Optimal. Leaf size=119 \[ -\frac{16 a q^3 x^q \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right ),\frac{1}{2} \left (4-\frac{1}{q}\right ),a x^q\right )}{d (1-2 q) \sqrt{d x}}-\frac{4 q \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{PolyLog}\left (3,a x^q\right )}{d \sqrt{d x}}+\frac{8 q^2 \log \left (1-a x^q\right )}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0780859, antiderivative size = 119, 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} \[ -\frac{4 q \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{PolyLog}\left (3,a x^q\right )}{d \sqrt{d x}}-\frac{16 a q^3 x^q \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right );\frac{1}{2} \left (4-\frac{1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt{d x}}+\frac{8 q^2 \log \left (1-a x^q\right )}{d \sqrt{d x}} \]
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)^{3/2}} \, dx &=-\frac{2 \text{Li}_3\left (a x^q\right )}{d \sqrt{d x}}+(2 q) \int \frac{\text{Li}_2\left (a x^q\right )}{(d x)^{3/2}} \, dx\\ &=-\frac{4 q \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_3\left (a x^q\right )}{d \sqrt{d x}}-\left (4 q^2\right ) \int \frac{\log \left (1-a x^q\right )}{(d x)^{3/2}} \, dx\\ &=\frac{8 q^2 \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_3\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (8 a q^3\right ) \int \frac{x^{-1+q}}{\sqrt{d x} \left (1-a x^q\right )} \, dx}{d}\\ &=\frac{8 q^2 \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_3\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (8 a q^3 \sqrt{x}\right ) \int \frac{x^{-\frac{3}{2}+q}}{1-a x^q} \, dx}{d \sqrt{d x}}\\ &=-\frac{16 a q^3 x^q \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right );\frac{1}{2} \left (4-\frac{1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt{d x}}+\frac{8 q^2 \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{4 q \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_3\left (a x^q\right )}{d \sqrt{d x}}\\ \end{align*}
Mathematica [C] time = 0.0248691, size = 50, normalized size = 0.42 \[ -\frac{x G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,1+\frac{1}{2 q} \\ 1,0,0,0,\frac{1}{2 q} \\\end{array}\right )}{q (d x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.372, size = 145, normalized size = 1.2 \begin{align*} -{\frac{1}{q}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{1}{2\,q}}} \left ( -8\,{\frac{{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{\sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}+4\,{\frac{{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{\sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-2\,{\frac{q \left ( 1-2\,q \right ){\it polylog} \left ( 3,a{x}^{q} \right ) }{ \left ( 2\,q-1 \right ) \sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-8\,{q}^{3}{x}^{q-1/2}a \left ( -a \right ) ^{-1/2\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,1/2\,{\frac{2\,q-1}{q}} \right ) \right ) \left ( dx \right ) ^{-{\frac{3}{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}{{\left (a^{2} d^{\frac{3}{2}}{\left (2 \, q + 1\right )} x^{2 \, q} - 2 \, a d^{\frac{3}{2}}{\left (2 \, q + 1\right )} x^{q} + d^{\frac{3}{2}}{\left (2 \, q + 1\right )}\right )} x^{\frac{3}{2}}}\,{d x} - \frac{2 \,{\left (\frac{2 \,{\left ({\left (2 \, q^{2} + q\right )} a x x^{q} -{\left (2 \, q^{2} + q\right )} x\right )}{\rm Li}_2\left (a x^{q}\right )}{x^{\frac{3}{2}}} - \frac{4 \,{\left ({\left (2 \, q^{3} + q^{2}\right )} a x x^{q} -{\left (2 \, q^{3} + q^{2}\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac{3}{2}}} + \frac{{\left (a{\left (2 \, q + 1\right )} x x^{q} -{\left (2 \, q + 1\right )} x\right )}{\rm Li}_{3}(a x^{q})}{x^{\frac{3}{2}}} + \frac{8 \,{\left (2 \, q^{4} x -{\left (2 \, q^{4} + q^{3}\right )} a x x^{q}\right )}}{x^{\frac{3}{2}}}\right )}}{a d^{\frac{3}{2}}{\left (2 \, q + 1\right )} x^{q} - d^{\frac{3}{2}}{\left (2 \, q + 1\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^{2} x^{2}}, 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 \frac{\operatorname{Li}_{3}\left (a x^{q}\right )}{\left (d x\right )^{\frac{3}{2}}}\, 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 \frac{{\rm Li}_{3}(a x^{q})}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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