Optimal. Leaf size=124 \[ -\frac{16 a q^3 \sqrt{d x} x^{q+1} \text{Hypergeometric2F1}\left (1,\frac{q+\frac{3}{2}}{q},\frac{1}{2} \left (\frac{3}{q}+4\right ),a x^q\right )}{27 (2 q+3)}-\frac{4 q (d x)^{3/2} \text{PolyLog}\left (2,a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}\left (3,a x^q\right )}{3 d}-\frac{8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d} \]
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Rubi [A] time = 0.0703269, antiderivative size = 124, 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} \[ -\frac{4 q (d x)^{3/2} \text{PolyLog}\left (2,a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{PolyLog}\left (3,a x^q\right )}{3 d}-\frac{16 a q^3 \sqrt{d x} x^{q+1} \, _2F_1\left (1,\frac{q+\frac{3}{2}}{q};\frac{1}{2} \left (4+\frac{3}{q}\right );a x^q\right )}{27 (2 q+3)}-\frac{8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 20
Rule 364
Rubi steps
\begin{align*} \int \sqrt{d x} \text{Li}_3\left (a x^q\right ) \, dx &=\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^q\right )}{3 d}-\frac{1}{3} (2 q) \int \sqrt{d x} \text{Li}_2\left (a x^q\right ) \, dx\\ &=-\frac{4 q (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^q\right )}{3 d}-\frac{1}{9} \left (4 q^2\right ) \int \sqrt{d x} \log \left (1-a x^q\right ) \, dx\\ &=-\frac{8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac{4 q (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^q\right )}{3 d}-\frac{\left (8 a q^3\right ) \int \frac{x^{-1+q} (d x)^{3/2}}{1-a x^q} \, dx}{27 d}\\ &=-\frac{8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac{4 q (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^q\right )}{3 d}-\frac{\left (8 a q^3 \sqrt{d x}\right ) \int \frac{x^{\frac{1}{2}+q}}{1-a x^q} \, dx}{27 \sqrt{x}}\\ &=-\frac{16 a q^3 x^{1+q} \sqrt{d x} \, _2F_1\left (1,\frac{\frac{3}{2}+q}{q};\frac{1}{2} \left (4+\frac{3}{q}\right );a x^q\right )}{27 (3+2 q)}-\frac{8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac{4 q (d x)^{3/2} \text{Li}_2\left (a x^q\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_3\left (a x^q\right )}{3 d}\\ \end{align*}
Mathematica [C] time = 0.0262881, size = 50, normalized size = 0.4 \[ -\frac{x \sqrt{d x} G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,1-\frac{3}{2 q} \\ 1,0,0,0,-\frac{3}{2 q} \\\end{array}\right )}{q} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.381, size = 145, normalized size = 1.2 \begin{align*} -{\frac{1}{q}\sqrt{dx} \left ( -a \right ) ^{-{\frac{3}{2\,q}}} \left ({\frac{8\,{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{27}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{3}{2\,q}}}}+{\frac{4\,{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{9}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{3}{2\,q}}}}-2\,{\frac{q{x}^{3/2} \left ( 1+2/3\,q \right ){\it polylog} \left ( 3,a{x}^{q} \right ) }{3+2\,q} \left ( -a \right ) ^{3/2\,{q}^{-1}}}+{\frac{8\,{q}^{3}a}{27}{x}^{{\frac{3}{2}}+q} \left ( -a \right ) ^{{\frac{3}{2\,q}}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{3+2\,q}{2\,q}} \right ) } \right ){\frac{1}{\sqrt{x}}}} \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 \, \sqrt{d} q^{4} \int \frac{\sqrt{x}}{27 \,{\left (a^{2}{\left (2 \, q - 3\right )} x^{2 \, q} - 2 \, a{\left (2 \, q - 3\right )} x^{q} + 2 \, q - 3\right )}}\,{d x} - \frac{2 \,{\left (18 \,{\left ({\left (2 \, q^{2} - 3 \, q\right )} a \sqrt{d} x x^{q} -{\left (2 \, q^{2} - 3 \, q\right )} \sqrt{d} x\right )} \sqrt{x}{\rm Li}_2\left (a x^{q}\right ) + 12 \,{\left ({\left (2 \, q^{3} - 3 \, q^{2}\right )} a \sqrt{d} x x^{q} -{\left (2 \, q^{3} - 3 \, q^{2}\right )} \sqrt{d} x\right )} \sqrt{x} \log \left (-a x^{q} + 1\right ) - 27 \,{\left (a \sqrt{d}{\left (2 \, q - 3\right )} x x^{q} - \sqrt{d}{\left (2 \, q - 3\right )} x\right )} \sqrt{x}{\rm Li}_{3}(a x^{q}) + 8 \,{\left (2 \, \sqrt{d} q^{4} x -{\left (2 \, q^{4} - 3 \, q^{3}\right )} a \sqrt{d} x x^{q}\right )} \sqrt{x}\right )}}{81 \,{\left (a{\left (2 \, q - 3\right )} x^{q} - 2 \, q + 3\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 (\sqrt{d x}{\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 \sqrt{d x} \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 \sqrt{d x}{\rm Li}_{3}(a x^{q})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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