Optimal. Leaf size=100 \[ \frac {8 a q^2 \sqrt {d x} x^{q+1} \, _2F_1\left (1,\frac {q+\frac {3}{2}}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{9 (2 q+3)}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{3 d}+\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d} \]
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Rubi [A] time = 0.05, antiderivative size = 100, 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} \[ \frac {2 (d x)^{3/2} \text {PolyLog}\left (2,a x^q\right )}{3 d}+\frac {8 a q^2 \sqrt {d x} x^{q+1} \, _2F_1\left (1,\frac {q+\frac {3}{2}}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{9 (2 q+3)}+\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \sqrt {d x} \text {Li}_2\left (a x^q\right ) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{3 d}+\frac {1}{3} (2 q) \int \sqrt {d x} \log \left (1-a x^q\right ) \, dx\\ &=\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{3 d}+\frac {\left (4 a q^2\right ) \int \frac {x^{-1+q} (d x)^{3/2}}{1-a x^q} \, dx}{9 d}\\ &=\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{3 d}+\frac {\left (4 a q^2 \sqrt {d x}\right ) \int \frac {x^{\frac {1}{2}+q}}{1-a x^q} \, dx}{9 \sqrt {x}}\\ &=\frac {8 a q^2 x^{1+q} \sqrt {d x} \, _2F_1\left (1,\frac {\frac {3}{2}+q}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{9 (3+2 q)}+\frac {4 q (d x)^{3/2} \log \left (1-a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 82, normalized size = 0.82 \[ \frac {2 x \sqrt {d x} \left (4 a q^2 x^q \, _2F_1\left (1,\frac {q+\frac {3}{2}}{q};2+\frac {3}{2 q};a x^q\right )+(2 q+3) \left (3 \text {Li}_2\left (a x^q\right )+2 q \log \left (1-a x^q\right )\right )\right )}{9 (2 q+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d x} {\rm Li}_2\left (a x^{q}\right ), x\right ) \]
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 \[ \int \sqrt {d x} {\rm Li}_2\left (a x^{q}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 121, normalized size = 1.21 \[ -\frac {\sqrt {d x}\, \left (-a \right )^{-\frac {3}{2 q}} \left (-\frac {4 q^{2} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{9}-\frac {2 q \,x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (1+\frac {2 q}{3}\right ) \polylog \left (2, a \,x^{q}\right )}{3+2 q}-\frac {4 q^{2} x^{\frac {3}{2}+q} a \left (-a \right )^{\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {3+2 q}{2 q}\right )}{9}\right )}{\sqrt {x}\, q} \]
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 \[ 8 \, \sqrt {d} q^{3} \int \frac {\sqrt {x}}{9 \, {\left ({\left (2 \, a^{2} q - 3 \, a^{2}\right )} x^{2 \, q} - 2 \, {\left (2 \, a q - 3 \, a\right )} x^{q} + 2 \, q - 3\right )}}\,{d x} + \frac {2 \, {\left (9 \, {\left ({\left (2 \, a \sqrt {d} q - 3 \, a \sqrt {d}\right )} x x^{q} - {\left (2 \, \sqrt {d} q - 3 \, \sqrt {d}\right )} x\right )} \sqrt {x} {\rm Li}_2\left (a x^{q}\right ) + 6 \, {\left ({\left (2 \, a \sqrt {d} q^{2} - 3 \, a \sqrt {d} q\right )} x x^{q} - {\left (2 \, \sqrt {d} q^{2} - 3 \, \sqrt {d} q\right )} x\right )} \sqrt {x} \log \left (-a x^{q} + 1\right ) + 4 \, {\left (2 \, \sqrt {d} q^{3} x - {\left (2 \, a \sqrt {d} q^{3} - 3 \, a \sqrt {d} q^{2}\right )} x x^{q}\right )} \sqrt {x}\right )}}{27 \, {\left ({\left (2 \, a q - 3 \, a\right )} x^{q} - 2 \, q + 3\right )}} \]
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 \[ \int \sqrt {d\,x}\,\mathrm {polylog}\left (2,a\,x^q\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x} \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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