Optimal. Leaf size=124 \[ -\frac {16 a q^3 x^{1+q} \sqrt {d x} \, _2F_1\left (1,\frac {\frac {3}{2}+q}{q};\frac {1}{2} \left (4+\frac {3}{q}\right );a x^q\right )}{27 (3+2 q)}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {PolyLog}\left (2,a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {PolyLog}\left (3,a x^q\right )}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 124, 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 = {6726, 2505, 20,
371} \begin {gather*} -\frac {16 a q^3 \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 )}{27 (2 q+3)}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \sqrt {d x} \text {Li}_3\left (a x^q\right ) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {1}{3} (2 q) \int \sqrt {d x} \text {Li}_2\left (a x^q\right ) \, dx\\ &=-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {1}{9} \left (4 q^2\right ) \int \sqrt {d x} \log \left (1-a x^q\right ) \, dx\\ &=-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q} (d x)^{3/2}}{1-a x^q} \, dx}{27 d}\\ &=-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 d}-\frac {\left (8 a q^3 \sqrt {d x}\right ) \int \frac {x^{\frac {1}{2}+q}}{1-a x^q} \, dx}{27 \sqrt {x}}\\ &=-\frac {16 a q^3 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 )}{27 (3+2 q)}-\frac {8 q^2 (d x)^{3/2} \log \left (1-a x^q\right )}{27 d}-\frac {4 q (d x)^{3/2} \text {Li}_2\left (a x^q\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_3\left (a x^q\right )}{3 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 = 50, normalized size = 0.40 \begin {gather*} -\frac {x \sqrt {d x} G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1-\frac {3}{2 q} \\ 1,0,0,0,-\frac {3}{2 q} \\\end {array}\right )}{q} \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 = 0.25, size = 145, normalized size = 1.17
method | result | size |
meijerg | \(-\frac {\sqrt {d x}\, \left (-a \right )^{-\frac {3}{2 q}} \left (\frac {8 q^{3} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{27}+\frac {4 q^{2} x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \polylog \left (2, a \,x^{q}\right )}{9}-\frac {2 q \,x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (1+\frac {2 q}{3}\right ) \polylog \left (3, a \,x^{q}\right )}{3+2 q}+\frac {8 q^{3} x^{\frac {3}{2}+q} a \left (-a \right )^{\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {3+2 q}{2 q}\right )}{27}\right )}{\sqrt {x}\, q}\) | \(145\) |
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 \sqrt {d x} \operatorname {Li}_{3}\left (a x^{q}\right )\, 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 \sqrt {d\,x}\,\mathrm {polylog}\left (3,a\,x^q\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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