Optimal. Leaf size=101 \[ \frac {8 a d q^2 \sqrt {d x} x^{q+2} \, _2F_1\left (1,\frac {q+\frac {5}{2}}{q};\frac {1}{2} \left (4+\frac {5}{q}\right );a x^q\right )}{25 (2 q+5)}+\frac {2 (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{5 d}+\frac {4 q (d x)^{5/2} \log \left (1-a x^q\right )}{25 d} \]
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Rubi [A] time = 0.06, antiderivative size = 101, 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} \[ \frac {2 (d x)^{5/2} \text {PolyLog}\left (2,a x^q\right )}{5 d}+\frac {8 a d q^2 \sqrt {d x} x^{q+2} \, _2F_1\left (1,\frac {q+\frac {5}{2}}{q};\frac {1}{2} \left (4+\frac {5}{q}\right );a x^q\right )}{25 (2 q+5)}+\frac {4 q (d x)^{5/2} \log \left (1-a x^q\right )}{25 d} \]
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}_2\left (a x^q\right ) \, dx &=\frac {2 (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{5 d}+\frac {1}{5} (2 q) \int (d x)^{3/2} \log \left (1-a x^q\right ) \, dx\\ &=\frac {4 q (d x)^{5/2} \log \left (1-a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{5 d}+\frac {\left (4 a q^2\right ) \int \frac {x^{-1+q} (d x)^{5/2}}{1-a x^q} \, dx}{25 d}\\ &=\frac {4 q (d x)^{5/2} \log \left (1-a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{5 d}+\frac {\left (4 a d q^2 \sqrt {d x}\right ) \int \frac {x^{\frac {3}{2}+q}}{1-a x^q} \, dx}{25 \sqrt {x}}\\ &=\frac {8 a d q^2 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 )}{25 (5+2 q)}+\frac {4 q (d x)^{5/2} \log \left (1-a x^q\right )}{25 d}+\frac {2 (d x)^{5/2} \text {Li}_2\left (a x^q\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.81 \[ \frac {2 x (d x)^{3/2} \left (4 a q^2 x^q \, _2F_1\left (1,\frac {q+\frac {5}{2}}{q};2+\frac {5}{2 q};a x^q\right )+(2 q+5) \left (5 \text {Li}_2\left (a x^q\right )+2 q \log \left (1-a x^q\right )\right )\right )}{25 (2 q+5)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x} d x {\rm Li}_2\left (a x^{q}\right ), x\right ) \]
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 \[ \int \left (d x\right )^{\frac {3}{2}} {\rm Li}_2\left (a x^{q}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 121, normalized size = 1.20 \[ -\frac {\left (d x \right )^{\frac {3}{2}} \left (-a \right )^{-\frac {5}{2 q}} \left (-\frac {4 q^{2} x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2 q}} \ln \left (1-a \,x^{q}\right )}{25}-\frac {2 q \,x^{\frac {5}{2}} \left (-a \right )^{\frac {5}{2 q}} \left (1+\frac {2 q}{5}\right ) \polylog \left (2, a \,x^{q}\right )}{5+2 q}-\frac {4 q^{2} x^{\frac {5}{2}+q} a \left (-a \right )^{\frac {5}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {5+2 q}{2 q}\right )}{25}\right )}{x^{\frac {3}{2}} q} \]
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 \[ 8 \, d^{\frac {3}{2}} q^{3} \int \frac {x^{\frac {3}{2}}}{25 \, {\left ({\left (2 \, a^{2} q - 5 \, a^{2}\right )} x^{2 \, q} - 2 \, {\left (2 \, a q - 5 \, a\right )} x^{q} + 2 \, q - 5\right )}}\,{d x} + \frac {2 \, {\left (25 \, {\left ({\left (2 \, a d^{\frac {3}{2}} q - 5 \, a d^{\frac {3}{2}}\right )} x x^{q} - {\left (2 \, d^{\frac {3}{2}} q - 5 \, d^{\frac {3}{2}}\right )} x\right )} x^{\frac {3}{2}} {\rm Li}_2\left (a x^{q}\right ) + 10 \, {\left ({\left (2 \, a d^{\frac {3}{2}} q^{2} - 5 \, a d^{\frac {3}{2}} q\right )} x x^{q} - {\left (2 \, d^{\frac {3}{2}} q^{2} - 5 \, d^{\frac {3}{2}} q\right )} x\right )} x^{\frac {3}{2}} \log \left (-a x^{q} + 1\right ) + 4 \, {\left (2 \, d^{\frac {3}{2}} q^{3} x - {\left (2 \, a d^{\frac {3}{2}} q^{3} - 5 \, a d^{\frac {3}{2}} q^{2}\right )} x x^{q}\right )} x^{\frac {3}{2}}\right )}}{125 \, {\left ({\left (2 \, a q - 5 \, a\right )} x^{q} - 2 \, q + 5\right )}} \]
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 \[ \int {\left (d\,x\right )}^{3/2}\,\mathrm {polylog}\left (2,a\,x^q\right ) \,d x \]
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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