Optimal. Leaf size=97 \[ -\frac{8 a q^2 x^q \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right ),\frac{1}{2} \left (4-\frac{1}{q}\right ),a x^q\right )}{d (1-2 q) \sqrt{d x}}-\frac{2 \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}+\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0599145, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac{2 \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}-\frac{8 a q^2 x^q \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right );\frac{1}{2} \left (4-\frac{1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt{d x}}+\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 20
Rule 364
Rubi steps
\begin{align*} \int \frac{\text{Li}_2\left (a x^q\right )}{(d x)^{3/2}} \, dx &=-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-(2 q) \int \frac{\log \left (1-a x^q\right )}{(d x)^{3/2}} \, dx\\ &=\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (4 a q^2\right ) \int \frac{x^{-1+q}}{\sqrt{d x} \left (1-a x^q\right )} \, dx}{d}\\ &=\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (4 a q^2 \sqrt{x}\right ) \int \frac{x^{-\frac{3}{2}+q}}{1-a x^q} \, dx}{d \sqrt{d x}}\\ &=-\frac{8 a q^2 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{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}\\ \end{align*}
Mathematica [C] time = 0.0290786, size = 48, normalized size = 0.49 \[ -\frac{x G_{4,4}^{1,4}\left (-a x^q|\begin{array}{c} 1,1,1,1+\frac{1}{2 q} \\ 1,0,0,\frac{1}{2 q} \\\end{array}\right )}{q (d x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.224, size = 121, normalized size = 1.3 \begin{align*} -{\frac{1}{q}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{1}{2\,q}}} \left ( -4\,{\frac{{q}^{2}\ln \left ( 1-a{x}^{q} \right ) }{\sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-2\,{\frac{q \left ( 1-2\,q \right ){\it polylog} \left ( 2,a{x}^{q} \right ) }{ \left ( 2\,q-1 \right ) \sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-4\,{q}^{2}{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*} 8 \, q^{3} \int \frac{1}{{\left (2 \, d^{\frac{3}{2}} q +{\left (2 \, a^{2} d^{\frac{3}{2}} q + a^{2} d^{\frac{3}{2}}\right )} x^{2 \, q} - 2 \,{\left (2 \, a d^{\frac{3}{2}} q + a d^{\frac{3}{2}}\right )} x^{q} + d^{\frac{3}{2}}\right )} x^{\frac{3}{2}}}\,{d x} + \frac{2 \,{\left (\frac{{\left ({\left (2 \, a \sqrt{d} q + a \sqrt{d}\right )} x x^{q} -{\left (2 \, \sqrt{d} q + \sqrt{d}\right )} x\right )}{\rm Li}_2\left (a x^{q}\right )}{x^{\frac{3}{2}}} - \frac{2 \,{\left ({\left (2 \, a \sqrt{d} q^{2} + a \sqrt{d} q\right )} x x^{q} -{\left (2 \, \sqrt{d} q^{2} + \sqrt{d} q\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac{3}{2}}} + \frac{4 \,{\left (2 \, \sqrt{d} q^{3} x -{\left (2 \, a \sqrt{d} q^{3} + a \sqrt{d} q^{2}\right )} x x^{q}\right )}}{x^{\frac{3}{2}}}\right )}}{2 \, d^{2} q + d^{2} -{\left (2 \, a d^{2} q + a d^{2}\right )} x^{q}} \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 Li}_2\left (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(-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}_2\left (a x^{q}\right )}{\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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