Optimal. Leaf size=29 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]
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Rubi [A] time = 0.0196157, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 63, 205} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{b}{x^5}-a}}{\sqrt{a}}\right )}{5 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 205
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 \tan ^{-1}\left (\frac{\sqrt{-a+\frac{b}{x^5}}}{\sqrt{a}}\right )}{5 \sqrt{a}}\\ \end{align*}
Mathematica [B] time = 0.0343614, size = 65, normalized size = 2.24 \[ \frac{2 \sqrt{a x^5-b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{5/2}}{\sqrt{a x^5-b}}\right )}{5 \sqrt{a} x^{5/2} \sqrt{\frac{b}{x^5}-a}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, 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 = 5.16746, size = 250, normalized size = 8.62 \begin{align*} \left [-\frac{\sqrt{-a} \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 \, a}, -\frac{\arctan \left (\frac{2 \, \sqrt{a} x^{5} \sqrt{-\frac{a x^{5} - b}{x^{5}}}}{2 \, a x^{5} - b}\right )}{5 \, \sqrt{a}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.56956, size = 61, normalized size = 2.1 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{for}\: \frac{\left |{a x^{5}}\right |}{\left |{b}\right |} > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{a} x^{\frac{5}{2}}}{\sqrt{b}} \right )}}{5 \sqrt{a}} & \text{otherwise} \end{cases} \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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