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

Optimal. Leaf size=557 \[ \frac{e p \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )}{3 d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}+\frac{3 e p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^2}+\frac{\sqrt [3]{a} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac{e \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d}-\frac{\sqrt{3} \sqrt [3]{a} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}+\frac{3 p}{d x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.668387, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.783, Rules used = {2466, 2455, 263, 325, 292, 31, 634, 617, 204, 628, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ \frac{e p \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )}{3 d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}+\frac{3 e p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^2}+\frac{\sqrt [3]{a} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac{e \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d}-\frac{\sqrt{3} \sqrt [3]{a} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}+\frac{3 p}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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 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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 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 2393

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

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x^2}-\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}+\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x^2} \, dx}{d}-\frac{e \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x} \, dx}{d^2}+\frac{e^2 \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d+e x} \, dx}{d^2}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{e \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac{1}{x^3}\right )}{3 d^2}-\frac{(3 b p) \int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^5} \, dx}{d}+\frac{(3 b e p) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x^3}\right ) x^4} \, dx}{d^2}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}-\frac{(3 b p) \int \frac{1}{x^2 \left (b+a x^3\right )} \, dx}{d}-\frac{(b e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx,x,\frac{1}{x^3}\right )}{3 d^2}+\frac{(3 b e p) \int \left (\frac{\log (d+e x)}{b x}-\frac{a x^2 \log (d+e x)}{b \left (b+a x^3\right )}\right ) \, dx}{d^2}\\ &=\frac{3 p}{d x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}+\frac{(3 a p) \int \frac{x}{b+a x^3} \, dx}{d}+\frac{(3 e p) \int \frac{\log (d+e x)}{x} \, dx}{d^2}-\frac{(3 a e p) \int \frac{x^2 \log (d+e x)}{b+a x^3} \, dx}{d^2}\\ &=\frac{3 p}{d x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}-\frac{\left (a^{2/3} p\right ) \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{\sqrt [3]{b} d}+\frac{\left (a^{2/3} p\right ) \int \frac{\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{\sqrt [3]{b} d}-\frac{(3 a e p) \int \left (\frac{\log (d+e x)}{3 a^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac{\log (d+e x)}{3 a^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac{\log (d+e x)}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}\right ) \, dx}{d^2}-\frac{\left (3 e^2 p\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{d^2}\\ &=\frac{3 p}{d x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}+\frac{3 e p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2}+\frac{\left (3 a^{2/3} p\right ) \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 d}+\frac{\left (\sqrt [3]{a} p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{b} d}-\frac{\left (\sqrt [3]{a} e p\right ) \int \frac{\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^2}-\frac{\left (\sqrt [3]{a} e p\right ) \int \frac{\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^2}-\frac{\left (\sqrt [3]{a} e p\right ) \int \frac{\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^2}\\ &=\frac{3 p}{d x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}+\frac{\sqrt [3]{a} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}+\frac{3 e p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2}+\frac{\left (3 \sqrt [3]{a} p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} d}+\frac{\left (e^2 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{d+e x} \, dx}{d^2}+\frac{\left (e^2 p\right ) \int \frac{\log \left (\frac{e \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{d^2}+\frac{\left (e^2 p\right ) \int \frac{\log \left (\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{d^2}\\ &=\frac{3 p}{d x}-\frac{\sqrt{3} \sqrt [3]{a} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}+\frac{\sqrt [3]{a} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}+\frac{3 e p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}\\ &=\frac{3 p}{d x}-\frac{\sqrt{3} \sqrt [3]{a} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^2}-\frac{\sqrt [3]{a} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac{3 e p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac{e p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}+\frac{\sqrt [3]{a} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d}+\frac{e p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^2}-\frac{e p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac{e p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}+\frac{3 e p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2}\\ \end{align*}

Mathematica [C]  time = 0.183896, size = 429, normalized size = 0.77 \[ \frac{4 a x^3 \left (-3 e p x \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )-3 e p x \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )-3 e p x \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )+e p x \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )+9 e p x \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+3 e x \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )-3 d \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )+e x \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )-3 e p x \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )-3 e p x \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} d}\right )-3 e p x \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} d}\right )+9 e p x \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+9 b d p \, _2F_1\left (1,\frac{4}{3};\frac{7}{3};-\frac{b}{a x^3}\right )}{12 a d^2 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b*d*p*Hypergeometric2F1[1, 4/3, 7/3, -(b/(a*x^3))] + 4*a*x^3*(-3*d*Log[c*(a + b/x^3)^p] + e*x*Log[c*(a + b/
x^3)^p]*Log[-(b/(a*x^3))] + 3*e*x*Log[c*(a + b/x^3)^p]*Log[d + e*x] + 9*e*p*x*Log[-((e*x)/d)]*Log[d + e*x] - 3
*e*p*x*Log[(e*((-1)^(1/3)*b^(1/3) - a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x] - 3*e*p*x*Log
[(e*(b^(1/3) + a^(1/3)*x))/(-(a^(1/3)*d) + b^(1/3)*e)]*Log[d + e*x] - 3*e*p*x*Log[(e*((-1)^(2/3)*b^(1/3) + a^(
1/3)*x))/(-(a^(1/3)*d) + (-1)^(2/3)*b^(1/3)*e)]*Log[d + e*x] + e*p*x*PolyLog[2, 1 + b/(a*x^3)] - 3*e*p*x*PolyL
og[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)] - 3*e*p*x*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^
(1/3)*b^(1/3)*e)] - 3*e*p*x*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)] + 9*e*p*x*PolyL
og[2, 1 + (e*x)/d]))/(12*a*d^2*x^4)

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Maple [F]  time = 0.709, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( ex+d \right ) }\ln \left ( c \left ( a+{\frac{b}{{x}^{3}}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x^3)^p)/x^2/(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{\log \left (c \left (\frac{a x^{3} + b}{x^{3}}\right )^{p}\right )}{e x^{3} + d x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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