Optimal. Leaf size=89 \[ \frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 d^{5/2}}-\frac {8 a}{9 d^2 \sqrt {d x}}-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6591, 2395, 51, 63, 206} \[ -\frac {2 \text {PolyLog}(2,a x)}{3 d (d x)^{3/2}}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 d^{5/2}}-\frac {8 a}{9 d^2 \sqrt {d x}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(a x)}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}-\frac {2}{3} \int \frac {\log (1-a x)}{(d x)^{5/2}} \, dx\\ &=\frac {4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}+\frac {(4 a) \int \frac {1}{(d x)^{3/2} (1-a x)} \, dx}{9 d}\\ &=-\frac {8 a}{9 d^2 \sqrt {d x}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}+\frac {\left (4 a^2\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{9 d^2}\\ &=-\frac {8 a}{9 d^2 \sqrt {d x}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}+\frac {\left (8 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{9 d^3}\\ &=-\frac {8 a}{9 d^2 \sqrt {d x}}+\frac {8 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 d^{5/2}}+\frac {4 \log (1-a x)}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2(a x)}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 57, normalized size = 0.64 \[ -\frac {2 x \left (-4 a^{3/2} x^{3/2} \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+3 \text {Li}_2(a x)+4 a x-2 \log (1-a x)\right )}{9 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 150, normalized size = 1.69 \[ \left [\frac {2 \, {\left (2 \, a d x^{2} \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{9 \, d^{3} x^{2}}, -\frac {2 \, {\left (4 \, a d x^{2} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + {\left (4 \, a x + 3 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{9 \, d^{3} x^{2}}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 76, normalized size = 0.85 \[ -\frac {2 \polylog \left (2, a x \right )}{3 d \left (d x \right )^{\frac {3}{2}}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{9 d \left (d x \right )^{\frac {3}{2}}}+\frac {8 a^{2} \arctanh \left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{9 d^{2} \sqrt {a d}}-\frac {8 a}{9 d^{2} \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 89, normalized size = 1.00 \[ -\frac {2 \, {\left (\frac {2 \, a^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} d} + \frac {4 \, a d x + 3 \, d {\rm Li}_2\left (a x\right ) - 2 \, d \log \left (-a d x + d\right ) + 2 \, d \log \relax (d)}{\left (d x\right )^{\frac {3}{2}} d}\right )}}{9 \, d} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a x\right )}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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