Optimal. Leaf size=118 \[ -\frac {8 a (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};a x^2\right )}{d^3 (m+1)^3 (m+3)}-\frac {2 \text {Li}_2\left (a x^2\right ) (d x)^{m+1}}{d (m+1)^2}+\frac {\text {Li}_3\left (a x^2\right ) (d x)^{m+1}}{d (m+1)}-\frac {4 \log \left (1-a x^2\right ) (d x)^{m+1}}{d (m+1)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 118, 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 = {6591, 2455, 16, 364} \[ -\frac {2 (d x)^{m+1} \text {PolyLog}\left (2,a x^2\right )}{d (m+1)^2}+\frac {(d x)^{m+1} \text {PolyLog}\left (3,a x^2\right )}{d (m+1)}-\frac {8 a (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};a x^2\right )}{d^3 (m+1)^3 (m+3)}-\frac {4 \log \left (1-a x^2\right ) (d x)^{m+1}}{d (m+1)^3} \]
Antiderivative was successfully verified.
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Rule 16
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int (d x)^m \text {Li}_3\left (a x^2\right ) \, dx &=\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {2 \int (d x)^m \text {Li}_2\left (a x^2\right ) \, dx}{1+m}\\ &=-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {4 \int (d x)^m \log \left (1-a x^2\right ) \, dx}{(1+m)^2}\\ &=-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {(8 a) \int \frac {x (d x)^{1+m}}{1-a x^2} \, dx}{d (1+m)^3}\\ &=-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {(8 a) \int \frac {(d x)^{2+m}}{1-a x^2} \, dx}{d^2 (1+m)^3}\\ &=-\frac {8 a (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};a x^2\right )}{d^3 (1+m)^3 (3+m)}-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 126, normalized size = 1.07 \[ -\frac {2 x \Gamma \left (\frac {m+3}{2}\right ) (d x)^m \left (2 a (m+1) x^2 \Gamma \left (\frac {m+1}{2}\right ) \, _2\tilde {F}_1\left (1,\frac {m+3}{2};\frac {m+5}{2};a x^2\right )+m^2 \left (-\text {Li}_3\left (a x^2\right )\right )-2 m \text {Li}_3\left (a x^2\right )+2 (m+1) \text {Li}_2\left (a x^2\right )-\text {Li}_3\left (a x^2\right )+4 \log \left (1-a x^2\right )\right )}{(m+1)^4 \Gamma \left (\frac {m+1}{2}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} {\rm polylog}\left (3, a x^{2}\right ), x\right ) \]
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 \[ \int \left (d x\right )^{m} {\rm Li}_{3}(a x^{2})\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 218, normalized size = 1.85 \[ -\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (8 m +24\right )}{\left (m +3\right ) \left (1+m \right )^{4} a}-\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (4 m +12\right ) \ln \left (-a \,x^{2}+1\right )}{\left (m +3\right ) \left (1+m \right )^{3} a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (-2 m -6\right ) \polylog \left (2, a \,x^{2}\right )}{\left (m +3\right ) \left (1+m \right )^{2} a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \polylog \left (3, a \,x^{2}\right )}{\left (1+m \right ) a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (-4 m -12\right ) \Phi \left (a \,x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{\left (m +3\right ) \left (1+m \right )^{3} a}\right )}{2} \]
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 \[ 8 \, a d^{m} \int \frac {x^{2} x^{m}}{{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a x^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1}\,{d x} - \frac {2 \, d^{m} {\left (m + 1\right )} x x^{m} {\rm Li}_2\left (a x^{2}\right ) - {\left (m^{2} + 2 \, m + 1\right )} d^{m} x x^{m} {\rm Li}_{3}(a x^{2}) + 4 \, d^{m} x x^{m} \log \left (-a x^{2} + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
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 \[ \int \mathrm {polylog}\left (3,a\,x^2\right )\,{\left (d\,x\right )}^m \,d x \]
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 \[ \int \left (d x\right )^{m} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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