Optimal. Leaf size=93 \[ \frac {8 a q^2 x^q \sqrt {d x} \, _2F_1\left (1,\frac {\frac {1}{2}+q}{q};\frac {1}{2} \left (4+\frac {1}{q}\right );a x^q\right )}{d (1+2 q)}+\frac {4 q \sqrt {d x} \log \left (1-a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {PolyLog}\left (2,a x^q\right )}{d} \]
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Rubi [A]
time = 0.04, antiderivative size = 93, 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 = {6726, 2505, 20,
371} \begin {gather*} \frac {8 a q^2 \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 {2 \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {4 q \sqrt {d x} \log \left (1-a x^q\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^q\right )}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+(2 q) \int \frac {\log \left (1-a x^q\right )}{\sqrt {d x}} \, dx\\ &=\frac {4 q \sqrt {d x} \log \left (1-a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {\left (4 a q^2\right ) \int \frac {x^{-1+q} \sqrt {d x}}{1-a x^q} \, dx}{d}\\ &=\frac {4 q \sqrt {d x} \log \left (1-a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}+\frac {\left (4 a q^2 \sqrt {d x}\right ) \int \frac {x^{-\frac {1}{2}+q}}{1-a x^q} \, dx}{d \sqrt {x}}\\ &=\frac {8 a q^2 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 {4 q \sqrt {d x} \log \left (1-a x^q\right )}{d}+\frac {2 \sqrt {d x} \text {Li}_2\left (a x^q\right )}{d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 0.02, size = 48, normalized size = 0.52 \begin {gather*} -\frac {x G_{4,4}^{1,4}\left (-a x^q|\begin {array}{c} 1,1,1,1-\frac {1}{2 q} \\ 1,0,0,-\frac {1}{2 q} \\\end {array}\right )}{q \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.52, size = 109, normalized size = 1.17
method | result | size |
meijerg | \(-\frac {\sqrt {x}\, \left (-a \right )^{-\frac {1}{2 q}} \left (-4 q^{2} \sqrt {x}\, \left (-a \right )^{\frac {1}{2 q}} \ln \left (1-a \,x^{q}\right )-2 q \sqrt {x}\, \left (-a \right )^{\frac {1}{2 q}} \polylog \left (2, a \,x^{q}\right )-4 q^{2} x^{\frac {1}{2}+q} a \left (-a \right )^{\frac {1}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {1+2 q}{2 q}\right )\right )}{\sqrt {d x}\, q}\) | \(109\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{\sqrt {d x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{\sqrt {d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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