Optimal. Leaf size=134 \[ -\frac{8 \sqrt{d x} \text{PolyLog}\left (2,a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{PolyLog}\left (3,a x^2\right )}{d}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{64 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}-\frac{64 \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}+\frac{128 \sqrt{d x}}{d} \]
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Rubi [A] time = 0.101164, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{8 \sqrt{d x} \text{PolyLog}\left (2,a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{PolyLog}\left (3,a x^2\right )}{d}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{64 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}-\frac{64 \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}+\frac{128 \sqrt{d x}}{d} \]
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 \frac{\text{Li}_3\left (a x^2\right )}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-4 \int \frac{\text{Li}_2\left (a x^2\right )}{\sqrt{d x}} \, dx\\ &=-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-16 \int \frac{\log \left (1-a x^2\right )}{\sqrt{d x}} \, dx\\ &=-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-\frac{(64 a) \int \frac{x \sqrt{d x}}{1-a x^2} \, dx}{d}\\ &=-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-\frac{(64 a) \int \frac{(d x)^{3/2}}{1-a x^2} \, dx}{d^2}\\ &=\frac{128 \sqrt{d x}}{d}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-64 \int \frac{1}{\sqrt{d x} \left (1-a x^2\right )} \, dx\\ &=\frac{128 \sqrt{d x}}{d}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-\frac{128 \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{d}\\ &=\frac{128 \sqrt{d x}}{d}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}-64 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )-64 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )\\ &=\frac{128 \sqrt{d x}}{d}-\frac{64 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}-\frac{64 \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{a} \sqrt{d}}-\frac{32 \sqrt{d x} \log \left (1-a x^2\right )}{d}-\frac{8 \sqrt{d x} \text{Li}_2\left (a x^2\right )}{d}+\frac{2 \sqrt{d x} \text{Li}_3\left (a x^2\right )}{d}\\ \end{align*}
Mathematica [C] time = 0.0861366, size = 68, normalized size = 0.51 \[ -\frac{5 x \text{Gamma}\left (\frac{5}{4}\right ) \left (64 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},a x^2\right )+4 \text{PolyLog}\left (2,a x^2\right )-\text{PolyLog}\left (3,a x^2\right )+16 \log \left (1-a x^2\right )-64\right )}{2 \text{Gamma}\left (\frac{9}{4}\right ) \sqrt{d x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.191, size = 147, normalized size = 1.1 \begin{align*} -{\frac{1}{2}\sqrt{x} \left ( 256\,{\frac{\sqrt{x} \left ( -a \right ) ^{5/4}}{a}}+64\,{\frac{\sqrt{x} \left ( -a \right ) ^{5/4} \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 ) }{a\sqrt [4]{a{x}^{2}}}}-64\,{\frac{\sqrt{x} \left ( -a \right ) ^{5/4}\ln \left ( -a{x}^{2}+1 \right ) }{a}}-16\,{\frac{\sqrt{x} \left ( -a \right ) ^{5/4}{\it polylog} \left ( 2,a{x}^{2} \right ) }{a}}+4\,{\frac{\sqrt{x} \left ( -a \right ) ^{5/4}{\it polylog} \left ( 3,a{x}^{2} \right ) }{a}} \right ){\frac{1}{\sqrt{dx}}}{\frac{1}{\sqrt [4]{-a}}}} \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 = 2.8615, size = 521, normalized size = 3.89 \begin{align*} \frac{2 \,{\left (64 \, d \left (\frac{1}{a d^{2}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{d^{2} \sqrt{\frac{1}{a d^{2}}} + d x} a d \left (\frac{1}{a d^{2}}\right )^{\frac{3}{4}} - \sqrt{d x} a d \left (\frac{1}{a d^{2}}\right )^{\frac{3}{4}}\right ) - 16 \, d \left (\frac{1}{a d^{2}}\right )^{\frac{1}{4}} \log \left (d \left (\frac{1}{a d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 16 \, d \left (\frac{1}{a d^{2}}\right )^{\frac{1}{4}} \log \left (-d \left (\frac{1}{a d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 16 \, \sqrt{d x}{\left (\log \left (-a x^{2} + 1\right ) - 4\right )} - 4 \, \sqrt{d x}{\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 ) + \sqrt{d x}{\rm polylog}\left (3, a x^{2}\right )\right )}}{d} \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^{2}\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^{2})}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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