Optimal. Leaf size=106 \[ \frac {8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{25 d^{7/2}}-\frac {8 a^2}{25 d^3 \sqrt {d x}}-\frac {8 a}{75 d^2 (d x)^{3/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 106, 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 = {6591, 2395, 51, 63, 206} \[ -\frac {2 \text {PolyLog}(2,a x)}{5 d (d x)^{5/2}}-\frac {8 a^2}{25 d^3 \sqrt {d x}}+\frac {8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{25 d^{7/2}}-\frac {8 a}{75 d^2 (d x)^{3/2}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}} \]
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)^{7/2}} \, dx &=-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}-\frac {2}{5} \int \frac {\log (1-a x)}{(d x)^{7/2}} \, dx\\ &=\frac {4 \log (1-a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}+\frac {(4 a) \int \frac {1}{(d x)^{5/2} (1-a x)} \, dx}{25 d}\\ &=-\frac {8 a}{75 d^2 (d x)^{3/2}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}+\frac {\left (4 a^2\right ) \int \frac {1}{(d x)^{3/2} (1-a x)} \, dx}{25 d^2}\\ &=-\frac {8 a}{75 d^2 (d x)^{3/2}}-\frac {8 a^2}{25 d^3 \sqrt {d x}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}+\frac {\left (4 a^3\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{25 d^3}\\ &=-\frac {8 a}{75 d^2 (d x)^{3/2}}-\frac {8 a^2}{25 d^3 \sqrt {d x}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}+\frac {\left (8 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{25 d^4}\\ &=-\frac {8 a}{75 d^2 (d x)^{3/2}}-\frac {8 a^2}{25 d^3 \sqrt {d x}}+\frac {8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{25 d^{7/2}}+\frac {4 \log (1-a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_2(a x)}{5 d (d x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 65, normalized size = 0.61 \[ -\frac {2 x \left (-12 a^{5/2} x^{5/2} \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+12 a^2 x^2+15 \text {Li}_2(a x)+4 a x-6 \log (1-a x)\right )}{75 (d x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 170, normalized size = 1.60 \[ \left [\frac {2 \, {\left (6 \, a^{2} d x^{3} \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - {\left (12 \, a^{2} x^{2} + 4 \, a x + 15 \, {\rm Li}_2\left (a x\right ) - 6 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{75 \, d^{4} x^{3}}, -\frac {2 \, {\left (12 \, a^{2} d x^{3} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + {\left (12 \, a^{2} x^{2} + 4 \, a x + 15 \, {\rm Li}_2\left (a x\right ) - 6 \, \log \left (-a x + 1\right )\right )} \sqrt {d x}\right )}}{75 \, d^{4} x^{3}}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 89, normalized size = 0.84 \[ -\frac {2 \polylog \left (2, a x \right )}{5 d \left (d x \right )^{\frac {5}{2}}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{25 d \left (d x \right )^{\frac {5}{2}}}-\frac {8 a}{75 d^{2} \left (d x \right )^{\frac {3}{2}}}-\frac {8 a^{2}}{25 d^{3} \sqrt {d x}}+\frac {8 a^{3} \arctanh \left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{25 d^{3} \sqrt {a d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 108, normalized size = 1.02 \[ -\frac {2 \, {\left (\frac {6 \, a^{3} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} d^{2}} + \frac {12 \, a^{2} d^{2} x^{2} + 4 \, a d^{2} x + 15 \, d^{2} {\rm Li}_2\left (a x\right ) - 6 \, d^{2} \log \left (-a d x + d\right ) + 6 \, d^{2} \log \relax (d)}{\left (d x\right )^{\frac {5}{2}} d^{2}}\right )}}{75 \, d} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x\right )}{{\left (d\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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