Optimal. Leaf size=125 \[ -\frac {16 a}{375 d^2 (d x)^{3/2}}-\frac {16 a^2}{125 d^3 \sqrt {d x}}+\frac {16 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 d^{7/2}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {PolyLog}(2,a x)}{25 d (d x)^{5/2}}-\frac {2 \text {PolyLog}(3,a x)}{5 d (d x)^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 125, 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, 53,
65, 212} \begin {gather*} \frac {16 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 d^{7/2}}-\frac {16 a^2}{125 d^3 \sqrt {d x}}-\frac {16 a}{375 d^2 (d x)^{3/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}} \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)^{7/2}} \, dx &=-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {2}{5} \int \frac {\text {Li}_2(a x)}{(d x)^{7/2}} \, dx\\ &=-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}-\frac {4}{25} \int \frac {\log (1-a x)}{(d x)^{7/2}} \, dx\\ &=\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {(8 a) \int \frac {1}{(d x)^{5/2} (1-a x)} \, dx}{125 d}\\ &=-\frac {16 a}{375 d^2 (d x)^{3/2}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {\left (8 a^2\right ) \int \frac {1}{(d x)^{3/2} (1-a x)} \, dx}{125 d^2}\\ &=-\frac {16 a}{375 d^2 (d x)^{3/2}}-\frac {16 a^2}{125 d^3 \sqrt {d x}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {\left (8 a^3\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{125 d^3}\\ &=-\frac {16 a}{375 d^2 (d x)^{3/2}}-\frac {16 a^2}{125 d^3 \sqrt {d x}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}+\frac {\left (16 a^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{125 d^4}\\ &=-\frac {16 a}{375 d^2 (d x)^{3/2}}-\frac {16 a^2}{125 d^3 \sqrt {d x}}+\frac {16 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 d^{7/2}}+\frac {8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac {4 \text {Li}_2(a x)}{25 d (d x)^{5/2}}-\frac {2 \text {Li}_3(a x)}{5 d (d x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.58 \begin {gather*} -\frac {2 x \left (8 a x+24 a^2 x^2-24 a^{5/2} x^{5/2} \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-12 \log (1-a x)+30 \text {PolyLog}(2,a x)+75 \text {PolyLog}(3,a x)\right )}{375 (d x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 135, normalized size = 1.08
method | result | size |
meijerg | \(\frac {x^{\frac {7}{2}} \left (-a \right )^{\frac {7}{2}} \left (-\frac {16}{375 x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2}}}-\frac {16 a}{125 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}}}-\frac {8 \sqrt {x}\, a^{2} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{125 \left (-a \right )^{\frac {3}{2}} \sqrt {a x}}+\frac {8 \ln \left (-a x +1\right )}{125 x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2}} a}-\frac {4 \polylog \left (2, a x \right )}{25 x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2}} a}-\frac {2 \polylog \left (3, a x \right )}{5 x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2}} a}\right )}{\left (d x \right )^{\frac {7}{2}} a}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 118, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (\frac {12 \, 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 {24 \, a^{2} d^{2} x^{2} + 8 \, a d^{2} x + 30 \, d^{2} {\rm Li}_2\left (a x\right ) - 12 \, d^{2} \log \left (-a d x + d\right ) + 12 \, d^{2} \log \left (d\right ) + 75 \, d^{2} {\rm Li}_{3}(a x)}{\left (d x\right )^{\frac {5}{2}} d^{2}}\right )}}{375 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 195, normalized size = 1.56 \begin {gather*} \left [\frac {2 \, {\left (12 \, 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 ) - 2 \, {\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} - 75 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}, -\frac {2 \, {\left (24 \, a^{2} d x^{3} \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + 2 \, {\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} + 75 \, \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}\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 {7}{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 )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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