\(\int \frac {\log (c (a+b x^3)^p)}{x^2 (d+e x)} \, dx\) [238]

Optimal result

Integrand size = 23, antiderivative size = 510 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {e p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{3 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {2516, 2505, 298, 31, 648, 631, 210, 642, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\sqrt {3} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e p \operatorname {PolyLog}\left (2,\frac {b x^3}{a}+1\right )}{3 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d} \]

[In]

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

[Out]

-((Sqrt[3]*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(a^(1/3)*d)) - (b^(1/3)*p*Log[a^(1/3)
+ b^(1/3)*x])/(a^(1/3)*d) - (e*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d^2 -
(e*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])/d^2 - (e*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])/d^2 + (
b^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)*d) - Log[c*(a + b*x^3)^p]/(d*x) - (e*Log[
-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d^2) + (e*Log[d + e*x]*Log[c*(a + b*x^3)^p])/d^2 - (e*p*PolyLog[2, (b^(
1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^
(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/d^2 - (e*p*PolyLog[2
, 1 + (b*x^3)/a])/(3*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 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 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+b x^3\right )^p\right )}{d x^2}-\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (c \left (a+b x^3\right )^p\right )}{d^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx}{d^2} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^3\right )}{3 d^2}+\frac {(3 b p) \int \frac {x}{a+b x^3} \, dx}{d}-\frac {(3 b e p) \int \frac {x^2 \log (d+e x)}{a+b x^3} \, dx}{d^2} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {\left (b^{2/3} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{\sqrt [3]{a} d}+\frac {\left (b^{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}{\sqrt [3]{a} d}+\frac {(b e p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^3\right )}{3 d^2}-\frac {(3 b e p) \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}{d^2} \\ & = -\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^2}+\frac {\left (\sqrt [3]{b} 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]{a} d}+\frac {\left (3 b^{2/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 d}-\frac {\left (\sqrt [3]{b} e p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2}-\frac {\left (\sqrt [3]{b} e p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2}-\frac {\left (\sqrt [3]{b} e p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2} \\ & = -\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^2}+\frac {\left (3 \sqrt [3]{b} 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]{a} d}+\frac {\left (e^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}{d^2}+\frac {\left (e^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}{d^2}+\frac {\left (e^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}{d^2} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^2}+\frac {(e p) \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 )}{d^2}+\frac {(e p) \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 )}{d^2}+\frac {(e p) \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 )}{d^2} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}-\frac {e 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)}{d^2}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 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 = 395, normalized size of antiderivative = 0.77 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {9 b d p x^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )-2 a \left (3 e p x \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+3 e p x \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+3 e p x \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)+3 d \log \left (c \left (a+b x^3\right )^p\right )+e x \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-3 e x \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+e p x \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )\right )}{6 a d^2 x} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.57

method	result	size
parts	\(\frac {e \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{2}}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-3 p b \left (\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{3 d^{2} b}+\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{3 d^{2} b}\right )\)	\(292\)
risch	\(\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \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 )}{d^{2}}-\frac {p \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \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 )}{d^{2}}+\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 {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\)	\(415\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)