Optimal. Leaf size=117 \[ \frac{2 (d x)^{5/2} \text{PolyLog}(2,a x)}{5 d}+\frac{8 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{25 a^{5/2}}-\frac{8 d \sqrt{d x}}{25 a^2}-\frac{8 (d x)^{3/2}}{75 a}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}-\frac{8 (d x)^{5/2}}{125 d} \]
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Rubi [A] time = 0.0721405, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 (d x)^{5/2} \text{PolyLog}(2,a x)}{5 d}+\frac{8 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{25 a^{5/2}}-\frac{8 d \sqrt{d x}}{25 a^2}-\frac{8 (d x)^{3/2}}{75 a}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}-\frac{8 (d x)^{5/2}}{125 d} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int (d x)^{3/2} \text{Li}_2(a x) \, dx &=\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{2}{5} \int (d x)^{3/2} \log (1-a x) \, dx\\ &=\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{(4 a) \int \frac{(d x)^{5/2}}{1-a x} \, dx}{25 d}\\ &=-\frac{8 (d x)^{5/2}}{125 d}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{4}{25} \int \frac{(d x)^{3/2}}{1-a x} \, dx\\ &=-\frac{8 (d x)^{3/2}}{75 a}-\frac{8 (d x)^{5/2}}{125 d}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{(4 d) \int \frac{\sqrt{d x}}{1-a x} \, dx}{25 a}\\ &=-\frac{8 d \sqrt{d x}}{25 a^2}-\frac{8 (d x)^{3/2}}{75 a}-\frac{8 (d x)^{5/2}}{125 d}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{\left (4 d^2\right ) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{25 a^2}\\ &=-\frac{8 d \sqrt{d x}}{25 a^2}-\frac{8 (d x)^{3/2}}{75 a}-\frac{8 (d x)^{5/2}}{125 d}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}+\frac{(8 d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{25 a^2}\\ &=-\frac{8 d \sqrt{d x}}{25 a^2}-\frac{8 (d x)^{3/2}}{75 a}-\frac{8 (d x)^{5/2}}{125 d}+\frac{8 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{25 a^{5/2}}+\frac{4 (d x)^{5/2} \log (1-a x)}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_2(a x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.102173, size = 90, normalized size = 0.77 \[ \frac{2 (d x)^{3/2} \left (x^{5/2} \text{PolyLog}(2,a x)+\frac{2}{75} \sqrt{x} \left (15 x^2 \log (1-a x)-\frac{2 \left (3 a^2 x^2+5 a x+15\right )}{a^2}\right )+\frac{4 \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{5 a^{5/2}}\right )}{5 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 96, normalized size = 0.8 \begin{align*}{\frac{2\,{\it polylog} \left ( 2,ax \right ) }{5\,d} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{4}{25\,d} \left ( dx \right ) ^{{\frac{5}{2}}}\ln \left ({\frac{-adx+d}{d}} \right ) }-{\frac{8}{125\,d} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{8}{75\,a} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d}{25\,{a}^{2}}\sqrt{dx}}+{\frac{8\,{d}^{2}}{25\,{a}^{2}}{\it Artanh} \left ({a\sqrt{dx}{\frac{1}{\sqrt{ad}}}} \right ){\frac{1}{\sqrt{ad}}}} \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 [A] time = 2.82002, size = 470, normalized size = 4.02 \begin{align*} \left [\frac{2 \,{\left (30 \, d \sqrt{\frac{d}{a}} \log \left (\frac{a d x + 2 \, \sqrt{d x} a \sqrt{\frac{d}{a}} + d}{a x - 1}\right ) +{\left (75 \, a^{2} d x^{2}{\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt{d x}\right )}}{375 \, a^{2}}, -\frac{2 \,{\left (60 \, d \sqrt{-\frac{d}{a}} \arctan \left (\frac{\sqrt{d x} a \sqrt{-\frac{d}{a}}}{d}\right ) -{\left (75 \, a^{2} d x^{2}{\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt{d x}\right )}}{375 \, a^{2}}\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 \left (d x\right )^{\frac{3}{2}}{\rm Li}_2\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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