3.229 \(\int \frac{\sqrt{\frac{a}{x^2}}}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=24 \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0255874, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x^2]/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.47861, size = 24, normalized size = 1. \[ - \frac{2 x \sqrt{\frac{a}{x^{2}}} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a/x**2)**(1/2)/(x**3+1)**(1/2),x)

[Out]

-2*x*sqrt(a/x**2)*atanh(sqrt(x**3 + 1))/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.018103, size = 24, normalized size = 1. \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x^2]/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 19, normalized size = 0.8 \[ -{\frac{2\,x}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{\frac{a}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a/x^2)^(1/2)/(x^3+1)^(1/2),x)

[Out]

-2/3*x*arctanh((x^3+1)^(1/2))*(a/x^2)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a/x^2)/sqrt(x^3 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.285416, size = 1, normalized size = 0.04 \[ \left [\frac{1}{3} \, x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{x^{3} - 2 \, \sqrt{x^{3} + 1} + 2}{x^{3}}\right ), \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{3} + 1} a}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="fricas")

[Out]

[1/3*x*sqrt(a/x^2)*log((x^3 - 2*sqrt(x^3 + 1) + 2)/x^3), 2/3*sqrt(-a)*arctan(sqr
t(-a)*x*sqrt(a/x^2)/(sqrt(x^3 + 1)*a))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a/x**2)**(1/2)/(x**3+1)**(1/2),x)

[Out]

Integral(sqrt(a/x**2)/sqrt((x + 1)*(x**2 - x + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263721, size = 42, normalized size = 1.75 \[ -\frac{1}{3} \, \sqrt{a}{\left ({\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) -{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right )\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="giac")

[Out]

-1/3*sqrt(a)*(ln(sqrt(x^3 + 1) + 1) - ln(abs(sqrt(x^3 + 1) - 1)))*sign(x)