Optimal. Leaf size=125 \[ -\frac {16 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}-\frac {32 (d x)^{3/2}}{27 d} \]
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Rubi [A] time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6591, 2455, 16, 321, 329, 298, 205, 208} \[ \frac {2 (d x)^{3/2} \text {PolyLog}\left (2,a x^2\right )}{3 d}-\frac {16 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}-\frac {32 (d x)^{3/2}}{27 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 205
Rule 208
Rule 298
Rule 321
Rule 329
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \sqrt {d x} \text {Li}_2\left (a x^2\right ) \, dx &=\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {4}{3} \int \sqrt {d x} \log \left (1-a x^2\right ) \, dx\\ &=\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {(16 a) \int \frac {x (d x)^{3/2}}{1-a x^2} \, dx}{9 d}\\ &=\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {(16 a) \int \frac {(d x)^{5/2}}{1-a x^2} \, dx}{9 d^2}\\ &=-\frac {32 (d x)^{3/2}}{27 d}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {16}{9} \int \frac {\sqrt {d x}}{1-a x^2} \, dx\\ &=-\frac {32 (d x)^{3/2}}{27 d}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {32 \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {a x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{9 d}\\ &=-\frac {32 (d x)^{3/2}}{27 d}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}+\frac {(16 d) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{9 \sqrt {a}}-\frac {(16 d) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {a} x^2} \, dx,x,\sqrt {d x}\right )}{9 \sqrt {a}}\\ &=-\frac {32 (d x)^{3/2}}{27 d}-\frac {16 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {16 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \text {Li}_2\left (a x^2\right )}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 91, normalized size = 0.73 \[ \frac {2 \sqrt {d x} \left (\frac {4 \left (a^{3/4} x^{3/2} \left (3 \log \left (1-a x^2\right )-4\right )-6 \tan ^{-1}\left (\sqrt [4]{a} \sqrt {x}\right )+6 \tanh ^{-1}\left (\sqrt [4]{a} \sqrt {x}\right )\right )}{a^{3/4}}+9 x^{3/2} \text {Li}_2\left (a x^2\right )\right )}{27 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 172, normalized size = 1.38 \[ \frac {2}{27} \, \sqrt {d x} {\left (9 \, x {\rm Li}_2\left (a x^{2}\right ) + 12 \, x \log \left (-a x^{2} + 1\right ) - 16 \, x\right )} + \frac {32}{9} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a d \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} - \sqrt {d^{3} x + a d^{2} \sqrt {\frac {d^{2}}{a^{3}}}} a \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}}}{d^{2}}\right ) + \frac {8}{9} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (512 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) - \frac {8}{9} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-512 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) \]
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 \[ \int \sqrt {d x} {\rm Li}_2\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 139, normalized size = 1.11 \[ \frac {2 \left (d x \right )^{\frac {3}{2}} \polylog \left (2, a \,x^{2}\right )}{3 d}+\frac {8 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{9 d}-\frac {32 \left (d x \right )^{\frac {3}{2}}}{27 d}-\frac {16 d \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{9 a \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}+\frac {8 d \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{9 a \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 139, normalized size = 1.11 \[ -\frac {2 \, {\left (12 \, d^{2} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}}\right )} + 8 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, \log \relax (d) + 2\right )} - 9 \, \left (d x\right )^{\frac {3}{2}} {\rm Li}_2\left (a x^{2}\right ) - 12 \, \left (d x\right )^{\frac {3}{2}} \log \left (-a d^{2} x^{2} + d^{2}\right )\right )}}{27 \, d} \]
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 \[ \int \mathrm {polylog}\left (2,a\,x^2\right )\,\sqrt {d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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