Optimal. Leaf size=147 \[ -\frac{8 \text{PolyLog}\left (2,a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}+\frac{64 a^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}+\frac{64 a^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}} \]
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Rubi [A] time = 0.105684, antiderivative size = 147, 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, 325, 329, 212, 208, 205} \[ -\frac{8 \text{PolyLog}\left (2,a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}+\frac{64 a^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}+\frac{64 a^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 16
Rule 325
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\text{Li}_3\left (a x^2\right )}{(d x)^{9/2}} \, dx &=-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{4}{7} \int \frac{\text{Li}_2\left (a x^2\right )}{(d x)^{9/2}} \, dx\\ &=-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}-\frac{16}{49} \int \frac{\log \left (1-a x^2\right )}{(d x)^{9/2}} \, dx\\ &=\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{(64 a) \int \frac{x}{(d x)^{7/2} \left (1-a x^2\right )} \, dx}{343 d}\\ &=\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{(64 a) \int \frac{1}{(d x)^{5/2} \left (1-a x^2\right )} \, dx}{343 d^2}\\ &=-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{\left (64 a^2\right ) \int \frac{1}{\sqrt{d x} \left (1-a x^2\right )} \, dx}{343 d^4}\\ &=-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{\left (128 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{343 d^5}\\ &=-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}+\frac{\left (64 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{343 d^4}+\frac{\left (64 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{343 d^4}\\ &=-\frac{128 a}{1029 d^3 (d x)^{3/2}}+\frac{64 a^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}+\frac{64 a^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{343 d^{9/2}}+\frac{32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac{8 \text{Li}_2\left (a x^2\right )}{49 d (d x)^{7/2}}-\frac{2 \text{Li}_3\left (a x^2\right )}{7 d (d x)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0961169, size = 84, normalized size = 0.57 \[ -\frac{\text{Gamma}\left (-\frac{3}{4}\right ) \sqrt{d x} \left (192 a^2 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},a x^2\right )-84 \text{PolyLog}\left (2,a x^2\right )-147 \text{PolyLog}\left (3,a x^2\right )-64 a x^2+48 \log \left (1-a x^2\right )\right )}{686 d^5 x^4 \text{Gamma}\left (\frac{1}{4}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.191, size = 142, normalized size = 1. \begin{align*} -{\frac{1}{2}{x}^{{\frac{9}{2}}} \left ( -a \right ) ^{{\frac{7}{4}}} \left ( -{\frac{256}{1029}{x}^{-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{3}{4}}}}-{\frac{64\,a}{343}\sqrt{x} \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 ) \left ( -a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a{x}^{2}}}}}+{\frac{64\,\ln \left ( -a{x}^{2}+1 \right ) }{343\,a}{x}^{-{\frac{7}{2}}} \left ( -a \right ) ^{-{\frac{3}{4}}}}-{\frac{16\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{49\,a}{x}^{-{\frac{7}{2}}} \left ( -a \right ) ^{-{\frac{3}{4}}}}-{\frac{4\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{7\,a}{x}^{-{\frac{7}{2}}} \left ( -a \right ) ^{-{\frac{3}{4}}}} \right ) \left ( dx \right ) ^{-{\frac{9}{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 = 3.05735, size = 629, normalized size = 4.28 \begin{align*} -\frac{2 \,{\left (192 \, d^{5} x^{4} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{2} d^{13} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{3}{4}} - \sqrt{d^{10} \sqrt{\frac{a^{7}}{d^{18}}} + a^{4} d x} d^{13} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{3}{4}}}{a^{7}}\right ) - 48 \, d^{5} x^{4} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{1}{4}} \log \left (32 \, d^{5} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{1}{4}} + 32 \, \sqrt{d x} a^{2}\right ) + 48 \, d^{5} x^{4} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{1}{4}} \log \left (-32 \, d^{5} \left (\frac{a^{7}}{d^{18}}\right )^{\frac{1}{4}} + 32 \, \sqrt{d x} a^{2}\right ) + 16 \,{\left (4 \, a x^{2} - 3 \, \log \left (-a x^{2} + 1\right )\right )} \sqrt{d x} + 84 \, \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 ) + 147 \, \sqrt{d x}{\rm polylog}\left (3, a x^{2}\right )\right )}}{1029 \, d^{5} x^{4}} \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 \frac{{\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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