3.5.60 \(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x} \, dx\) [460]

Optimal. Leaf size=135 \[ 3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+9 b n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )-18 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e \sqrt [3]{x}}{d}\right )+18 b^3 n^3 \text {Li}_4\left (1+\frac {e \sqrt [3]{x}}{d}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2443, 2481, 2421, 2430, 6724} \begin {gather*} -18 b^2 n^2 \text {PolyLog}\left (3,\frac {e \sqrt [3]{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+9 b n \text {PolyLog}\left (2,\frac {e \sqrt [3]{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+18 b^3 n^3 \text {PolyLog}\left (4,\frac {e \sqrt [3]{x}}{d}+1\right )+3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \end {gather*}

Antiderivative was successfully verified.

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

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(333\) vs. \(2(135)=270\).
time = 0.10, size = 333, normalized size = 2.47 \begin {gather*} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (\left (\log \left (d+e \sqrt [3]{x}\right )-\log \left (1+\frac {e \sqrt [3]{x}}{d}\right )\right ) \log (x)-3 \text {Li}_2\left (-\frac {e \sqrt [3]{x}}{d}\right )\right )+9 b^2 n^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (\log ^2\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+2 \log \left (d+e \sqrt [3]{x}\right ) \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )-2 \text {Li}_3\left (1+\frac {e \sqrt [3]{x}}{d}\right )\right )+3 b^3 n^3 \left (\log ^3\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+3 \log ^2\left (d+e \sqrt [3]{x}\right ) \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )-6 \log \left (d+e \sqrt [3]{x}\right ) \text {Li}_3\left (1+\frac {e \sqrt [3]{x}}{d}\right )+6 \text {Li}_4\left (1+\frac {e \sqrt [3]{x}}{d}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^3*Log[x] + 3*b*n*(a - b*n*Log[d + e*x^(1/3)] + b*Log
[c*(d + e*x^(1/3))^n])^2*((Log[d + e*x^(1/3)] - Log[1 + (e*x^(1/3))/d])*Log[x] - 3*PolyLog[2, -((e*x^(1/3))/d)
]) + 9*b^2*n^2*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])*(Log[d + e*x^(1/3)]^2*Log[-((e*x^(1/3
))/d)] + 2*Log[d + e*x^(1/3)]*PolyLog[2, 1 + (e*x^(1/3))/d] - 2*PolyLog[3, 1 + (e*x^(1/3))/d]) + 3*b^3*n^3*(Lo
g[d + e*x^(1/3)]^3*Log[-((e*x^(1/3))/d)] + 3*Log[d + e*x^(1/3)]^2*PolyLog[2, 1 + (e*x^(1/3))/d] - 6*Log[d + e*
x^(1/3)]*PolyLog[3, 1 + (e*x^(1/3))/d] + 6*PolyLog[4, 1 + (e*x^(1/3))/d])

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )^{3}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________