Optimal. Leaf size=53 \[ \frac{2}{3} x \sqrt{b x-\frac{a}{x^2}}+\frac{2}{3} \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b x-\frac{a}{x^2}}}\right ) \]
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Rubi [A] time = 0.0679645, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1979, 2007, 2029, 203} \[ \frac{2}{3} x \sqrt{b x-\frac{a}{x^2}}+\frac{2}{3} \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b x-\frac{a}{x^2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2007
Rule 2029
Rule 203
Rubi steps
\begin{align*} \int \sqrt{\frac{-a+b x^3}{x^2}} \, dx &=\int \sqrt{-\frac{a}{x^2}+b x} \, dx\\ &=\frac{2}{3} x \sqrt{-\frac{a}{x^2}+b x}-a \int \frac{1}{x^2 \sqrt{-\frac{a}{x^2}+b x}} \, dx\\ &=\frac{2}{3} x \sqrt{-\frac{a}{x^2}+b x}+\frac{1}{3} (2 a) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{1}{x \sqrt{-\frac{a}{x^2}+b x}}\right )\\ &=\frac{2}{3} x \sqrt{-\frac{a}{x^2}+b x}+\frac{2}{3} \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{-\frac{a}{x^2}+b x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0403021, size = 73, normalized size = 1.38 \[ \frac{2 x \sqrt{b x-\frac{a}{x^2}} \left (\sqrt{b x^3-a}-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x^3-a}}{\sqrt{a}}\right )\right )}{3 \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 73, normalized size = 1.4 \begin{align*}{\frac{2\,x}{3}\sqrt{{\frac{b{x}^{3}-a}{{x}^{2}}}} \left ( \sqrt{b{x}^{3}-a}\sqrt{-a}+a{\it Artanh} \left ({\sqrt{b{x}^{3}-a}{\frac{1}{\sqrt{-a}}}} \right ) \right ){\frac{1}{\sqrt{b{x}^{3}-a}}}{\frac{1}{\sqrt{-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{b x^{3} - a}{x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.821793, size = 258, normalized size = 4.87 \begin{align*} \left [\frac{2}{3} \, x \sqrt{\frac{b x^{3} - a}{x^{2}}} + \frac{1}{3} \, \sqrt{-a} \log \left (\frac{b x^{3} - 2 \, \sqrt{-a} x \sqrt{\frac{b x^{3} - a}{x^{2}}} - 2 \, a}{x^{3}}\right ), \frac{2}{3} \, x \sqrt{\frac{b x^{3} - a}{x^{2}}} - \frac{2}{3} \, \sqrt{a} \arctan \left (\frac{x \sqrt{\frac{b x^{3} - a}{x^{2}}}}{\sqrt{a}}\right )\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 [A] time = 1.30164, size = 88, normalized size = 1.66 \begin{align*} -\frac{2}{3} \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b x^{3} - a}}{\sqrt{a}}\right ) - \sqrt{b x^{3} - a}\right )} \mathrm{sgn}\left (x\right ) + \frac{2}{3} \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{a}}\right ) - \sqrt{-a}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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