Optimal. Leaf size=85 \[ -\frac{4 \text{PolyLog}(2,a x)}{d \sqrt{d x}}-\frac{2 \text{PolyLog}(3,a x)}{d \sqrt{d x}}+\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0588706, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2395, 63, 206} \[ -\frac{4 \text{PolyLog}(2,a x)}{d \sqrt{d x}}-\frac{2 \text{PolyLog}(3,a x)}{d \sqrt{d x}}+\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_3(a x)}{(d x)^{3/2}} \, dx &=-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+2 \int \frac{\text{Li}_2(a x)}{(d x)^{3/2}} \, dx\\ &=-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}-4 \int \frac{\log (1-a x)}{(d x)^{3/2}} \, dx\\ &=\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+\frac{(8 a) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{d}\\ &=\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+\frac{(16 a) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d^2}\\ &=\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.101918, size = 58, normalized size = 0.68 \[ \frac{2 x \left (-2 \text{PolyLog}(2,a x)-\text{PolyLog}(3,a x)+4 \log (1-a x)+8 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 111, normalized size = 1.3 \begin{align*}{\frac{1}{a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{3}{2}}} \left ( -8\,{\frac{\sqrt{x}\sqrt{-a} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) }{\sqrt{ax}}}+8\,{\frac{\sqrt{-a}\ln \left ( -ax+1 \right ) }{\sqrt{x}a}}-4\,{\frac{\sqrt{-a}{\it polylog} \left ( 2,ax \right ) }{\sqrt{x}a}}-2\,{\frac{\sqrt{-a}{\it polylog} \left ( 3,ax \right ) }{\sqrt{x}a}} \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(-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 = 2.5892, size = 518, normalized size = 6.09 \begin{align*} \left [\frac{2 \,{\left (4 \, d x \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - 2 \, \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 ) + 4 \, \sqrt{d x} \log \left (-a x + 1\right ) - \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{d^{2} x}, -\frac{2 \,{\left (8 \, d x \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + 2 \, \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 ) - 4 \, \sqrt{d x} \log \left (-a x + 1\right ) + \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{d^{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\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)}{\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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