Optimal. Leaf size=130 \[ -\frac {a q^3 x^{1+q} (d x)^m \, _2F_1\left (1,\frac {1+m+q}{q};\frac {1+m+2 q}{q};a x^q\right )}{(1+m)^3 (1+m+q)}-\frac {q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \text {PolyLog}\left (2,a x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {PolyLog}\left (3,a x^q\right )}{d (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6726, 2505, 20,
371} \begin {gather*} -\frac {a q^3 x^{q+1} (d x)^m \, _2F_1\left (1,\frac {m+q+1}{q};\frac {m+2 q+1}{q};a x^q\right )}{(m+1)^3 (m+q+1)}-\frac {q (d x)^{m+1} \text {Li}_2\left (a x^q\right )}{d (m+1)^2}+\frac {(d x)^{m+1} \text {Li}_3\left (a x^q\right )}{d (m+1)}-\frac {q^2 (d x)^{m+1} \log \left (1-a x^q\right )}{d (m+1)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int (d x)^m \text {Li}_3\left (a x^q\right ) \, dx &=\frac {(d x)^{1+m} \text {Li}_3\left (a x^q\right )}{d (1+m)}-\frac {q \int (d x)^m \text {Li}_2\left (a x^q\right ) \, dx}{1+m}\\ &=-\frac {q (d x)^{1+m} \text {Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \log \left (1-a x^q\right ) \, dx}{(1+m)^2}\\ &=-\frac {q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \text {Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^q\right )}{d (1+m)}-\frac {\left (a q^3\right ) \int \frac {x^{-1+q} (d x)^{1+m}}{1-a x^q} \, dx}{d (1+m)^3}\\ &=-\frac {q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \text {Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^q\right )}{d (1+m)}-\frac {\left (a q^3 x^{-m} (d x)^m\right ) \int \frac {x^{m+q}}{1-a x^q} \, dx}{(1+m)^3}\\ &=-\frac {a q^3 x^{1+q} (d x)^m \, _2F_1\left (1,\frac {1+m+q}{q};\frac {1+m+2 q}{q};a x^q\right )}{(1+m)^3 (1+m+q)}-\frac {q^2 (d x)^{1+m} \log \left (1-a x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \text {Li}_2\left (a x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^q\right )}{d (1+m)}\\ \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 = 0.38 \begin {gather*} -\frac {x (d x)^m G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1-\frac {1+m}{q} \\ 1,0,0,0,-\frac {1+m}{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 = 0.28, size = 180, normalized size = 1.38
method | result | size |
meijerg | \(-\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-\frac {m}{q}-\frac {1}{q}} \left (\frac {q^{3} x^{1+m} \left (-a \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-a \,x^{q}\right )}{\left (1+m \right )^{3}}+\frac {q^{2} x^{1+m} \left (-a \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{1+m} \left (-a \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (3, a \,x^{q}\right )}{1+m}+\frac {q^{3} x^{1+m +q} a \left (-a \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{3}}\right )}{q}\) | \(180\) |
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 \left (d x\right )^{m} \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 {\left (d\,x\right )}^m\,\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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