\(\int \frac {\log (c (a+\frac {b}{x^3})^p)}{x^2 (d+e x)} \, dx\) [259]

Optimal result

Integrand size = 23, antiderivative size = 557 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {3 p}{d x}-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^2}-\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac {3 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}+\frac {\sqrt [3]{a} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d}+\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^3}\right )}{3 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}+\frac {3 e p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {2516, 2505, 269, 331, 298, 31, 648, 631, 210, 642, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {\sqrt [3]{a} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d}+\frac {e \log \left (-\frac {b}{a x^3}\right ) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d x}+\frac {e p \operatorname {PolyLog}\left (2,\frac {b}{a x^3}+1\right )}{3 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d}+\frac {3 e p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^2}+\frac {3 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}+\frac {3 p}{d x} \]

[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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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 269

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 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 331

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 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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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 2504

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 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 {\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 \text {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) \text {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 ) \text {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) \text {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) \text {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) \text {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] (verified)

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

Time = 0.10 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {3 b p \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {7}{3},-\frac {b}{a x^3}\right )}{4 a d x^4}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^2}+\frac {3 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x^3}}{a}\right )}{3 d^2}+\frac {3 e p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^2} \]

[In]

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

[Out]

(3*b*p*Hypergeometric2F1[1, 4/3, 7/3, -(b/(a*x^3))])/(4*a*d*x^4) - 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) + (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[-(
((-1)^(2/3)*e*(b^(1/3) - (-1)^(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 + (e
*p*PolyLog[2, (a + b/x^3)/a])/(3*d^2) + (3*e*p*PolyLog[2, (d + e*x)/d])/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

Maple [C] (warning: unable to verify)

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

Time = 1.97 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.62

method	result	size
parts	\(\frac {e \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{d x}-\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) e \ln \left (x \right )}{d^{2}}+3 p b \left (\frac {e \left (-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )}{3 b}+\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}\right )}{d^{2}}+\frac {1}{d b x}-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 d b \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d b \left (\frac {b}{a}\right )^{\frac {1}{3}}}-\frac {e \ln \left (x \right )^{2}}{2 d^{2} b}+\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a +b \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{3 d^{2} b}\right )\)	\(345\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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