Optimal. Leaf size=153 \[ -\frac {16 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}+\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}+\frac {16 (d x)^{5/2}}{1715 a}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}+\frac {16 (d x)^{7/2}}{2401 d} \]
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Rubi [A] time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6591, 2395, 50, 63, 206} \[ -\frac {4 (d x)^{7/2} \text {PolyLog}(2,a x)}{49 d}+\frac {2 (d x)^{7/2} \text {PolyLog}(3,a x)}{7 d}+\frac {16 d^2 \sqrt {d x}}{343 a^3}-\frac {16 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}+\frac {16 (d x)^{7/2}}{2401 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int (d x)^{5/2} \text {Li}_3(a x) \, dx &=\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {2}{7} \int (d x)^{5/2} \text {Li}_2(a x) \, dx\\ &=-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {4}{49} \int (d x)^{5/2} \log (1-a x) \, dx\\ &=-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {(8 a) \int \frac {(d x)^{7/2}}{1-a x} \, dx}{343 d}\\ &=\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {8}{343} \int \frac {(d x)^{5/2}}{1-a x} \, dx\\ &=\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {(8 d) \int \frac {(d x)^{3/2}}{1-a x} \, dx}{343 a}\\ &=\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (8 d^2\right ) \int \frac {\sqrt {d x}}{1-a x} \, dx}{343 a^2}\\ &=\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (8 d^3\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{343 a^3}\\ &=\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}-\frac {\left (16 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{343 a^3}\\ &=\frac {16 d^2 \sqrt {d x}}{343 a^3}+\frac {16 d (d x)^{3/2}}{1029 a^2}+\frac {16 (d x)^{5/2}}{1715 a}+\frac {16 (d x)^{7/2}}{2401 d}-\frac {16 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{343 a^{7/2}}-\frac {8 (d x)^{7/2} \log (1-a x)}{343 d}-\frac {4 (d x)^{7/2} \text {Li}_2(a x)}{49 d}+\frac {2 (d x)^{7/2} \text {Li}_3(a x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 98, normalized size = 0.64 \[ \frac {2 (d x)^{5/2} \left (-\frac {840 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \sqrt {x}}+\frac {8 \left (15 a^3 x^3+21 a^2 x^2+35 a x+105\right )}{a^3}-1470 x^3 \text {Li}_2(a x)+5145 x^3 \text {Li}_3(a x)-420 x^3 \log (1-a x)\right )}{36015 x^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.56, size = 279, normalized size = 1.82 \[ \left [\frac {2 \, {\left (5145 \, \sqrt {d x} a^{3} d^{2} x^{3} {\rm polylog}\left (3, a x\right ) + 420 \, d^{2} \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right ) - 2 \, {\left (735 \, a^{3} d^{2} x^{3} {\rm Li}_2\left (a x\right ) + 210 \, a^{3} d^{2} x^{3} \log \left (-a x + 1\right ) - 60 \, a^{3} d^{2} x^{3} - 84 \, a^{2} d^{2} x^{2} - 140 \, a d^{2} x - 420 \, d^{2}\right )} \sqrt {d x}\right )}}{36015 \, a^{3}}, \frac {2 \, {\left (5145 \, \sqrt {d x} a^{3} d^{2} x^{3} {\rm polylog}\left (3, a x\right ) + 840 \, d^{2} \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right ) - 2 \, {\left (735 \, a^{3} d^{2} x^{3} {\rm Li}_2\left (a x\right ) + 210 \, a^{3} d^{2} x^{3} \log \left (-a x + 1\right ) - 60 \, a^{3} d^{2} x^{3} - 84 \, a^{2} d^{2} x^{2} - 140 \, a d^{2} x - 420 \, d^{2}\right )} \sqrt {d x}\right )}}{36015 \, a^{3}}\right ] \]
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 \[ \int \left (d x\right )^{\frac {5}{2}} {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 149, normalized size = 0.97 \[ \frac {\left (d x \right )^{\frac {5}{2}} \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (360 a^{3} x^{3}+504 a^{2} x^{2}+840 a x +2520\right )}{108045 a^{4}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{343 a^{4} \sqrt {a x}}-\frac {8 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \ln \left (-a x +1\right )}{343 a}-\frac {4 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \polylog \left (2, a x \right )}{49 a}+\frac {2 x^{\frac {7}{2}} \left (-a \right )^{\frac {9}{2}} \polylog \left (3, a x \right )}{7 a}\right )}{x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 156, normalized size = 1.02 \[ \frac {2 \, {\left (\frac {420 \, d^{4} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a^{3}} - \frac {1470 \, \left (d x\right )^{\frac {7}{2}} a^{3} {\rm Li}_2\left (a x\right ) + 420 \, \left (d x\right )^{\frac {7}{2}} a^{3} \log \left (-a d x + d\right ) - 5145 \, \left (d x\right )^{\frac {7}{2}} a^{3} {\rm Li}_{3}(a x) - 168 \, \left (d x\right )^{\frac {5}{2}} a^{2} d - 60 \, {\left (7 \, a^{3} \log \relax (d) + 2 \, a^{3}\right )} \left (d x\right )^{\frac {7}{2}} - 280 \, \left (d x\right )^{\frac {3}{2}} a d^{2} - 840 \, \sqrt {d x} d^{3}}{a^{3}}\right )}}{36015 \, d} \]
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 \[ \int {\left (d\,x\right )}^{5/2}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]
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 \[ \int \left (d x\right )^{\frac {5}{2}} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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