Optimal. Leaf size=161 \[ -\frac{8 (d x)^{5/2} \text{PolyLog}\left (2,a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^2\right )}{5 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}+\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d} \]
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Rubi [A] time = 0.115868, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {6591, 2455, 16, 321, 329, 212, 208, 205} \[ -\frac{8 (d x)^{5/2} \text{PolyLog}\left (2,a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{PolyLog}\left (3,a x^2\right )}{5 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}+\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 16
Rule 321
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int (d x)^{3/2} \text{Li}_3\left (a x^2\right ) \, dx &=\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{4}{5} \int (d x)^{3/2} \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{16}{25} \int (d x)^{3/2} \log \left (1-a x^2\right ) \, dx\\ &=-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(64 a) \int \frac{x (d x)^{5/2}}{1-a x^2} \, dx}{125 d}\\ &=-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(64 a) \int \frac{(d x)^{7/2}}{1-a x^2} \, dx}{125 d^2}\\ &=\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{64}{125} \int \frac{(d x)^{3/2}}{1-a x^2} \, dx\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{\left (64 d^2\right ) \int \frac{1}{\sqrt{d x} \left (1-a x^2\right )} \, dx}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{(128 d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}-\frac{\left (64 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{125 a}-\frac{\left (64 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{125 a}\\ &=\frac{128 d \sqrt{d x}}{125 a}+\frac{128 (d x)^{5/2}}{625 d}-\frac{64 d^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{64 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 a^{5/4}}-\frac{32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac{8 (d x)^{5/2} \text{Li}_2\left (a x^2\right )}{25 d}+\frac{2 (d x)^{5/2} \text{Li}_3\left (a x^2\right )}{5 d}\\ \end{align*}
Mathematica [C] time = 0.0996985, size = 89, normalized size = 0.55 \[ -\frac{9 d \text{Gamma}\left (\frac{9}{4}\right ) \sqrt{d x} \left (320 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},a x^2\right )+100 a x^2 \text{PolyLog}\left (2,a x^2\right )-125 a x^2 \text{PolyLog}\left (3,a x^2\right )-64 a x^2+80 a x^2 \log \left (1-a x^2\right )-320\right )}{1250 a \text{Gamma}\left (\frac{13}{4}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.208, size = 155, normalized size = 1. \begin{align*} -{\frac{1}{2} \left ( dx \right ) ^{{\frac{3}{2}}} \left ({\frac{2304\,a{x}^{2}+11520}{5625\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{9}{4}}}}+{\frac{64}{125\,{a}^{2}}\sqrt{x} \left ( -a \right ) ^{{\frac{9}{4}}} \left ( \ln \left ( 1-\sqrt [4]{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt [4]{a{x}^{2}} \right ) -2\,\arctan \left ( \sqrt [4]{a{x}^{2}} \right ) \right ){\frac{1}{\sqrt [4]{a{x}^{2}}}}}-{\frac{64\,\ln \left ( -a{x}^{2}+1 \right ) }{125\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}}-{\frac{16\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{25\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}}+{\frac{4\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{5\,a}{x}^{{\frac{5}{2}}} \left ( -a \right ) ^{{\frac{9}{4}}}} \right ){x}^{-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{5}{4}}}} \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 = 2.91474, size = 606, normalized size = 3.76 \begin{align*} -\frac{2 \,{\left (100 \, \sqrt{d x} a d x^{2}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) - 125 \, \sqrt{d x} a d x^{2}{\rm polylog}\left (3, a x^{2}\right ) - 320 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{4} d \left (\frac{d^{6}}{a^{5}}\right )^{\frac{3}{4}} - \sqrt{d^{3} x + a^{2} \sqrt{\frac{d^{6}}{a^{5}}}} a^{4} \left (\frac{d^{6}}{a^{5}}\right )^{\frac{3}{4}}}{d^{6}}\right ) + 80 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \log \left (32 \, \sqrt{d x} d + 32 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}}\right ) - 80 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}} \log \left (32 \, \sqrt{d x} d - 32 \, a \left (\frac{d^{6}}{a^{5}}\right )^{\frac{1}{4}}\right ) + 16 \,{\left (5 \, a d x^{2} \log \left (-a x^{2} + 1\right ) - 4 \, a d x^{2} - 20 \, d\right )} \sqrt{d x}\right )}}{625 \, a} \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 \left (d x\right )^{\frac{3}{2}} \operatorname{Li}_{3}\left (a x^{2}\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 \left (d x\right )^{\frac{3}{2}}{\rm Li}_{3}(a x^{2})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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