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

Optimal. Leaf size=113 \[ \frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac{3 b n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{d}{e x}\right )}{d}-\frac{\log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d} \]

[Out]

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

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Rubi [A]  time = 0.207856, antiderivative size = 130, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2344, 2302, 30, 2317, 2374, 2383, 6589} \[ \frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{3 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{e x}{d}\right )}{d}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[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 6589

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 x^n\right )\right )^3}{x (d+e x)} \, dx &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{d+e x} \, dx}{d}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}+\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d n}+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e x}{d}\right )}{d}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e x}{d}\right )}{d}-\frac{6 b^3 n^3 \text{Li}_4\left (-\frac{e x}{d}\right )}{d}\\ \end{align*}

Mathematica [B]  time = 0.175878, size = 243, normalized size = 2.15 \[ \frac{-4 b^2 n^2 \left (6 \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 \log (x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) \left (\log (x)-3 \log \left (\frac{e x}{d}+1\right )\right )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )+6 b n \left (\log ^2(x)-2 \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2+b^3 n^3 \left (-24 \text{PolyLog}\left (4,-\frac{e x}{d}\right )-12 \log ^2(x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+24 \log (x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )-4 \log ^3(x) \log \left (\frac{e x}{d}+1\right )+\log ^4(x)\right )-4 \log (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n])^3 - 4*(a - b*n*Log[x] + b*Log[c*x^n])^3*Log[d + e*x] + 6*b*n*(a - b*
n*Log[x] + b*Log[c*x^n])^2*(Log[x]^2 - 2*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)])) - 4*b^2*n^2*(-a +
 b*n*Log[x] - b*Log[c*x^n])*(Log[x]^2*(Log[x] - 3*Log[1 + (e*x)/d]) - 6*Log[x]*PolyLog[2, -((e*x)/d)] + 6*Poly
Log[3, -((e*x)/d)]) + b^3*n^3*(Log[x]^4 - 4*Log[x]^3*Log[1 + (e*x)/d] - 12*Log[x]^2*PolyLog[2, -((e*x)/d)] + 2
4*Log[x]*PolyLog[3, -((e*x)/d)] - 24*PolyLog[4, -((e*x)/d)]))/(4*d)

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Maple [C]  time = 0.718, size = 9909, normalized size = 87.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a^{3}{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + \int \frac{b^{3} \log \left (c\right )^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left (x^{n}\right )}{e x^{2} + d x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}}{e x^{2} + d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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