Optimal. Leaf size=105 \[ -\frac {8 a q^2 x^{q-1} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{9 d^2 (3-2 q) \sqrt {d x}}-\frac {2 \text {Li}_2\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {4 q \log \left (1-a x^q\right )}{9 d (d x)^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac {2 \text {PolyLog}\left (2,a x^q\right )}{3 d (d x)^{3/2}}-\frac {8 a q^2 x^{q-1} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{9 d^2 (3-2 q) \sqrt {d x}}+\frac {4 q \log \left (1-a x^q\right )}{9 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^q\right )}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_2\left (a x^q\right )}{3 d (d x)^{3/2}}-\frac {1}{3} (2 q) \int \frac {\log \left (1-a x^q\right )}{(d x)^{5/2}} \, dx\\ &=\frac {4 q \log \left (1-a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (4 a q^2\right ) \int \frac {x^{-1+q}}{(d x)^{3/2} \left (1-a x^q\right )} \, dx}{9 d}\\ &=\frac {4 q \log \left (1-a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (4 a q^2 \sqrt {x}\right ) \int \frac {x^{-\frac {5}{2}+q}}{1-a x^q} \, dx}{9 d^2 \sqrt {d x}}\\ &=-\frac {8 a q^2 x^{-1+q} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{9 d^2 (3-2 q) \sqrt {d x}}+\frac {4 q \log \left (1-a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_2\left (a x^q\right )}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 48, normalized size = 0.46 \[ -\frac {x G_{4,4}^{1,4}\left (-a x^q|\begin {array}{c} 1,1,1,1+\frac {3}{2 q} \\ 1,0,0,\frac {3}{2 q} \\\end {array}\right )}{q (d x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\rm Li}_2\left (a x^{q}\right )}{d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{q}\right )}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 121, normalized size = 1.15 \[ -\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (-\frac {4 q^{2} \left (-a \right )^{-\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{9 x^{\frac {3}{2}}}-\frac {2 q \left (-a \right )^{-\frac {3}{2 q}} \left (1-\frac {2 q}{3}\right ) \polylog \left (2, a \,x^{q}\right )}{\left (-3+2 q \right ) x^{\frac {3}{2}}}-\frac {4 q^{2} x^{q -\frac {3}{2}} a \left (-a \right )^{-\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {-3+2 q}{2 q}\right )}{9}\right )}{\left (d x \right )^{\frac {5}{2}} q} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 8 \, q^{3} \int \frac {1}{9 \, {\left (2 \, d^{\frac {5}{2}} q + 3 \, d^{\frac {5}{2}} + {\left (2 \, a^{2} d^{\frac {5}{2}} q + 3 \, a^{2} d^{\frac {5}{2}}\right )} x^{2 \, q} - 2 \, {\left (2 \, a d^{\frac {5}{2}} q + 3 \, a d^{\frac {5}{2}}\right )} x^{q}\right )} x^{\frac {5}{2}}}\,{d x} + \frac {2 \, {\left (\frac {9 \, {\left ({\left (2 \, a \sqrt {d} q + 3 \, a \sqrt {d}\right )} x x^{q} - {\left (2 \, \sqrt {d} q + 3 \, \sqrt {d}\right )} x\right )} {\rm Li}_2\left (a x^{q}\right )}{x^{\frac {5}{2}}} - \frac {6 \, {\left ({\left (2 \, a \sqrt {d} q^{2} + 3 \, a \sqrt {d} q\right )} x x^{q} - {\left (2 \, \sqrt {d} q^{2} + 3 \, \sqrt {d} q\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac {5}{2}}} + \frac {4 \, {\left (2 \, \sqrt {d} q^{3} x - {\left (2 \, a \sqrt {d} q^{3} + 3 \, a \sqrt {d} q^{2}\right )} x x^{q}\right )}}{x^{\frac {5}{2}}}\right )}}{27 \, {\left (2 \, d^{3} q + 3 \, d^{3} - {\left (2 \, a d^{3} q + 3 \, a d^{3}\right )} x^{q}\right )}} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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