Optimal. Leaf size=43 \[ \frac{2}{3} \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0699796, antiderivative size = 43, 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 \[ \frac{2}{3} \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^3]/x,x]
[Out]
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Rubi in Sympy [A] time = 6.95605, size = 37, normalized size = 0.86 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \frac{2 \sqrt{a + b x^{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.101235, size = 48, normalized size = 1.12 \[ \frac{2}{3} \sqrt{a+b x^3} \left (1-\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^3]/x,x]
[Out]
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Maple [A] time = 0.158, size = 32, normalized size = 0.7 \[ -{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}}+{\frac{2}{3}\sqrt{b{x}^{3}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^(1/2)/x,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,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230027, size = 1, normalized size = 0.02 \[ \left [\frac{1}{3} \, \sqrt{a} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + \frac{2}{3} \, \sqrt{b x^{3} + a}, -\frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) + \frac{2}{3} \, \sqrt{b x^{3} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^3 + a)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.01579, size = 76, normalized size = 1.77 \[ - \frac{2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3} + \frac{2 a}{3 \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.269739, size = 49, normalized size = 1.14 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2}{3} \, \sqrt{b x^{3} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^3 + a)/x,x, algorithm="giac")
[Out]