3.5.83 \(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^3}{x} \, dx\) [483]

Optimal. Leaf size=139 \[ \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+9 b^3 n^3 \text {Li}_4\left (1+\frac {e x^{2/3}}{d}\right ) \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 139, 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*} -9 b^2 n^2 \text {PolyLog}\left (3,\frac {e x^{2/3}}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {9}{2} b n \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+9 b^3 n^3 \text {PolyLog}\left (4,\frac {e x^{2/3}}{d}+1\right )+\frac {3}{2} \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*Log[c*(d + e*x^(2/3))^n])^3*Log[-((e*x^(2/3))/d)])/2 + (9*b*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2*Pol
yLog[2, 1 + (e*x^(2/3))/d])/2 - 9*b^2*n^2*(a + b*Log[c*(d + e*x^(2/3))^n])*PolyLog[3, 1 + (e*x^(2/3))/d] + 9*b
^3*n^3*PolyLog[4, 1 + (e*x^(2/3))/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 x^{2/3}\right )^n\right )\right )^3}{x} \, dx &=\frac {3}{2} \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )-\frac {1}{2} (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,x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )-\frac {1}{2} (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 x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-\left (9 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 x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+\left (9 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )\\ &=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+9 b^3 n^3 \text {Li}_4\left (1+\frac {e x^{2/3}}{d}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(339\) vs. \(2(139)=278\).
time = 0.12, size = 339, normalized size = 2.44 \begin {gather*} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (\left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right ) \log (x)-\frac {3}{2} \text {Li}_2\left (-\frac {e x^{2/3}}{d}\right )\right )+\frac {9}{2} b^2 n^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (\log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+2 \log \left (d+e x^{2/3}\right ) \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-2 \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )\right )+\frac {3}{2} b^3 n^3 \left (\log ^3\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+3 \log ^2\left (d+e x^{2/3}\right ) \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-6 \log \left (d+e x^{2/3}\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+6 \text {Li}_4\left (1+\frac {e x^{2/3}}{d}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (127) = 254\).
time = 0.38, size = 297, normalized size = 2.14 \begin {gather*} \frac {3}{2} \, {\left (\log \left (x^{\frac {2}{3}} e + d\right )^{3} \log \left (-\frac {x^{\frac {2}{3}} e + d}{d} + 1\right ) + 3 \, {\rm Li}_2\left (\frac {x^{\frac {2}{3}} e + d}{d}\right ) \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 6 \, \log \left (x^{\frac {2}{3}} e + d\right ) {\rm Li}_{3}(\frac {x^{\frac {2}{3}} e + d}{d}) + 6 \, {\rm Li}_{4}(\frac {x^{\frac {2}{3}} e + d}{d})\right )} b^{3} n^{3} + a^{3} \log \left (x\right ) + \frac {9}{2} \, {\left (b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} {\left (\log \left (x^{\frac {2}{3}} e + d\right )^{2} \log \left (-\frac {x^{\frac {2}{3}} e + d}{d} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {x^{\frac {2}{3}} e + d}{d}\right ) \log \left (x^{\frac {2}{3}} e + d\right ) - 2 \, {\rm Li}_{3}(\frac {x^{\frac {2}{3}} e + d}{d})\right )} + \frac {9}{2} \, {\left (b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n\right )} {\left (\log \left (x^{\frac {2}{3}} e + d\right ) \log \left (-\frac {x^{\frac {2}{3}} e + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {x^{\frac {2}{3}} e + d}{d}\right )\right )} + {\left (b^{3} \log \left (c\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right )\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(log(x^(2/3)*e + d)^3*log(-(x^(2/3)*e + d)/d + 1) + 3*dilog((x^(2/3)*e + d)/d)*log(x^(2/3)*e + d)^2 - 6*lo
g(x^(2/3)*e + d)*polylog(3, (x^(2/3)*e + d)/d) + 6*polylog(4, (x^(2/3)*e + d)/d))*b^3*n^3 + a^3*log(x) + 9/2*(
b^3*n^2*log(c) + a*b^2*n^2)*(log(x^(2/3)*e + d)^2*log(-(x^(2/3)*e + d)/d + 1) + 2*dilog((x^(2/3)*e + d)/d)*log
(x^(2/3)*e + d) - 2*polylog(3, (x^(2/3)*e + d)/d)) + 9/2*(b^3*n*log(c)^2 + 2*a*b^2*n*log(c) + a^2*b*n)*(log(x^
(2/3)*e + d)*log(-(x^(2/3)*e + d)/d + 1) + dilog((x^(2/3)*e + d)/d)) + (b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^
2*b*log(c))*log(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((a+b*log(c*(d+e*x^(2/3))^n))^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\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**(2/3))**n))**3/x,x)

[Out]

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

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

[Out]

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

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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^{2/3}\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^(2/3))^n))^3/x,x)

[Out]

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

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