Optimal. Leaf size=108 \[ -\frac {16 a}{27 d^2 \sqrt {d x}}+\frac {16 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \text {PolyLog}(2,a x)}{9 d (d x)^{3/2}}-\frac {2 \text {PolyLog}(3,a x)}{3 d (d x)^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6726, 2442, 53,
65, 212} \begin {gather*} \frac {16 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}-\frac {16 a}{27 d^2 \sqrt {d x}}-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 212
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(a x)}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}+\frac {2}{3} \int \frac {\text {Li}_2(a x)}{(d x)^{5/2}} \, dx\\ &=-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}-\frac {4}{9} \int \frac {\log (1-a x)}{(d x)^{5/2}} \, dx\\ &=\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}+\frac {(8 a) \int \frac {1}{(d x)^{3/2} (1-a x)} \, dx}{27 d}\\ &=-\frac {16 a}{27 d^2 \sqrt {d x}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}+\frac {\left (8 a^2\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{27 d^2}\\ &=-\frac {16 a}{27 d^2 \sqrt {d x}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}+\frac {\left (16 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{27 d^3}\\ &=-\frac {16 a}{27 d^2 \sqrt {d x}}+\frac {16 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 d^{5/2}}+\frac {8 \log (1-a x)}{27 d (d x)^{3/2}}-\frac {4 \text {Li}_2(a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3(a x)}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 64, normalized size = 0.59 \begin {gather*} -\frac {2 x \left (8 a x-8 a^{3/2} x^{3/2} \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-4 \log (1-a x)+6 \text {PolyLog}(2,a x)+9 \text {PolyLog}(3,a x)\right )}{27 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 122, normalized size = 1.13
method | result | size |
meijerg | \(\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} \left (-\frac {16}{27 \sqrt {x}\, \sqrt {-a}}-\frac {8 \sqrt {x}\, a \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{27 \sqrt {-a}\, \sqrt {a x}}+\frac {8 \ln \left (-a x +1\right )}{27 x^{\frac {3}{2}} \sqrt {-a}\, a}-\frac {4 \polylog \left (2, a x \right )}{9 x^{\frac {3}{2}} \sqrt {-a}\, a}-\frac {2 \polylog \left (3, a x \right )}{3 x^{\frac {3}{2}} \sqrt {-a}\, a}\right )}{\left (d x \right )^{\frac {5}{2}} a}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 97, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (\frac {4 \, a^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} d} + \frac {8 \, a d x + 6 \, d {\rm Li}_2\left (a x\right ) - 4 \, d \log \left (-a d x + d\right ) + 4 \, d \log \left (d\right ) + 9 \, d {\rm Li}_{3}(a x)}{\left (d x\right )^{\frac {3}{2}} d}\right )}}{27 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 175, normalized size = 1.62 \begin {gather*} \left [\frac {2 \, {\left (4 \, a d x^{2} \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - 2 \, {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x} - 9 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}, -\frac {2 \, {\left (8 \, a d x^{2} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + 2 \, {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x} + 9 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{27 \, d^{3} x^{2}}\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 \frac {\operatorname {Li}_{3}\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}}\, 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 \frac {\mathrm {polylog}\left (3,a\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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