Optimal. Leaf size=130 \[ -\frac{a q^3 x^{q+1} (d x)^m \text{Hypergeometric2F1}\left (1,\frac{m+q+1}{q},\frac{m+2 q+1}{q},a x^q\right )}{(m+1)^3 (m+q+1)}-\frac{q (d x)^{m+1} \text{PolyLog}\left (2,a x^q\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (3,a x^q\right )}{d (m+1)}-\frac{q^2 (d x)^{m+1} \log \left (1-a x^q\right )}{d (m+1)^3} \]
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Rubi [A] time = 0.0758585, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2455, 20, 364} \[ -\frac{q (d x)^{m+1} \text{PolyLog}\left (2,a x^q\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (3,a x^q\right )}{d (m+1)}-\frac{a q^3 x^{q+1} (d x)^m \, _2F_1\left (1,\frac{m+q+1}{q};\frac{m+2 q+1}{q};a x^q\right )}{(m+1)^3 (m+q+1)}-\frac{q^2 (d x)^{m+1} \log \left (1-a x^q\right )}{d (m+1)^3} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 20
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \text{Li}_3\left (a x^q\right ) \, dx &=\frac{(d x)^{1+m} \text{Li}_3\left (a x^q\right )}{d (1+m)}-\frac{q \int (d x)^m \text{Li}_2\left (a x^q\right ) \, dx}{1+m}\\ &=-\frac{q (d x)^{1+m} \text{Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^q\right )}{d (1+m)}-\frac{q^2 \int (d x)^m \log \left (1-a x^q\right ) \, dx}{(1+m)^2}\\ &=-\frac{q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac{q (d x)^{1+m} \text{Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^q\right )}{d (1+m)}-\frac{\left (a q^3\right ) \int \frac{x^{-1+q} (d x)^{1+m}}{1-a x^q} \, dx}{d (1+m)^3}\\ &=-\frac{q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac{q (d x)^{1+m} \text{Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^q\right )}{d (1+m)}-\frac{\left (a q^3 x^{-m} (d x)^m\right ) \int \frac{x^{m+q}}{1-a x^q} \, dx}{(1+m)^3}\\ &=-\frac{a q^3 x^{1+q} (d x)^m \, _2F_1\left (1,\frac{1+m+q}{q};\frac{1+m+2 q}{q};a x^q\right )}{(1+m)^3 (1+m+q)}-\frac{q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac{q (d x)^{1+m} \text{Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^q\right )}{d (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0248751, size = 50, normalized size = 0.38 \[ -\frac{x (d x)^m G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,1-\frac{m+1}{q} \\ 1,0,0,0,-\frac{m+1}{q} \\\end{array}\right )}{q} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.529, size = 180, normalized size = 1.4 \begin{align*} -{\frac{ \left ( dx \right ) ^{m}{x}^{-m}}{q} \left ( -a \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{{q}^{3}{x}^{1+m}\ln \left ( 1-a{x}^{q} \right ) }{ \left ( 1+m \right ) ^{3}} \left ( -a \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{{q}^{2}{x}^{1+m}{\it polylog} \left ( 2,a{x}^{q} \right ) }{ \left ( 1+m \right ) ^{2}} \left ( -a \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{1+m}{\it polylog} \left ( 3,a{x}^{q} \right ) }{1+m} \left ( -a \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{{q}^{3}{x}^{1+m+q}a}{ \left ( 1+m \right ) ^{3}} \left ( -a \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{m} q^{3} \int -\frac{x^{m}}{m^{3} -{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a x^{q} + 3 \, m^{2} + 3 \, m + 1}\,{d x} + \frac{d^{m} q^{3} x x^{m} -{\left (m^{2} q + 2 \, m q + q\right )} d^{m} x x^{m}{\rm Li}_2\left (a x^{q}\right ) -{\left (m q^{2} + q^{2}\right )} d^{m} x x^{m} \log \left (-a x^{q} + 1\right ) +{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} d^{m} x x^{m}{\rm Li}_{3}(a x^{q})}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\rm polylog}\left (3, a x^{q}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \operatorname{Li}_{3}\left (a x^{q}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\rm Li}_{3}(a x^{q})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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