Optimal. Leaf size=146 \[ \frac {128 (d x)^{3/2}}{81 d}+\frac {64 \sqrt {d} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {64 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {PolyLog}\left (2,a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {PolyLog}\left (3,a x^2\right )}{3 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 146, normalized size of antiderivative = 1.00, 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 = {6726, 2505, 16,
327, 335, 304, 211, 214} \begin {gather*} \frac {64 \sqrt {d} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {64 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}+\frac {128 (d x)^{3/2}}{81 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 211
Rule 214
Rule 304
Rule 327
Rule 335
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \sqrt {d x} \text {Li}_3\left (a x^2\right ) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {4}{3} \int \sqrt {d x} \text {Li}_2\left (a x^2\right ) \, dx\\ &=-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {16}{9} \int \sqrt {d x} \log \left (1-a x^2\right ) \, dx\\ &=-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {(64 a) \int \frac {x (d x)^{3/2}}{1-a x^2} \, dx}{27 d}\\ &=-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {(64 a) \int \frac {(d x)^{5/2}}{1-a x^2} \, dx}{27 d^2}\\ &=\frac {128 (d x)^{3/2}}{81 d}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {64}{27} \int \frac {\sqrt {d x}}{1-a x^2} \, dx\\ &=\frac {128 (d x)^{3/2}}{81 d}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {128 \text {Subst}\left (\int \frac {x^2}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{27 d}\\ &=\frac {128 (d x)^{3/2}}{81 d}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}-\frac {(64 d) \text {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{27 \sqrt {a}}+\frac {(64 d) \text {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{27 \sqrt {a}}\\ &=\frac {128 (d x)^{3/2}}{81 d}+\frac {64 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {64 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/4}}-\frac {32 (d x)^{3/2} \log \left (1-a x^2\right )}{27 d}-\frac {8 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^2\right )}{3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.07, size = 68, normalized size = 0.47 \begin {gather*} -\frac {7 x \sqrt {d x} \Gamma \left (\frac {7}{4}\right ) \left (-64+64 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};a x^2\right )+48 \log \left (1-a x^2\right )+36 \text {PolyLog}\left (2,a x^2\right )-27 \text {PolyLog}\left (3,a x^2\right )\right )}{162 \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 147, normalized size = 1.01
method | result | size |
meijerg | \(-\frac {\sqrt {d x}\, \left (\frac {256 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}}}{81 a}+\frac {64 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{27 a \left (a \,x^{2}\right )^{\frac {3}{4}}}-\frac {64 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \ln \left (-a \,x^{2}+1\right )}{27 a}-\frac {16 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \polylog \left (2, a \,x^{2}\right )}{9 a}+\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \polylog \left (3, a \,x^{2}\right )}{3 a}\right )}{2 \sqrt {x}\, \left (-a \right )^{\frac {3}{4}}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 153, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (48 \, d^{2} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\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} \sqrt {a}}\right )} + 32 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, \log \left (d\right ) + 2\right )} - 36 \, \left (d x\right )^{\frac {3}{2}} {\rm Li}_2\left (a x^{2}\right ) - 48 \, \left (d x\right )^{\frac {3}{2}} \log \left (-a d^{2} x^{2} + d^{2}\right ) + 27 \, \left (d x\right )^{\frac {3}{2}} {\rm Li}_{3}(a x^{2})\right )}}{81 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 187, normalized size = 1.28 \begin {gather*} \frac {2}{3} \, \sqrt {d x} x {\rm polylog}\left (3, a x^{2}\right ) - \frac {8}{81} \, \sqrt {d x} {\left (9 \, x {\rm Li}_2\left (a x^{2}\right ) + 12 \, x \log \left (-a x^{2} + 1\right ) - 16 \, x\right )} - \frac {128}{27} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a d \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} - \sqrt {d^{3} x + a d^{2} \sqrt {\frac {d^{2}}{a^{3}}}} a \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}}}{d^{2}}\right ) - \frac {32}{27} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (32768 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 32768 \, \sqrt {d x} d\right ) + \frac {32}{27} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-32768 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 32768 \, \sqrt {d x} d\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {d x} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {polylog}\left (3,a\,x^2\right )\,\sqrt {d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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