Optimal. Leaf size=129 \[ -\frac {16 a q^3 x^{-1+q} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {PolyLog}\left (2,a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {PolyLog}\left (3,a x^q\right )}{3 d (d x)^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 129, 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 x^{q-1} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}} \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}_3\left (a x^q\right )}{(d x)^{5/2}} \, dx &=-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {1}{3} (2 q) \int \frac {\text {Li}_2\left (a x^q\right )}{(d x)^{5/2}} \, dx\\ &=-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}-\frac {1}{9} \left (4 q^2\right ) \int \frac {\log \left (1-a x^q\right )}{(d x)^{5/2}} \, dx\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q}}{(d x)^{3/2} \left (1-a x^q\right )} \, dx}{27 d}\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}+\frac {\left (8 a q^3 \sqrt {x}\right ) \int \frac {x^{-\frac {5}{2}+q}}{1-a x^q} \, dx}{27 d^2 \sqrt {d x}}\\ &=-\frac {16 a q^3 x^{-1+q} \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {3}{q}\right );\frac {1}{2} \left (4-\frac {3}{q}\right );a x^q\right )}{27 d^2 (3-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{27 d (d x)^{3/2}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{9 d (d x)^{3/2}}-\frac {2 \text {Li}_3\left (a x^q\right )}{3 d (d x)^{3/2}}\\ \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 = 0.39 \begin {gather*} -\frac {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 (d x)^{5/2}} \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 = 0.24, size = 145, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {x^{\frac {5}{2}} \left (-a \right )^{\frac {3}{2 q}} \left (-\frac {8 q^{3} \left (-a \right )^{-\frac {3}{2 q}} \ln \left (1-a \,x^{q}\right )}{27 x^{\frac {3}{2}}}+\frac {4 q^{2} \left (-a \right )^{-\frac {3}{2 q}} \polylog \left (2, a \,x^{q}\right )}{9 x^{\frac {3}{2}}}-\frac {2 q \left (-a \right )^{-\frac {3}{2 q}} \left (1-\frac {2 q}{3}\right ) \polylog \left (3, a \,x^{q}\right )}{\left (-3+2 q \right ) x^{\frac {3}{2}}}-\frac {8 q^{3} x^{q -\frac {3}{2}} a \left (-a \right )^{-\frac {3}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {-3+2 q}{2 q}\right )}{27}\right )}{\left (d x \right )^{\frac {5}{2}} q}\) | \(145\) |
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 \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{\left (d x\right )^{\frac {5}{2}}}\, 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 (3,a\,x^q\right )}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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