3.1.12 \(\int x^2 \text {PolyLog}(3,a x) \, dx\) [12]

Optimal. Leaf size=78 \[ \frac {x}{27 a^2}+\frac {x^2}{54 a}+\frac {x^3}{81}+\frac {\log (1-a x)}{27 a^3}-\frac {1}{27} x^3 \log (1-a x)-\frac {1}{9} x^3 \text {PolyLog}(2,a x)+\frac {1}{3} x^3 \text {PolyLog}(3,a x) \]

[Out]

1/27*x/a^2+1/54*x^2/a+1/81*x^3+1/27*ln(-a*x+1)/a^3-1/27*x^3*ln(-a*x+1)-1/9*x^3*polylog(2,a*x)+1/3*x^3*polylog(
3,a*x)

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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6726, 2442, 45} \begin {gather*} \frac {\log (1-a x)}{27 a^3}+\frac {x}{27 a^2}-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{27} x^3 \log (1-a x)+\frac {x^2}{54 a}+\frac {x^3}{81} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[3, a*x],x]

[Out]

x/(27*a^2) + x^2/(54*a) + x^3/81 + Log[1 - a*x]/(27*a^3) - (x^3*Log[1 - a*x])/27 - (x^3*PolyLog[2, a*x])/9 + (
x^3*PolyLog[3, a*x])/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2442

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

Rule 6726

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[(d*x)^(m + 1)*(PolyLog[n
, a*(b*x^p)^q]/(d*(m + 1))), x] - Dist[p*(q/(m + 1)), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rubi steps

\begin {align*} \int x^2 \text {Li}_3(a x) \, dx &=\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{3} \int x^2 \text {Li}_2(a x) \, dx\\ &=-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{9} \int x^2 \log (1-a x) \, dx\\ &=-\frac {1}{27} x^3 \log (1-a x)-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{27} a \int \frac {x^3}{1-a x} \, dx\\ &=-\frac {1}{27} x^3 \log (1-a x)-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{27} a \int \left (-\frac {1}{a^3}-\frac {x}{a^2}-\frac {x^2}{a}-\frac {1}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac {x}{27 a^2}+\frac {x^2}{54 a}+\frac {x^3}{81}+\frac {\log (1-a x)}{27 a^3}-\frac {1}{27} x^3 \log (1-a x)-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 78, normalized size = 1.00 \begin {gather*} \frac {6 a x+3 a^2 x^2+2 a^3 x^3+6 \log (1-a x)-6 a^3 x^3 \log (1-a x)-18 a^3 x^3 \text {PolyLog}(2,a x)+54 a^3 x^3 \text {PolyLog}(3,a x)}{162 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*PolyLog[3, a*x],x]

[Out]

(6*a*x + 3*a^2*x^2 + 2*a^3*x^3 + 6*Log[1 - a*x] - 6*a^3*x^3*Log[1 - a*x] - 18*a^3*x^3*PolyLog[2, a*x] + 54*a^3
*x^3*PolyLog[3, a*x])/(162*a^3)

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Maple [A]
time = 0.13, size = 69, normalized size = 0.88

method	result	size
meijerg	\(\frac {\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{324}+\frac {\left (-4 a^{3} x^{3}+4\right ) \ln \left (-a x +1\right )}{108}-\frac {a^{3} x^{3} \polylog \left (2, a x \right )}{9}+\frac {a^{3} x^{3} \polylog \left (3, a x \right )}{3}}{a^{3}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(3,a*x),x,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/324*a*x*(4*a^2*x^2+6*a*x+12)+1/108*(-4*a^3*x^3+4)*ln(-a*x+1)-1/9*a^3*x^3*polylog(2,a*x)+1/3*a^3*x^3*p
olylog(3,a*x))

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Maxima [A]
time = 0.26, size = 69, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 54 \, a^{3} x^{3} {\rm Li}_{3}(a x) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*polylog(3,a*x),x, algorithm="maxima")

[Out]

-1/162*(18*a^3*x^3*dilog(a*x) - 54*a^3*x^3*polylog(3, a*x) - 2*a^3*x^3 - 3*a^2*x^2 - 6*a*x + 6*(a^3*x^3 - 1)*l
og(-a*x + 1))/a^3

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Fricas [A]
time = 0.35, size = 69, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 54 \, a^{3} x^{3} {\rm polylog}\left (3, a x\right ) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*polylog(3,a*x),x, algorithm="fricas")

[Out]

-1/162*(18*a^3*x^3*dilog(a*x) - 54*a^3*x^3*polylog(3, a*x) - 2*a^3*x^3 - 3*a^2*x^2 - 6*a*x + 6*(a^3*x^3 - 1)*l
og(-a*x + 1))/a^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {Li}_{3}\left (a x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*polylog(3,a*x),x)

[Out]

Integral(x**2*polylog(3, a*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(x^2*polylog(3,a*x),x, algorithm="giac")

[Out]

integrate(x^2*polylog(3, a*x), x)

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Mupad [B]
time = 0.95, size = 63, normalized size = 0.81 \begin {gather*} \frac {\ln \left (a\,x-1\right )}{27\,a^3}-\frac {x^3\,\ln \left (1-a\,x\right )}{27}+\frac {x}{27\,a^2}+\frac {x^3}{81}-\frac {x^3\,\mathrm {polylog}\left (2,a\,x\right )}{9}+\frac {x^3\,\mathrm {polylog}\left (3,a\,x\right )}{3}+\frac {x^2}{54\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(3, a*x),x)

[Out]

log(a*x - 1)/(27*a^3) - (x^3*log(1 - a*x))/27 + x/(27*a^2) + x^3/81 - (x^3*polylog(2, a*x))/9 + (x^3*polylog(3
, a*x))/3 + x^2/(54*a)

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