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

Optimal. Leaf size=71 \[ \frac{a q^2 x^{q+3} \text{Hypergeometric2F1}\left (1,\frac{q+3}{q},\frac{3}{q}+2,a x^q\right )}{9 (q+3)}+\frac{1}{3} x^3 \text{PolyLog}\left (2,a x^q\right )+\frac{1}{9} q x^3 \log \left (1-a x^q\right ) \]

[Out]

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

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Rubi [A]  time = 0.0373625, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 364} \[ \frac{1}{3} x^3 \text{PolyLog}\left (2,a x^q\right )+\frac{a q^2 x^{q+3} \, _2F_1\left (1,\frac{q+3}{q};2+\frac{3}{q};a x^q\right )}{9 (q+3)}+\frac{1}{9} q x^3 \log \left (1-a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[2, a*x^q],x]

[Out]

(a*q^2*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(9*(3 + q)) + (q*x^3*Log[1 - a*x^q])/9 + (x^
3*PolyLog[2, 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}_2\left (a x^q\right ) \, dx &=\frac{1}{3} x^3 \text{Li}_2\left (a x^q\right )+\frac{1}{3} q \int x^2 \log \left (1-a x^q\right ) \, dx\\ &=\frac{1}{9} q x^3 \log \left (1-a x^q\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^q\right )+\frac{1}{9} \left (a q^2\right ) \int \frac{x^{2+q}}{1-a x^q} \, dx\\ &=\frac{a q^2 x^{3+q} \, _2F_1\left (1,\frac{3+q}{q};2+\frac{3}{q};a x^q\right )}{9 (3+q)}+\frac{1}{9} q x^3 \log \left (1-a x^q\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^q\right )\\ \end{align*}

Mathematica [A]  time = 0.0439455, size = 69, normalized size = 0.97 \[ \frac{q x^3 \left (a q x^q \text{Hypergeometric2F1}\left (1,\frac{q+3}{q},\frac{3}{q}+2,a x^q\right )+(q+3) \log \left (1-a x^q\right )\right )}{9 (q+3)}+\frac{1}{3} x^3 \text{PolyLog}\left (2,a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*PolyLog[2, a*x^q],x]

[Out]

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

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Maple [C]  time = 0.26, size = 108, normalized size = 1.5 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-3\,{q}^{-1}} \left ( -{\frac{{q}^{2}{x}^{3}\ln \left ( 1-a{x}^{q} \right ) }{9} \left ( -a \right ) ^{3\,{q}^{-1}}}-{\frac{q{x}^{3}{\it polylog} \left ( 2,a{x}^{q} \right ) }{3+q} \left ( -a \right ) ^{3\,{q}^{-1}} \left ( 1+{\frac{q}{3}} \right ) }-{\frac{{q}^{2}{x}^{3+q}a}{9} \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(2,a*x^q),x)

[Out]

-(-a)^(-3/q)/q*(-1/9*q^2*x^3*(-a)^(3/q)*ln(1-a*x^q)-q/(3+q)*x^3*(-a)^(3/q)*(1+1/3*q)*polylog(2,a*x^q)-1/9*q^2*
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}{27} \, q^{2} x^{3} + \frac{1}{9} \, q x^{3} \log \left (-a x^{q} + 1\right ) + \frac{1}{3} \, x^{3}{\rm Li}_2\left (a x^{q}\right ) - q^{2} \int \frac{x^{2}}{9 \,{\left (a x^{q} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2}{\rm Li}_2\left (a x^{q}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*dilog(a*x^q), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Li}_2\left (a x^{q}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*dilog(a*x^q), x)