3.122 \(\int \frac{(a+b \log (c x^n))^3}{x (d+e x)^2} \, dx\)

Optimal. Leaf size=217 \[ \frac{6 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{3 b n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac{6 b^3 n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^2}+\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{d}{e x}\right )}{d^2}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac{\log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.396486, antiderivative size = 234, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2347, 2344, 2302, 30, 2317, 2374, 2383, 6589, 2318} \[ \frac{6 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac{3 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac{6 b^3 n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^2}-\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{e x}{d}\right )}{d^2}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^2 (d+e x)}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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]

Rule 2318

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

Rubi steps

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

Mathematica [A]  time = 0.471825, size = 432, normalized size = 1.99 \[ \frac{4 b^2 n^2 \left (6 (d+e x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 (\log (x)-1) (d+e x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \left (\log ^2(x) (d+e x)-3 \log (x) \left ((d+e x) \log \left (\frac{e x}{d}+1\right )+e x\right )+6 (d+e x) \log \left (\frac{e x}{d}+1\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+6 b n \left (-2 (d+e x) \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )+\log ^2(x) (d+e x)+2 (d+e x) \log (d+e x)-2 e x \log (x)\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2+b^3 n^3 \left (-4 \left (6 (d+e x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 \log (x) (d+e x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (\frac{e x}{d}+1\right )\right )\right )-4 (d+e x) \left (6 \text{PolyLog}\left (4,-\frac{e x}{d}\right )+3 \log ^2(x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )-6 \log (x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\log ^3(x) \log \left (\frac{e x}{d}+1\right )\right )+\log ^4(x) (d+e x)\right )+4 \log (x) (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3-4 (d+e x) \log (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3}{4 d^2 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.924, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{x \left ( ex+d \right ) ^{2}}}\, dx \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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3}{\left (\frac{1}{d e x + d^{2}} - \frac{\log \left (e x + d\right )}{d^{2}} + \frac{\log \left (x\right )}{d^{2}}\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^{2} x^{3} + 2 \, d e x^{2} + d^{2} 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)^2,x, algorithm="maxima")

[Out]

a^3*(1/(d*e*x + d^2) - log(e*x + d)/d^2 + log(x)/d^2) + 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^2*x^3 + 2*d*e*x^2 + d^2*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^{2} x^{3} + 2 \, d e x^{2} + d^{2} 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)^2,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^2*x^3 + 2*d*e*x^2 + d^2*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 )^{2}}\, 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)**2,x)

[Out]

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

[Out]

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