Optimal. Leaf size=51 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x}}\right ) \]
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Rubi [A] time = 0.112344, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(a + b*x^3)/x^2],x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{a + b x^{3}}{x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x**3+a)/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0830906, size = 66, normalized size = 1.29 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2 x \sqrt{\frac{a}{x^2}+b x} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(a + b*x^3)/x^2],x]
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Maple [A] time = 0.011, size = 55, normalized size = 1.1 \[{\frac{2\,x}{3}\sqrt{{\frac{b{x}^{3}+a}{{x}^{2}}}} \left ( -\sqrt{a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) +\sqrt{b{x}^{3}+a} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x^3+a)/x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)/x^2),x, algorithm="maxima")
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Fricas [A] time = 0.238227, size = 1, normalized size = 0.02 \[ \left [\frac{2}{3} \, x \sqrt{\frac{b x^{3} + a}{x^{2}}} + \frac{1}{3} \, \sqrt{a} \log \left (\frac{b x^{3} - 2 \, \sqrt{a} x \sqrt{\frac{b x^{3} + a}{x^{2}}} + 2 \, a}{x^{3}}\right ), \frac{2}{3} \, x \sqrt{\frac{b x^{3} + a}{x^{2}}} - \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{x \sqrt{\frac{b x^{3} + a}{x^{2}}}}{\sqrt{-a}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)/x^2),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x**3+a)/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222289, size = 93, normalized size = 1.82 \[ \frac{2}{3} \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{3} + a}\right )}{\rm sign}\left (x\right ) - \frac{2 \,{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)/x^2),x, algorithm="giac")
[Out]