3.52 \(\int x^2 \text{PolyLog}(3,a x^q) \, dx\)

Optimal. Leaf size=88 \[ -\frac{a q^3 x^{q+3} \text{Hypergeometric2F1}\left (1,\frac{q+3}{q},\frac{3}{q}+2,a x^q\right )}{27 (q+3)}-\frac{1}{9} q x^3 \text{PolyLog}\left (2,a x^q\right )+\frac{1}{3} x^3 \text{PolyLog}\left (3,a x^q\right )-\frac{1}{27} q^2 x^3 \log \left (1-a x^q\right ) \]

[Out]

-(a*q^3*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(27*(3 + q)) - (q^2*x^3*Log[1 - a*x^q])/27
- (q*x^3*PolyLog[2, a*x^q])/9 + (x^3*PolyLog[3, a*x^q])/3

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Rubi [A]  time = 0.0515477, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 364} \[ -\frac{1}{9} q x^3 \text{PolyLog}\left (2,a x^q\right )+\frac{1}{3} x^3 \text{PolyLog}\left (3,a x^q\right )-\frac{a q^3 x^{q+3} \, _2F_1\left (1,\frac{q+3}{q};2+\frac{3}{q};a x^q\right )}{27 (q+3)}-\frac{1}{27} q^2 x^3 \log \left (1-a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[3, a*x^q],x]

[Out]

-(a*q^3*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(27*(3 + q)) - (q^2*x^3*Log[1 - a*x^q])/27
- (q*x^3*PolyLog[2, a*x^q])/9 + (x^3*PolyLog[3, a*x^q])/3

Rule 6591

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]

Rule 2455

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^2 \text{Li}_3\left (a x^q\right ) \, dx &=\frac{1}{3} x^3 \text{Li}_3\left (a x^q\right )-\frac{1}{3} q \int x^2 \text{Li}_2\left (a x^q\right ) \, dx\\ &=-\frac{1}{9} q x^3 \text{Li}_2\left (a x^q\right )+\frac{1}{3} x^3 \text{Li}_3\left (a x^q\right )-\frac{1}{9} q^2 \int x^2 \log \left (1-a x^q\right ) \, dx\\ &=-\frac{1}{27} q^2 x^3 \log \left (1-a x^q\right )-\frac{1}{9} q x^3 \text{Li}_2\left (a x^q\right )+\frac{1}{3} x^3 \text{Li}_3\left (a x^q\right )-\frac{1}{27} \left (a q^3\right ) \int \frac{x^{2+q}}{1-a x^q} \, dx\\ &=-\frac{a q^3 x^{3+q} \, _2F_1\left (1,\frac{3+q}{q};2+\frac{3}{q};a x^q\right )}{27 (3+q)}-\frac{1}{27} q^2 x^3 \log \left (1-a x^q\right )-\frac{1}{9} q x^3 \text{Li}_2\left (a x^q\right )+\frac{1}{3} x^3 \text{Li}_3\left (a x^q\right )\\ \end{align*}

Mathematica [C]  time = 0.0089413, size = 41, normalized size = 0.47 \[ -\frac{x^3 G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,\frac{q-3}{q} \\ 1,0,0,0,-\frac{3}{q} \\\end{array}\right )}{q} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*PolyLog[3, a*x^q],x]

[Out]

-((x^3*MeijerG[{{1, 1, 1, 1, (-3 + q)/q}, {}}, {{1}, {0, 0, 0, -3/q}}, -(a*x^q)])/q)

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Maple [C]  time = 0.345, size = 132, normalized size = 1.5 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-3\,{q}^{-1}} \left ({\frac{{q}^{3}{x}^{3}\ln \left ( 1-a{x}^{q} \right ) }{27} \left ( -a \right ) ^{3\,{q}^{-1}}}+{\frac{{q}^{2}{x}^{3}{\it polylog} \left ( 2,a{x}^{q} \right ) }{9} \left ( -a \right ) ^{3\,{q}^{-1}}}-{\frac{q{x}^{3}{\it polylog} \left ( 3,a{x}^{q} \right ) }{3+q} \left ( -a \right ) ^{3\,{q}^{-1}} \left ( 1+{\frac{q}{3}} \right ) }+{\frac{{q}^{3}{x}^{3+q}a}{27} \left ( -a \right ) ^{3\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{3+q}{q}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(3,a*x^q),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{81} \, q^{3} x^{3} - \frac{1}{27} \, q^{2} x^{3} \log \left (-a x^{q} + 1\right ) - \frac{1}{9} \, q x^{3}{\rm Li}_2\left (a x^{q}\right ) + q^{3} \int \frac{x^{2}}{27 \,{\left (a x^{q} - 1\right )}}\,{d x} + \frac{1}{3} \, x^{3}{\rm Li}_{3}(a x^{q}) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2}{\rm polylog}\left (3, a x^{q}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*polylog(3, a*x^q), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*polylog(3, a*x**q), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Li}_{3}(a x^{q})\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*polylog(3, a*x^q), x)