Optimal. Leaf size=121 \[ \frac {16 \sqrt {d x}}{27 a}+\frac {16 (d x)^{3/2}}{81 d}-\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/2}}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {PolyLog}(2,a x)}{9 d}+\frac {2 (d x)^{3/2} \text {PolyLog}(3,a x)}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6726, 2442, 52,
65, 212} \begin {gather*} -\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/2}}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}+\frac {16 \sqrt {d x}}{27 a}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}+\frac {16 (d x)^{3/2}}{81 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int \sqrt {d x} \text {Li}_3(a x) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {2}{3} \int \sqrt {d x} \text {Li}_2(a x) \, dx\\ &=-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {4}{9} \int \sqrt {d x} \log (1-a x) \, dx\\ &=-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {(8 a) \int \frac {(d x)^{3/2}}{1-a x} \, dx}{27 d}\\ &=\frac {16 (d x)^{3/2}}{81 d}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {8}{27} \int \frac {\sqrt {d x}}{1-a x} \, dx\\ &=\frac {16 \sqrt {d x}}{27 a}+\frac {16 (d x)^{3/2}}{81 d}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {(8 d) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{27 a}\\ &=\frac {16 \sqrt {d x}}{27 a}+\frac {16 (d x)^{3/2}}{81 d}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}-\frac {16 \text {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{27 a}\\ &=\frac {16 \sqrt {d x}}{27 a}+\frac {16 (d x)^{3/2}}{81 d}-\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/2}}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \text {Li}_2(a x)}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3(a x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 73, normalized size = 0.60 \begin {gather*} \frac {2}{81} \sqrt {d x} \left (4 \left (\frac {6}{a}+2 x-\frac {6 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {x}}-3 x \log (1-a x)\right )-18 x \text {PolyLog}(2,a x)+27 x \text {PolyLog}(3,a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 133, normalized size = 1.10
method | result | size |
meijerg | \(\frac {\sqrt {d x}\, \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (40 a x +120\right )}{405 a^{2}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{27 a^{2} \sqrt {a x}}-\frac {8 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \ln \left (-a x +1\right )}{27 a}-\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \polylog \left (2, a x \right )}{9 a}+\frac {2 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \polylog \left (3, a x \right )}{3 a}\right )}{\sqrt {x}\, \sqrt {-a}\, a}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 122, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (\frac {12 \, d^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a} - \frac {18 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_2\left (a x\right ) + 12 \, \left (d x\right )^{\frac {3}{2}} a \log \left (-a d x + d\right ) - 27 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_{3}(a x) - 4 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, a \log \left (d\right ) + 2 \, a\right )} - 24 \, \sqrt {d x} d}{a}\right )}}{81 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 173, normalized size = 1.43 \begin {gather*} \left [\frac {2 \, {\left (27 \, \sqrt {d x} a x {\rm polylog}\left (3, a x\right ) - 2 \, {\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 12 \, \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right )\right )}}{81 \, a}, \frac {2 \, {\left (27 \, \sqrt {d x} a x {\rm polylog}\left (3, a x\right ) - 2 \, {\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 24 \, \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right )\right )}}{81 \, a}\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\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 \sqrt {d\,x}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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