Optimal. Leaf size=115 \[ -\frac {16 a q^3 \sqrt {d x} x^q \, _2F_1\left (1,\frac {q+\frac {1}{2}}{q};\frac {1}{2} \left (4+\frac {1}{q}\right );a x^q\right )}{d (2 q+1)}-\frac {4 q \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}-\frac {8 q^2 \sqrt {d x} \log \left (1-a x^q\right )}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 115, 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} \[ -\frac {4 q \sqrt {d x} \text {PolyLog}\left (2,a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {PolyLog}\left (3,a x^q\right )}{d}-\frac {16 a q^3 \sqrt {d x} x^q \, _2F_1\left (1,\frac {q+\frac {1}{2}}{q};\frac {1}{2} \left (4+\frac {1}{q}\right );a x^q\right )}{d (2 q+1)}-\frac {8 q^2 \sqrt {d x} \log \left (1-a x^q\right )}{d} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^q\right )}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}-(2 q) \int \frac {\text {Li}_2\left (a x^q\right )}{\sqrt {d x}} \, dx\\ &=-\frac {4 q \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}-\left (4 q^2\right ) \int \frac {\log \left (1-a x^q\right )}{\sqrt {d x}} \, dx\\ &=-\frac {8 q^2 \sqrt {d x} \log \left (1-a x^q\right )}{d}-\frac {4 q \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}-\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q} \sqrt {d x}}{1-a x^q} \, dx}{d}\\ &=-\frac {8 q^2 \sqrt {d x} \log \left (1-a x^q\right )}{d}-\frac {4 q \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}-\frac {\left (8 a q^3 \sqrt {d x}\right ) \int \frac {x^{-\frac {1}{2}+q}}{1-a x^q} \, dx}{d \sqrt {x}}\\ &=-\frac {16 a q^3 x^q \sqrt {d x} \, _2F_1\left (1,\frac {\frac {1}{2}+q}{q};\frac {1}{2} \left (4+\frac {1}{q}\right );a x^q\right )}{d (1+2 q)}-\frac {8 q^2 \sqrt {d x} \log \left (1-a x^q\right )}{d}-\frac {4 q \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_3\left (a x^q\right )}{d}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.43 \[ -\frac {x G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1-\frac {1}{2 q} \\ 1,0,0,0,-\frac {1}{2 q} \\\end {array}\right )}{q \sqrt {d x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\rm polylog}\left (3, a x^{q}\right )}{d x}, x\right ) \]
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 \[ \int \frac {{\rm Li}_{3}(a x^{q})}{\sqrt {d x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 133, normalized size = 1.16 \[ -\frac {\sqrt {x}\, \left (-a \right )^{-\frac {1}{2 q}} \left (8 q^{3} \sqrt {x}\, \left (-a \right )^{\frac {1}{2 q}} \ln \left (1-a \,x^{q}\right )+4 q^{2} \sqrt {x}\, \left (-a \right )^{\frac {1}{2 q}} \polylog \left (2, a \,x^{q}\right )-2 q \sqrt {x}\, \left (-a \right )^{\frac {1}{2 q}} \polylog \left (3, a \,x^{q}\right )+8 q^{3} x^{\frac {1}{2}+q} a \left (-a \right )^{\frac {1}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {1+2 q}{2 q}\right )\right )}{\sqrt {d x}\, q} \]
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 \[ -16 \, q^{4} \int \frac {1}{{\left (a^{2} \sqrt {d} {\left (2 \, q - 1\right )} x^{2 \, q} - 2 \, a \sqrt {d} {\left (2 \, q - 1\right )} x^{q} + \sqrt {d} {\left (2 \, q - 1\right )}\right )} \sqrt {x}}\,{d x} - \frac {2 \, {\left (\frac {2 \, {\left ({\left (2 \, q^{2} - q\right )} a x x^{q} - {\left (2 \, q^{2} - q\right )} x\right )} {\rm Li}_2\left (a x^{q}\right )}{\sqrt {x}} + \frac {4 \, {\left ({\left (2 \, q^{3} - q^{2}\right )} a x x^{q} - {\left (2 \, q^{3} - q^{2}\right )} x\right )} \log \left (-a x^{q} + 1\right )}{\sqrt {x}} - \frac {{\left (a {\left (2 \, q - 1\right )} x x^{q} - {\left (2 \, q - 1\right )} x\right )} {\rm Li}_{3}(a x^{q})}{\sqrt {x}} + \frac {8 \, {\left (2 \, q^{4} x - {\left (2 \, q^{4} - q^{3}\right )} a x x^{q}\right )}}{\sqrt {x}}\right )}}{a \sqrt {d} {\left (2 \, q - 1\right )} x^{q} - \sqrt {d} {\left (2 \, q - 1\right )}} \]
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 \[ \int \frac {\mathrm {polylog}\left (3,a\,x^q\right )}{\sqrt {d\,x}} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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