Optimal. Leaf size=136 \[ -\frac {16 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/2}}+\frac {16 d \sqrt {d x}}{125 a^2}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}+\frac {16 (d x)^{3/2}}{375 a}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}+\frac {16 (d x)^{5/2}}{625 d} \]
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Rubi [A] time = 0.08, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {4 (d x)^{5/2} \text {PolyLog}(2,a x)}{25 d}+\frac {2 (d x)^{5/2} \text {PolyLog}(3,a x)}{5 d}-\frac {16 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/2}}+\frac {16 d \sqrt {d x}}{125 a^2}+\frac {16 (d x)^{3/2}}{375 a}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}+\frac {16 (d x)^{5/2}}{625 d} \]
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)^{3/2} \text {Li}_3(a x) \, dx &=\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {2}{5} \int (d x)^{3/2} \text {Li}_2(a x) \, dx\\ &=-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {4}{25} \int (d x)^{3/2} \log (1-a x) \, dx\\ &=-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {(8 a) \int \frac {(d x)^{5/2}}{1-a x} \, dx}{125 d}\\ &=\frac {16 (d x)^{5/2}}{625 d}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {8}{125} \int \frac {(d x)^{3/2}}{1-a x} \, dx\\ &=\frac {16 (d x)^{3/2}}{375 a}+\frac {16 (d x)^{5/2}}{625 d}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {(8 d) \int \frac {\sqrt {d x}}{1-a x} \, dx}{125 a}\\ &=\frac {16 d \sqrt {d x}}{125 a^2}+\frac {16 (d x)^{3/2}}{375 a}+\frac {16 (d x)^{5/2}}{625 d}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {\left (8 d^2\right ) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{125 a^2}\\ &=\frac {16 d \sqrt {d x}}{125 a^2}+\frac {16 (d x)^{3/2}}{375 a}+\frac {16 (d x)^{5/2}}{625 d}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}-\frac {(16 d) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{125 a^2}\\ &=\frac {16 d \sqrt {d x}}{125 a^2}+\frac {16 (d x)^{3/2}}{375 a}+\frac {16 (d x)^{5/2}}{625 d}-\frac {16 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/2}}-\frac {8 (d x)^{5/2} \log (1-a x)}{125 d}-\frac {4 (d x)^{5/2} \text {Li}_2(a x)}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3(a x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 88, normalized size = 0.65 \[ \frac {2 d \sqrt {d x} \left (4 \left (-\frac {30 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \sqrt {x}}+\frac {30}{a^2}-15 x^2 \log (1-a x)+\frac {10 x}{a}+6 x^2\right )-150 x^2 \text {Li}_2(a x)+375 x^2 \text {Li}_3(a x)\right )}{1875} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.53, size = 229, normalized size = 1.68 \[ \left [\frac {2 \, {\left (375 \, \sqrt {d x} a^{2} d x^{2} {\rm polylog}\left (3, a x\right ) + 60 \, d \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right ) - 2 \, {\left (75 \, a^{2} d x^{2} {\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt {d x}\right )}}{1875 \, a^{2}}, \frac {2 \, {\left (375 \, \sqrt {d x} a^{2} d x^{2} {\rm polylog}\left (3, a x\right ) + 120 \, d \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right ) - 2 \, {\left (75 \, a^{2} d x^{2} {\rm Li}_2\left (a x\right ) + 30 \, a^{2} d x^{2} \log \left (-a x + 1\right ) - 12 \, a^{2} d x^{2} - 20 \, a d x - 60 \, d\right )} \sqrt {d x}\right )}}{1875 \, a^{2}}\right ] \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 141, normalized size = 1.04 \[ \frac {\left (d x \right )^{\frac {3}{2}} \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (168 a^{2} x^{2}+280 a x +840\right )}{13125 a^{3}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{125 a^{3} \sqrt {a x}}-\frac {8 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \ln \left (-a x +1\right )}{125 a}-\frac {4 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \polylog \left (2, a x \right )}{25 a}+\frac {2 x^{\frac {5}{2}} \left (-a \right )^{\frac {7}{2}} \polylog \left (3, a x \right )}{5 a}\right )}{x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 143, normalized size = 1.05 \[ \frac {2 \, {\left (\frac {60 \, d^{3} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a^{2}} - \frac {150 \, \left (d x\right )^{\frac {5}{2}} a^{2} {\rm Li}_2\left (a x\right ) + 60 \, \left (d x\right )^{\frac {5}{2}} a^{2} \log \left (-a d x + d\right ) - 375 \, \left (d x\right )^{\frac {5}{2}} a^{2} {\rm Li}_{3}(a x) - 12 \, {\left (5 \, a^{2} \log \relax (d) + 2 \, a^{2}\right )} \left (d x\right )^{\frac {5}{2}} - 40 \, \left (d x\right )^{\frac {3}{2}} a d - 120 \, \sqrt {d x} d^{2}}{a^{2}}\right )}}{1875 \, d} \]
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 \[ \int {\left (d\,x\right )}^{3/2}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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