3.666 \(\int \frac{1}{\sqrt{x}+\sqrt{x^3}} \, dx\)

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]

ArcTan[Sqrt[x]] + (Sqrt[x^3]*ArcTan[Sqrt[x]])/x^(3/2) + ArcTanh[Sqrt[x]] - (Sqrt
[x^3]*ArcTanh[Sqrt[x]])/x^(3/2)

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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]

ArcTan[Sqrt[x]] + (Sqrt[x^3]*ArcTan[Sqrt[x]])/x^(3/2) + ArcTanh[Sqrt[x]] - (Sqrt
[x^3]*ArcTanh[Sqrt[x]])/x^(3/2)

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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]

-sqrt(x) + sqrt(x**3)/3 + atan(sqrt(x)) + 2*Integral((x - sqrt(x**6))/x, (x, sqr
t(x))) - 2*Integral((x/4 - sqrt(x**6)/4)/(x - 1), (x, sqrt(x))) - 2*Integral((x/
4 - sqrt(x**6)/4)/(x + 1), (x, sqrt(x))) - sqrt(x**3)/x + sqrt(x**3)*atan(sqrt(x
))/x**(3/2)

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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]

Integrate[(Sqrt[x] + Sqrt[x^3])^(-1), x]

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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]

2/((x^3)^(1/2)/x^(3/2))^(1/2)*arctan(((x^3)^(1/2)/x^(3/2))^(1/2)*x^(1/2))

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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]

2*arctan(sqrt(x))

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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]

2*arctan(sqrt(x))

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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]

Integral(1/(sqrt(x) + sqrt(x**3)), x)

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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]

2*arctan(sqrt(x))