Optimal. Leaf size=97 \[ -\frac{4 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{2 \sqrt{d x} \text{PolyLog}(3,a x)}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}+\frac{16 \sqrt{d x}}{d} \]
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Rubi [A] time = 0.0621491, antiderivative size = 97, 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, 50, 63, 206} \[ -\frac{4 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{2 \sqrt{d x} \text{PolyLog}(3,a x)}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}+\frac{16 \sqrt{d x}}{d} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_3(a x)}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}-2 \int \frac{\text{Li}_2(a x)}{\sqrt{d x}} \, dx\\ &=-\frac{4 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}-4 \int \frac{\log (1-a x)}{\sqrt{d x}} \, dx\\ &=-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{4 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}-\frac{(8 a) \int \frac{\sqrt{d x}}{1-a x} \, dx}{d}\\ &=\frac{16 \sqrt{d x}}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{4 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}-8 \int \frac{1}{\sqrt{d x} (1-a x)} \, dx\\ &=\frac{16 \sqrt{d x}}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{4 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}-\frac{16 \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}\\ &=\frac{16 \sqrt{d x}}{d}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{4 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_3(a x)}{d}\\ \end{align*}
Mathematica [A] time = 0.138969, size = 57, normalized size = 0.59 \[ \frac{2 x \left (-2 \text{PolyLog}(2,a x)+\text{PolyLog}(3,a x)-4 \log (1-a x)-\frac{8 \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{x}}+8\right )}{\sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 127, normalized size = 1.3 \begin{align*}{\frac{1}{a}\sqrt{x}\sqrt{-a} \left ( 16\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}}{a}}+8\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) }{a\sqrt{ax}}}-8\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}\ln \left ( -ax+1 \right ) }{a}}-4\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}{\it polylog} \left ( 2,ax \right ) }{a}}+2\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}{\it polylog} \left ( 3,ax \right ) }{a}} \right ){\frac{1}{\sqrt{dx}}}} \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 = 2.72649, size = 547, normalized size = 5.64 \begin{align*} \left [-\frac{2 \,{\left (2 \, \sqrt{d x} a{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - \sqrt{d x} a{\rm polylog}\left (3, a x\right ) + 4 \, \sqrt{d x}{\left (a \log \left (-a x + 1\right ) - 2 \, a\right )} - 4 \, \sqrt{a d} \log \left (\frac{a d x - 2 \, \sqrt{a d} \sqrt{d x} + d}{a x - 1}\right )\right )}}{a d}, -\frac{2 \,{\left (2 \, \sqrt{d x} a{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - \sqrt{d x} a{\rm polylog}\left (3, a x\right ) + 4 \, \sqrt{d x}{\left (a \log \left (-a x + 1\right ) - 2 \, a\right )} - 8 \, \sqrt{-a d} \arctan \left (\frac{\sqrt{-a d} \sqrt{d x}}{a d x}\right )\right )}}{a d}\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 )}{\sqrt{d x}}\, 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)}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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