Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]
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Rubi [A] time = 0.0194627, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^5}} x} \, dx &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^5}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^5}}\right )}{5 b}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}}\\ \end{align*}
Mathematica [B] time = 0.0207192, size = 59, normalized size = 2.19 \[ \frac{2 \sqrt{a x^5+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{5/2}}{\sqrt{a x^5+b}}\right )}{5 \sqrt{a} x^{5/2} \sqrt{a+\frac{b}{x^5}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{a+{\frac{b}{{x}^{5}}}}}}}\, dx \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 [B] time = 5.05926, size = 246, normalized size = 9.11 \begin{align*} \left [\frac{\log \left (-8 \, a^{2} x^{10} - 8 \, a b x^{5} - b^{2} - 4 \,{\left (2 \, a x^{10} + b x^{5}\right )} \sqrt{a} \sqrt{\frac{a x^{5} + b}{x^{5}}}\right )}{10 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{5} \sqrt{\frac{a x^{5} + b}{x^{5}}}}{2 \, a x^{5} + b}\right )}{5 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45086, size = 24, normalized size = 0.89 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} \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{1}{\sqrt{a + \frac{b}{x^{5}}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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