Optimal. Leaf size=129 \[ -\frac{16 a q^3 x^{q-1} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (2-\frac{3}{q}\right ),\frac{1}{2} \left (4-\frac{3}{q}\right ),a x^q\right )}{27 d^2 (3-2 q) \sqrt{d x}}-\frac{4 q \text{PolyLog}\left (2,a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{PolyLog}\left (3,a x^q\right )}{3 d (d x)^{3/2}}+\frac{8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}} \]
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Rubi [A] time = 0.0749586, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac{4 q \text{PolyLog}\left (2,a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{PolyLog}\left (3,a x^q\right )}{3 d (d x)^{3/2}}-\frac{16 a q^3 x^{q-1} \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{3}{q}\right );\frac{1}{2} \left (4-\frac{3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt{d x}}+\frac{8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 20
Rule 364
Rubi steps
\begin{align*} \int \frac{\text{Li}_3\left (a x^q\right )}{(d x)^{5/2}} \, dx &=-\frac{2 \text{Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac{1}{3} (2 q) \int \frac{\text{Li}_2\left (a x^q\right )}{(d x)^{5/2}} \, dx\\ &=-\frac{4 q \text{Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}-\frac{1}{9} \left (4 q^2\right ) \int \frac{\log \left (1-a x^q\right )}{(d x)^{5/2}} \, dx\\ &=\frac{8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac{\left (8 a q^3\right ) \int \frac{x^{-1+q}}{(d x)^{3/2} \left (1-a x^q\right )} \, dx}{27 d}\\ &=\frac{8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac{\left (8 a q^3 \sqrt{x}\right ) \int \frac{x^{-\frac{5}{2}+q}}{1-a x^q} \, dx}{27 d^2 \sqrt{d x}}\\ &=-\frac{16 a q^3 x^{-1+q} \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{3}{q}\right );\frac{1}{2} \left (4-\frac{3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt{d x}}+\frac{8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0240634, size = 50, normalized size = 0.39 \[ -\frac{x G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,1+\frac{3}{2 q} \\ 1,0,0,0,\frac{3}{2 q} \\\end{array}\right )}{q (d x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.381, size = 145, normalized size = 1.1 \begin{align*} -{\frac{1}{q}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{3}{2\,q}}} \left ( -{\frac{8\,{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{27} \left ( -a \right ) ^{-{\frac{3}{2\,q}}}{x}^{-{\frac{3}{2}}}}+{\frac{4\,{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{9} \left ( -a \right ) ^{-{\frac{3}{2\,q}}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{q \left ( 1-2/3\,q \right ){\it polylog} \left ( 3,a{x}^{q} \right ) }{ \left ( -3+2\,q \right ){x}^{3/2}} \left ( -a \right ) ^{-3/2\,{q}^{-1}}}-{\frac{8\,{q}^{3}a}{27}{x}^{q-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{3}{2\,q}}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{-3+2\,q}{2\,q}} \right ) } \right ) \left ( dx \right ) ^{-{\frac{5}{2}}}} \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*} 16 \, q^{4} \int \frac{1}{27 \,{\left (a^{2} d^{\frac{5}{2}}{\left (2 \, q + 3\right )} x^{2 \, q} - 2 \, a d^{\frac{5}{2}}{\left (2 \, q + 3\right )} x^{q} + d^{\frac{5}{2}}{\left (2 \, q + 3\right )}\right )} x^{\frac{5}{2}}}\,{d x} - \frac{2 \,{\left (\frac{18 \,{\left ({\left (2 \, q^{2} + 3 \, q\right )} a x x^{q} -{\left (2 \, q^{2} + 3 \, q\right )} x\right )}{\rm Li}_2\left (a x^{q}\right )}{x^{\frac{5}{2}}} - \frac{12 \,{\left ({\left (2 \, q^{3} + 3 \, q^{2}\right )} a x x^{q} -{\left (2 \, q^{3} + 3 \, q^{2}\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac{5}{2}}} + \frac{27 \,{\left (a{\left (2 \, q + 3\right )} x x^{q} -{\left (2 \, q + 3\right )} x\right )}{\rm Li}_{3}(a x^{q})}{x^{\frac{5}{2}}} + \frac{8 \,{\left (2 \, q^{4} x -{\left (2 \, q^{4} + 3 \, q^{3}\right )} a x x^{q}\right )}}{x^{\frac{5}{2}}}\right )}}{81 \,{\left (a d^{\frac{5}{2}}{\left (2 \, q + 3\right )} x^{q} - d^{\frac{5}{2}}{\left (2 \, q + 3\right )}\right )}} \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 (\frac{\sqrt{d x}{\rm polylog}\left (3, a x^{q}\right )}{d^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x^{q})}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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