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

Optimal. Leaf size=51 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.112344, antiderivative size = 51, 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 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x}}\right ) \]

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]*ArcTanh[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 = 0.0830906, size = 66, normalized size = 1.29 \[ \frac{2}{3} x \sqrt{\frac{a}{x^2}+b x}-\frac{2 x \sqrt{\frac{a}{x^2}+b x} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 \sqrt{\frac{b x^3}{a}+1}} \]

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 = 0.011, size = 55, normalized size = 1.1 \[{\frac{2\,x}{3}\sqrt{{\frac{b{x}^{3}+a}{{x}^{2}}}} \left ( -\sqrt{a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) +\sqrt{b{x}^{3}+a} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

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*(-a^(1/2)*arctanh((b*x^3+a)^(1/2)/a^(1/2))+(b*x^3+a)
^(1/2))/(b*x^3+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 = 0.238227, 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 = 0.222289, size = 93, normalized size = 1.82 \[ \frac{2}{3} \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{3} + a}\right )}{\rm sign}\left (x\right ) - \frac{2 \,{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{-a}} \]

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