Optimal. Leaf size=52 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
[Out]
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Rubi [A] time = 0.213682, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x] + Sqrt[x^3])^(-1),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \sqrt{x} + \frac{\sqrt{x^{3}}}{3} + \operatorname{atan}{\left (\sqrt{x} \right )} + 2 \int ^{\sqrt{x}} \frac{x - \sqrt{x^{6}}}{x}\, dx - 2 \int ^{\sqrt{x}} \frac{\frac{x}{4} - \frac{\sqrt{x^{6}}}{4}}{x - 1}\, dx - 2 \int ^{\sqrt{x}} \frac{\frac{x}{4} - \frac{\sqrt{x^{6}}}{4}}{x + 1}\, dx - \frac{\sqrt{x^{3}}}{x} + \frac{\sqrt{x^{3}} \operatorname{atan}{\left (\sqrt{x} \right )}}{x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**(1/2)+(x**3)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0863081, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x}+\sqrt{x^3}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(Sqrt[x] + Sqrt[x^3])^(-1),x]
[Out]
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Maple [A] time = 0.014, size = 30, normalized size = 0.6 \[ 2\,{1\arctan \left ( \sqrt{{\frac{\sqrt{{x}^{3}}}{{x}^{3/2}}}}\sqrt{x} \right ){\frac{1}{\sqrt{{\frac{\sqrt{{x}^{3}}}{{x}^{3/2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^(1/2)+(x^3)^(1/2)),x)
[Out]
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Maxima [A] time = 1.16592, size = 8, normalized size = 0.15 \[ 2 \, \arctan \left (\sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3) + sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269641, size = 8, normalized size = 0.15 \[ 2 \, \arctan \left (\sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3) + sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x} + \sqrt{x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**(1/2)+(x**3)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.266514, size = 8, normalized size = 0.15 \[ 2 \, \arctan \left (\sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3) + sqrt(x)),x, algorithm="giac")
[Out]