Optimal. Leaf size=132 \[ \frac {64 a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {64 a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6591, 2455, 16, 329, 212, 208, 205} \[ -\frac {8 \text {PolyLog}\left (2,a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {PolyLog}\left (3,a x^2\right )}{3 d (d x)^{3/2}}+\frac {64 a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {64 a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 205
Rule 208
Rule 212
Rule 329
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {4}{3} \int \frac {\text {Li}_2\left (a x^2\right )}{(d x)^{5/2}} \, dx\\ &=-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}-\frac {16}{9} \int \frac {\log \left (1-a x^2\right )}{(d x)^{5/2}} \, dx\\ &=\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {(64 a) \int \frac {x}{(d x)^{3/2} \left (1-a x^2\right )} \, dx}{27 d}\\ &=\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {(64 a) \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )} \, dx}{27 d^2}\\ &=\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {(128 a) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{27 d^3}\\ &=\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}+\frac {(64 a) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{27 d^2}+\frac {(64 a) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{27 d^2}\\ &=\frac {64 a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {64 a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {32 \log \left (1-a x^2\right )}{27 d (d x)^{3/2}}-\frac {8 \text {Li}_2\left (a x^2\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^2\right )}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 71, normalized size = 0.54 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) \left (64 a x^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};a x^2\right )-12 \text {Li}_2\left (a x^2\right )-9 \text {Li}_3\left (a x^2\right )+16 \log \left (1-a x^2\right )\right )}{54 \Gamma \left (\frac {5}{4}\right ) (d x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.61, size = 211, normalized size = 1.60 \[ -\frac {2 \, {\left (64 \, d^{3} x^{2} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a d^{7} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {3}{4}} - \sqrt {d^{6} \sqrt {\frac {a^{3}}{d^{10}}} + a^{2} d x} d^{7} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {3}{4}}}{a^{3}}\right ) - 16 \, d^{3} x^{2} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (32 \, d^{3} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a\right ) + 16 \, d^{3} x^{2} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (-32 \, d^{3} \left (\frac {a^{3}}{d^{10}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a\right ) + 4 \, \sqrt {d x} {\left (3 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right )\right )} + 9 \, \sqrt {d x} {\rm polylog}\left (3, a x^{2}\right )\right )}}{27 \, d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 131, normalized size = 0.99 \[ -\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{4}} \left (-\frac {64 \sqrt {x}\, \left (-a \right )^{\frac {1}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{27 \left (a \,x^{2}\right )^{\frac {1}{4}}}+\frac {64 \left (-a \right )^{\frac {1}{4}} \ln \left (-a \,x^{2}+1\right )}{27 x^{\frac {3}{2}} a}-\frac {16 \left (-a \right )^{\frac {1}{4}} \polylog \left (2, a \,x^{2}\right )}{9 x^{\frac {3}{2}} a}-\frac {4 \left (-a \right )^{\frac {1}{4}} \polylog \left (3, a \,x^{2}\right )}{3 x^{\frac {3}{2}} a}\right )}{2 \left (d x \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 134, normalized size = 1.02 \[ \frac {2 \, {\left (\frac {32 \, a \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d} - \frac {16 \, a \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d} - \frac {12 \, {\rm Li}_2\left (a x^{2}\right ) - 16 \, \log \left (-a d^{2} x^{2} + d^{2}\right ) + 32 \, \log \relax (d) + 9 \, {\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac {3}{2}}}\right )}}{27 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (3,a\,x^2\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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