p]))/b - 12*d^3*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] + Log[(e*(a^(1/3) + b^(
1/3)*x))/(-(b^(1/3)*d) + a^(1/3)*e)]*Log[d + e*x] + 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] + PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)] + PolyLog[2, (b^(
1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)] + PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a
^(1/3)*e)]))/(12*e^4)

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Maple [C]  time = 0.665, size = 912, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2/e*x^3+3/4*d*p*x^2/e^2+1/3*ln(c)/e*x^3-ln(c)*d^3/e^4*ln(e*
x+d)-1/2*ln(c)/e^2*x^2*d+ln(c)/e^3*x*d^2-49/12*p/e^4*d^3+1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^3*d^3/e^4*ln(e*x+d)+1/
4*I*Pi*csgn(I*c*(b*x^3+a)^p)^3/e^2*x^2*d-1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^3/e^3*x*d^2+1/6*I*Pi*csgn(I*c*(b*x^3+a
)^p)^2*csgn(I*c)/e*x^3-ln((b*x^3+a)^p)*d^3/e^4*ln(e*x+d)-1/2*ln((b*x^3+a)^p)/e^2*x^2*d+ln((b*x^3+a)^p)/e^3*x*d
^2-1/6*I*Pi*csgn(I*c*(b*x^3+a)^p)^3/e*x^3+1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)*d^3/e^4
*ln(e*x+d)+1/6/b*p/e*sum((2*_R^2-7*_R*d+11*d^2)/(_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+1/3*ln((b*x^3+a)^p)/e*x^3+p/e^4*d^3*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*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)
^p)*csgn(I*c)/e^2*x^2*d-1/3*p*x^3/e-1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)/e^3*x*d^2-3*d
^2*p*x/e^3-1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2*d^3/e^4*ln(e*x+d)-1/4*I*Pi*csgn(I*(b*x^3+a)^p)
*csgn(I*c*(b*x^3+a)^p)^2/e^2*x^2*d-1/6*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)/e*x^3-1/4*I*Pi
*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)/e^2*x^2*d+1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)/e^3*x*d^2-1/2*I*Pi*csg
n(I*c*(b*x^3+a)^p)^2*csgn(I*c)*d^3/e^4*ln(e*x+d)+1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2/e^3*x*d^
2

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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(x^3*log(c*(b*x^3+a)^p)/(e*x+d),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{x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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