3.388 \(\int \sqrt{\frac{-a+b x^3}{x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2}{3} x \sqrt{b x-\frac{a}{x^2}}+\frac{2}{3} \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b x-\frac{a}{x^2}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.119749, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2}{3} x \sqrt{b x-\frac{a}{x^2}}+\frac{2}{3} \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b x-\frac{a}{x^2}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-a + b*x^3)/x^2],x]

[Out]

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

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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]

Integral(sqrt((-a + b*x**3)/x**2), x)

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Mathematica [A]  time = 0.0929416, size = 70, normalized size = 1.32 \[ \frac{2}{3} x \sqrt{b x-\frac{a}{x^2}}-\frac{2 x \sqrt{b x-\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{b x^3}{a}}\right )}{3 \sqrt{1-\frac{b x^3}{a}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(-a + b*x^3)/x^2],x]

[Out]

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

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Maple [A]  time = 0.034, size = 73, normalized size = 1.4 \[{\frac{2\,x}{3}\sqrt{{\frac{b{x}^{3}-a}{{x}^{2}}}} \left ( \sqrt{b{x}^{3}-a}\sqrt{-a}+a{\it Artanh} \left ({1\sqrt{b{x}^{3}-a}{\frac{1}{\sqrt{-a}}}} \right ) \right ){\frac{1}{\sqrt{b{x}^{3}-a}}}{\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3-a)/x^2)^(1/2),x)

[Out]

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

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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")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22992, 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")

[Out]

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

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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]

Timed out

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GIAC/XCAS [A]  time = 0.223384, size = 88, normalized size = 1.66 \[ -\frac{2}{3} \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b x^{3} - a}}{\sqrt{a}}\right ) - \sqrt{b x^{3} - a}\right )}{\rm sign}\left (x\right ) + \frac{2}{3} \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{a}}\right ) - \sqrt{-a}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 - a)/x^2),x, algorithm="giac")

[Out]

-2/3*(sqrt(a)*arctan(sqrt(b*x^3 - a)/sqrt(a)) - sqrt(b*x^3 - a))*sign(x) + 2/3*(
sqrt(a)*arctan(sqrt(-a)/sqrt(a)) - sqrt(-a))*sign(x)