Optimal. Leaf size=125 \[ -\frac{4 \text{PolyLog}(2,a x)}{25 d (d x)^{5/2}}-\frac{2 \text{PolyLog}(3,a x)}{5 d (d x)^{5/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}} \]
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Rubi [A] time = 0.0763217, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6591, 2395, 51, 63, 206} \[ -\frac{4 \text{PolyLog}(2,a x)}{25 d (d x)^{5/2}}-\frac{2 \text{PolyLog}(3,a x)}{5 d (d x)^{5/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_3(a x)}{(d x)^{7/2}} \, dx &=-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{2}{5} \int \frac{\text{Li}_2(a x)}{(d x)^{7/2}} \, dx\\ &=-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}-\frac{4}{25} \int \frac{\log (1-a x)}{(d x)^{7/2}} \, dx\\ &=\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{(8 a) \int \frac{1}{(d x)^{5/2} (1-a x)} \, dx}{125 d}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (8 a^2\right ) \int \frac{1}{(d x)^{3/2} (1-a x)} \, dx}{125 d^2}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (8 a^3\right ) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{125 d^3}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}+\frac{\left (16 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{125 d^4}\\ &=-\frac{16 a}{375 d^2 (d x)^{3/2}}-\frac{16 a^2}{125 d^3 \sqrt{d x}}+\frac{16 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{125 d^{7/2}}+\frac{8 \log (1-a x)}{125 d (d x)^{5/2}}-\frac{4 \text{Li}_2(a x)}{25 d (d x)^{5/2}}-\frac{2 \text{Li}_3(a x)}{5 d (d x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.13028, size = 72, normalized size = 0.58 \[ -\frac{2 x \left (30 \text{PolyLog}(2,a x)+75 \text{PolyLog}(3,a x)+24 a^2 x^2-24 a^{5/2} x^{5/2} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+8 a x-12 \log (1-a x)\right )}{375 (d x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 135, normalized size = 1.1 \begin{align*}{\frac{1}{a}{x}^{{\frac{7}{2}}} \left ( -a \right ) ^{{\frac{7}{2}}} \left ( -{\frac{16}{375}{x}^{-{\frac{3}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,a}{125}{\frac{1}{\sqrt{x}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{a}^{2}}{125}\sqrt{x} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) \left ( -a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax}}}}+{\frac{8\,\ln \left ( -ax+1 \right ) }{125\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{\it polylog} \left ( 2,ax \right ) }{25\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{\it polylog} \left ( 3,ax \right ) }{5\,a}{x}^{-{\frac{5}{2}}} \left ( -a \right ) ^{-{\frac{3}{2}}}} \right ) \left ( dx \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.85994, size = 629, normalized size = 5.03 \begin{align*} \left [\frac{2 \,{\left (12 \, a^{2} d x^{3} \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - 4 \,{\left (6 \, a^{2} x^{2} + 2 \, a x - 3 \, \log \left (-a x + 1\right )\right )} \sqrt{d x} - 30 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 75 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}, -\frac{2 \,{\left (24 \, a^{2} d x^{3} \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + 4 \,{\left (6 \, a^{2} x^{2} + 2 \, a x - 3 \, \log \left (-a x + 1\right )\right )} \sqrt{d x} + 30 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) + 75 \, \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{375 \, d^{4} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x)}{\left (d x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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