Optimal. Leaf size=108 \[ -\frac{4 \text{PolyLog}(2,a x)}{9 d (d x)^{3/2}}-\frac{2 \text{PolyLog}(3,a x)}{3 d (d x)^{3/2}}+\frac{16 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 d^{5/2}}-\frac{16 a}{27 d^2 \sqrt{d x}}+\frac{8 \log (1-a x)}{27 d (d x)^{3/2}} \]
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Rubi [A] time = 0.0655397, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6591, 2395, 51, 63, 206} \[ -\frac{4 \text{PolyLog}(2,a x)}{9 d (d x)^{3/2}}-\frac{2 \text{PolyLog}(3,a x)}{3 d (d x)^{3/2}}+\frac{16 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 d^{5/2}}-\frac{16 a}{27 d^2 \sqrt{d x}}+\frac{8 \log (1-a x)}{27 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_3(a x)}{(d x)^{5/2}} \, dx &=-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}+\frac{2}{3} \int \frac{\text{Li}_2(a x)}{(d x)^{5/2}} \, dx\\ &=-\frac{4 \text{Li}_2(a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}-\frac{4}{9} \int \frac{\log (1-a x)}{(d x)^{5/2}} \, dx\\ &=\frac{8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac{4 \text{Li}_2(a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}+\frac{(8 a) \int \frac{1}{(d x)^{3/2} (1-a x)} \, dx}{27 d}\\ &=-\frac{16 a}{27 d^2 \sqrt{d x}}+\frac{8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac{4 \text{Li}_2(a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}+\frac{\left (8 a^2\right ) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{27 d^2}\\ &=-\frac{16 a}{27 d^2 \sqrt{d x}}+\frac{8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac{4 \text{Li}_2(a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}+\frac{\left (16 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{27 d^3}\\ &=-\frac{16 a}{27 d^2 \sqrt{d x}}+\frac{16 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{27 d^{5/2}}+\frac{8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac{4 \text{Li}_2(a x)}{9 d (d x)^{3/2}}-\frac{2 \text{Li}_3(a x)}{3 d (d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0961699, size = 64, normalized size = 0.59 \[ -\frac{2 x \left (6 \text{PolyLog}(2,a x)+9 \text{PolyLog}(3,a x)-8 a^{3/2} x^{3/2} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+8 a x-4 \log (1-a x)\right )}{27 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 122, normalized size = 1.1 \begin{align*}{\frac{1}{a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{5}{2}}} \left ( -{\frac{16}{27}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-a}}}}-{\frac{8\,a}{27}\sqrt{x} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{ax}}}}+{\frac{8\,\ln \left ( -ax+1 \right ) }{27\,a}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-a}}}}-{\frac{4\,{\it polylog} \left ( 2,ax \right ) }{9\,a}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-a}}}}-{\frac{2\,{\it polylog} \left ( 3,ax \right ) }{3\,a}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-a}}}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.78928, size = 575, normalized size = 5.32 \begin{align*} \left [\frac{2 \,{\left (4 \, a d x^{2} \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - 4 \,{\left (2 \, a x - \log \left (-a x + 1\right )\right )} \sqrt{d x} - 6 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 9 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}, -\frac{2 \,{\left (8 \, a d x^{2} \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + 4 \,{\left (2 \, a x - \log \left (-a x + 1\right )\right )} \sqrt{d x} + 6 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) + 9 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}\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\right )}{\left (d x\right )^{\frac{5}{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)}{\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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