\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^3}{x} \, dx\) [505]

Optimal result

Integrand size = 24, antiderivative size = 135 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e}{d \sqrt [3]{x}}\right )-18 b^3 n^3 \operatorname {PolyLog}\left (4,1+\frac {e}{d \sqrt [3]{x}}\right ) \]

[Out]

-3*(a+b*ln(c*(d+e/x^(1/3))^n))^3*ln(-e/d/x^(1/3))-9*b*n*(a+b*ln(c*(d+e/x^(1/3))^n))^2*polylog(2,1+e/d/x^(1/3))
+18*b^2*n^2*(a+b*ln(c*(d+e/x^(1/3))^n))*polylog(3,1+e/d/x^(1/3))-18*b^3*n^3*polylog(4,1+e/d/x^(1/3))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=18 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-9 b n \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-18 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {e}{d \sqrt [3]{x}}+1\right ) \]

[In]

Int[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x,x]

[Out]

-3*(a + b*Log[c*(d + e/x^(1/3))^n])^3*Log[-(e/(d*x^(1/3)))] - 9*b*n*(a + b*Log[c*(d + e/x^(1/3))^n])^2*PolyLog
[2, 1 + e/(d*x^(1/3))] + 18*b^2*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*PolyLog[3, 1 + e/(d*x^(1/3))] - 18*b^3*n^
3*PolyLog[4, 1 + e/(d*x^(1/3))]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2443

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

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

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 6724

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

Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+(9 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+(9 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+\left (18 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )-\left (18 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )-18 b^3 n^3 \text {Li}_4\left (1+\frac {e}{d \sqrt [3]{x}}\right ) \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x,x]

[Out]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x, x]

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{3}}{x}d x\]

[In]

int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x,x)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*ln(c*(d+e/x**(1/3))**n))**3/x,x)

[Out]

Integral((a + b*log(c*(d + e/x**(1/3))**n))**3/x, x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x,x, algorithm="maxima")

[Out]

b^3*log((d*x^(1/3) + e)^n)^3*log(x) - integrate(((b^3*d*x + b^3*e*x^(2/3))*log(x^(1/3*n))^3 + (b^3*d*n*x*log(x
) - 3*(b^3*d*log(c) + a*b^2*d)*x + 3*(b^3*d*x + b^3*e*x^(2/3))*log(x^(1/3*n)) - 3*(b^3*e*log(c) + a*b^2*e)*x^(
2/3))*log((d*x^(1/3) + e)^n)^2 - 3*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*x^(2/3))*log(x^(1/3*
n))^2 - (b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x - 3*((b^3*d*x + b^3*e*x^(2/3))*log(
x^(1/3*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x - 2*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c)
+ a*b^2*e)*x^(2/3))*log(x^(1/3*n)) + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(2/3))*log((d*x^(1/3) +
e)^n) + 3*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(
2/3))*log(x^(1/3*n)) - (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(2/3))/(d*x^2 + e*x^
(5/3)), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^n) + a)^3/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^3}{x} \,d x \]

[In]

int((a + b*log(c*(d + e/x^(1/3))^n))^3/x,x)

[Out]

int((a + b*log(c*(d + e/x^(1/3))^n))^3/x, x)