Optimal. Leaf size=88 \[ -\frac {a q^3 x^{q+2} \, _2F_1\left (1,\frac {q+2}{q};2 \left (1+\frac {1}{q}\right );a x^q\right )}{8 (q+2)}-\frac {1}{4} q x^2 \text {Li}_2\left (a x^q\right )+\frac {1}{2} x^2 \text {Li}_3\left (a x^q\right )-\frac {1}{8} q^2 x^2 \log \left (1-a x^q\right ) \]
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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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6591, 2455, 364} \[ -\frac {1}{4} q x^2 \text {PolyLog}\left (2,a x^q\right )+\frac {1}{2} x^2 \text {PolyLog}\left (3,a x^q\right )-\frac {a q^3 x^{q+2} \, _2F_1\left (1,\frac {q+2}{q};2 \left (1+\frac {1}{q}\right );a x^q\right )}{8 (q+2)}-\frac {1}{8} q^2 x^2 \log \left (1-a x^q\right ) \]
Antiderivative was successfully verified.
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Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int x \text {Li}_3\left (a x^q\right ) \, dx &=\frac {1}{2} x^2 \text {Li}_3\left (a x^q\right )-\frac {1}{2} q \int x \text {Li}_2\left (a x^q\right ) \, dx\\ &=-\frac {1}{4} q x^2 \text {Li}_2\left (a x^q\right )+\frac {1}{2} x^2 \text {Li}_3\left (a x^q\right )-\frac {1}{4} q^2 \int x \log \left (1-a x^q\right ) \, dx\\ &=-\frac {1}{8} q^2 x^2 \log \left (1-a x^q\right )-\frac {1}{4} q x^2 \text {Li}_2\left (a x^q\right )+\frac {1}{2} x^2 \text {Li}_3\left (a x^q\right )-\frac {1}{8} \left (a q^3\right ) \int \frac {x^{1+q}}{1-a x^q} \, dx\\ &=-\frac {a q^3 x^{2+q} \, _2F_1\left (1,\frac {2+q}{q};2 \left (1+\frac {1}{q}\right );a x^q\right )}{8 (2+q)}-\frac {1}{8} q^2 x^2 \log \left (1-a x^q\right )-\frac {1}{4} q x^2 \text {Li}_2\left (a x^q\right )+\frac {1}{2} x^2 \text {Li}_3\left (a x^q\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 41, normalized size = 0.47 \[ -\frac {x^2 G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,\frac {q-2}{q} \\ 1,0,0,0,-\frac {2}{q} \\\end {array}\right )}{q} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm polylog}\left (3, a x^{q}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Li}_{3}(a x^{q})\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 132, normalized size = 1.50 \[ -\frac {\left (-a \right )^{-\frac {2}{q}} \left (\frac {q^{3} x^{2} \left (-a \right )^{\frac {2}{q}} \ln \left (1-a \,x^{q}\right )}{8}+\frac {q^{2} x^{2} \left (-a \right )^{\frac {2}{q}} \polylog \left (2, a \,x^{q}\right )}{4}-\frac {q \,x^{2} \left (-a \right )^{\frac {2}{q}} \left (1+\frac {q}{2}\right ) \polylog \left (3, a \,x^{q}\right )}{2+q}+\frac {q^{3} x^{2+q} a \left (-a \right )^{\frac {2}{q}} \Phi \left (a \,x^{q}, 1, \frac {2+q}{q}\right )}{8}\right )}{q} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, q^{3} x^{2} - \frac {1}{8} \, q^{2} x^{2} \log \left (-a x^{q} + 1\right ) - \frac {1}{4} \, q x^{2} {\rm Li}_2\left (a x^{q}\right ) + q^{3} \int \frac {x}{8 \, {\left (a x^{q} - 1\right )}}\,{d x} + \frac {1}{2} \, x^{2} {\rm Li}_{3}(a x^{q}) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {polylog}\left (3,a\,x^q\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {Li}_{3}\left (a x^{q}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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