3.1.52 \(\int x^2 \text {PolyLog}(3,a x^q) \, dx\) [52]

Optimal. Leaf size=88 \[ -\frac {a q^3 x^{3+q} \, _2F_1\left (1,\frac {3+q}{q};2+\frac {3}{q};a x^q\right )}{27 (3+q)}-\frac {1}{27} q^2 x^3 \log \left (1-a x^q\right )-\frac {1}{9} q x^3 \text {PolyLog}\left (2,a x^q\right )+\frac {1}{3} x^3 \text {PolyLog}\left (3,a x^q\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 = {6726, 2505, 371} \begin {gather*} -\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}{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} q^2 x^3 \log \left (1-a x^q\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[3, a*x^q],x]

[Out]

-1/27*(a*q^3*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2505

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 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\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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 0.01, size = 41, normalized size = 0.47 \begin {gather*} -\frac {x^3 G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,\frac {-3+q}{q} \\ 1,0,0,0,-\frac {3}{q} \\\end {array}\right )}{q} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 5.
time = 0.20, size = 132, normalized size = 1.50

method	result	size
meijerg	\(-\frac {\left (-a \right )^{-\frac {3}{q}} \left (\frac {q^{3} x^{3} \left (-a \right )^{\frac {3}{q}} \ln \left (1-a \,x^{q}\right )}{27}+\frac {q^{2} x^{3} \left (-a \right )^{\frac {3}{q}} \polylog \left (2, a \,x^{q}\right )}{9}-\frac {q \,x^{3} \left (-a \right )^{\frac {3}{q}} \left (1+\frac {q}{3}\right ) \polylog \left (3, a \,x^{q}\right )}{3+q}+\frac {q^{3} x^{3+q} a \left (-a \right )^{\frac {3}{q}} \Phi \left (a \,x^{q}, 1, \frac {3+q}{q}\right )}{27}\right )}{q}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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^q),x, algorithm="fricas")

[Out]

integral(x^2*polylog(3, a*x^q), x)

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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^{q}\right )\, dx \end {gather*}

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 = 0.00, 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^q),x, algorithm="giac")

[Out]

integrate(x^2*polylog(3, a*x^q), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {polylog}\left (3,a\,x^q\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(3, a*x^q),x)

[Out]

int(x^2*polylog(3, a*x^q), x)

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