3.3.39 \(\int \frac {\log (c (a+b x^3)^p)}{x^3 (d+e x)} \, dx\) [239]

Optimal. Leaf size=674 \[ -\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 674, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 17, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {2516, 2505, 206, 31, 648, 631, 210, 642, 298, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \begin {gather*} \frac {e^2 p \text {PolyLog}\left (2,\frac {b x^3}{a}+1\right )}{3 d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}-\frac {\sqrt {3} b^{2/3} p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^3}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 206

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

Rule 210

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

Rule 266

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

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.27, size = 542, normalized size = 0.80 \begin {gather*} \frac {-\frac {6 \sqrt {3} b^{2/3} d^2 p \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {18 b d e p x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a}\right )}{a}+\frac {6 b^{2/3} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)-\frac {3 b^{2/3} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac {6 d^2 \log \left (c \left (a+b x^3\right )^p\right )}{x^2}+\frac {12 d e \log \left (c \left (a+b x^3\right )^p\right )}{x}+4 e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-12 e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+12 e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+12 e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+12 e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+4 e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{12 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[3]*b^(2/3)*d^2*p*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) - (18*b*d*e*p*x^2*Hypergeometr
ic2F1[2/3, 1, 5/3, -((b*x^3)/a)])/a + (6*b^(2/3)*d^2*p*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + 12*e^2*p*Log[(e*((-
1)^(1/3)*a^(1/3) - b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x] + 12*e^2*p*Log[(e*(a^(1/3) + b
^(1/3)*x))/(-(b^(1/3)*d) + a^(1/3)*e)]*Log[d + e*x] + 12*e^2*p*Log[(e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(-(b^(
1/3)*d) + (-1)^(2/3)*a^(1/3)*e)]*Log[d + e*x] - (3*b^(2/3)*d^2*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
])/a^(2/3) - (6*d^2*Log[c*(a + b*x^3)^p])/x^2 + (12*d*e*Log[c*(a + b*x^3)^p])/x + 4*e^2*Log[-((b*x^3)/a)]*Log[
c*(a + b*x^3)^p] - 12*e^2*Log[d + e*x]*Log[c*(a + b*x^3)^p] + 12*e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)
*d - a^(1/3)*e)] + 12*e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)] + 12*e^2*p*Poly
Log[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)] + 4*e^2*p*PolyLog[2, 1 + (b*x^3)/a])/(12*d^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.61, size = 1025, normalized size = 1.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(1025\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)*e^2/d^3*ln(e*x+d)+p*e^2/d^3*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dil
og((-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))-1/2*ln((b*x^3+a)^p)/d/x^2-1/2*I*Pi*
csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)*e/d^2/x+1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)
*csgn(I*c)*e^2/d^3*ln(e*x+d)+1/2*p/d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/4*p/d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a
/b)^(2/3))-p*e^2/d^3*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^3*b+a))-1/2*I*Pi*csgn(I*(b*x^3
+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)*e^2/d^3*ln(x)-ln(c)*e^2/d^3*ln(e*x+d)+ln(c)*e^2/d^3*ln(x)+ln(c)*e/d^2/x
+1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)*e^2/d^3*ln(x)+p/d^2*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/2*p/d^2*e/(a
/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-1/2*ln(c)/d/x^2-ln((b*x^3+a)^p)*e^2/d^3*ln(e*x+d)+ln((b*x^3+a)^p)*
e^2/d^3*ln(x)+ln((b*x^3+a)^p)*e/d^2/x-p/d^2*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/2*
p/d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a
)^p)^2*e^2/d^3*ln(e*x+d)-1/4*I*Pi*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)/d/x^2-1/4*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*
c*(b*x^3+a)^p)^2/d/x^2+1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2*e/d^2/x+1/4*I*Pi*csgn(I*c*(b*x^3+a
)^p)^3/d/x^2+1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^3*e^2/d^3*ln(e*x+d)+1/4*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a
)^p)*csgn(I*c)/d/x^2+1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)*e/d^2/x+1/2*I*Pi*csgn(I*(b*x^3+a)^p)*csgn(I*c*
(b*x^3+a)^p)^2*e^2/d^3*ln(x)-1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^3*e^2/d^3*ln(x)-1/2*I*Pi*csgn(I*c*(b*x^3+a)^p)^3*e
/d^2/x

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x^3 + a)^p*c)/(x^4*e + d*x^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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