Optimal. Leaf size=125 \[ -\frac {16 a d q^3 \sqrt {d x} x^{q+2} \, _2F_1\left (1,\frac {q+\frac {5}{2}}{q};\frac {1}{2} \left (4+\frac {5}{q}\right );a x^q\right )}{125 (2 q+5)}-\frac {4 q (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}-\frac {8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d} \]
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Rubi [A] time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac {4 q (d x)^{5/2} \text {PolyLog}\left (2,a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {PolyLog}\left (3,a x^q\right )}{5 d}-\frac {16 a d q^3 \sqrt {d x} x^{q+2} \, _2F_1\left (1,\frac {q+\frac {5}{2}}{q};\frac {1}{2} \left (4+\frac {5}{q}\right );a x^q\right )}{125 (2 q+5)}-\frac {8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int (d x)^{3/2} \text {Li}_3\left (a x^q\right ) \, dx &=\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}-\frac {1}{5} (2 q) \int (d x)^{3/2} \text {Li}_2\left (a x^q\right ) \, dx\\ &=-\frac {4 q (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}-\frac {1}{25} \left (4 q^2\right ) \int (d x)^{3/2} \log \left (1-a x^q\right ) \, dx\\ &=-\frac {8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac {4 q (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}-\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q} (d x)^{5/2}}{1-a x^q} \, dx}{125 d}\\ &=-\frac {8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac {4 q (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}-\frac {\left (8 a d q^3 \sqrt {d x}\right ) \int \frac {x^{\frac {3}{2}+q}}{1-a x^q} \, dx}{125 \sqrt {x}}\\ &=-\frac {16 a d q^3 x^{2+q} \sqrt {d x} \, _2F_1\left (1,\frac {\frac {5}{2}+q}{q};\frac {1}{2} \left (4+\frac {5}{q}\right );a x^q\right )}{125 (5+2 q)}-\frac {8 q^2 (d x)^{5/2} \log \left (1-a x^q\right )}{125 d}-\frac {4 q (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_3\left (a x^q\right )}{5 d}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 50, normalized size = 0.40 \[ -\frac {x (d x)^{3/2} G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1-\frac {5}{2 q} \\ 1,0,0,0,-\frac {5}{2 q} \\\end {array}\right )}{q} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x} d x {\rm polylog}\left (3, a x^{q}\right ), x\right ) \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} {\rm Li}_{3}(a x^{q})\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 145, normalized size = 1.16 \[ -\frac {\left (d x \right )^{\frac {3}{2}} \left (-a \right )^{-\frac {5}{2 q}} \left (\frac {8 q^{3} x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2 q}} \ln \left (1-a \,x^{q}\right )}{125}+\frac {4 q^{2} x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2 q}} \polylog \left (2, a \,x^{q}\right )}{25}-\frac {2 q \,x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2 q}} \left (1+\frac {2 q}{5}\right ) \polylog \left (3, a \,x^{q}\right )}{5+2 q}+\frac {8 q^{3} x^{\frac {5}{2}+q} a \left (-a \right )^{\frac {5}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {5+2 q}{2 q}\right )}{125}\right )}{x^{\frac {3}{2}} q} \]
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 \[ -16 \, d^{\frac {3}{2}} q^{4} \int \frac {x^{\frac {3}{2}}}{125 \, {\left (a^{2} {\left (2 \, q - 5\right )} x^{2 \, q} - 2 \, a {\left (2 \, q - 5\right )} x^{q} + 2 \, q - 5\right )}}\,{d x} - \frac {2 \, {\left (50 \, {\left ({\left (2 \, q^{2} - 5 \, q\right )} a d^{\frac {3}{2}} x x^{q} - {\left (2 \, q^{2} - 5 \, q\right )} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}} {\rm Li}_2\left (a x^{q}\right ) + 20 \, {\left ({\left (2 \, q^{3} - 5 \, q^{2}\right )} a d^{\frac {3}{2}} x x^{q} - {\left (2 \, q^{3} - 5 \, q^{2}\right )} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}} \log \left (-a x^{q} + 1\right ) - 125 \, {\left (a d^{\frac {3}{2}} {\left (2 \, q - 5\right )} x x^{q} - d^{\frac {3}{2}} {\left (2 \, q - 5\right )} x\right )} x^{\frac {3}{2}} {\rm Li}_{3}(a x^{q}) + 8 \, {\left (2 \, d^{\frac {3}{2}} q^{4} x - {\left (2 \, q^{4} - 5 \, q^{3}\right )} a d^{\frac {3}{2}} x x^{q}\right )} x^{\frac {3}{2}}\right )}}{625 \, {\left (a {\left (2 \, q - 5\right )} x^{q} - 2 \, q + 5\right )}} \]
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 \[ \int {\left (d\,x\right )}^{3/2}\,\mathrm {polylog}\left (3,a\,x^q\right ) \,d x \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{3}\left (a x^{q}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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