3.2057 \(\int \sqrt{a+\frac{b}{x^4}} x \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]

[Out]

(Sqrt[a + b/x^4]*x^2)/2 - (Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/2

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Rubi [A]  time = 0.106822, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]*x,x]

[Out]

(Sqrt[a + b/x^4]*x^2)/2 - (Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^4]*x^2)])/2

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Rubi in Sympy [A]  time = 8.44782, size = 41, normalized size = 0.84 \[ - \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{2} + \frac{x^{2} \sqrt{a + \frac{b}{x^{4}}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b/x**4)**(1/2),x)

[Out]

-sqrt(b)*atanh(sqrt(b)/(x**2*sqrt(a + b/x**4)))/2 + x**2*sqrt(a + b/x**4)/2

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Mathematica [A]  time = 0.0908435, size = 66, normalized size = 1.35 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}} \left (\sqrt{a x^4+b}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )\right )}{2 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^4]*x,x]

[Out]

(Sqrt[a + b/x^4]*x^2*(Sqrt[b + a*x^4] - Sqrt[b]*ArcTanh[Sqrt[b + a*x^4]/Sqrt[b]]
))/(2*Sqrt[b + a*x^4])

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Maple [A]  time = 0.017, size = 65, normalized size = 1.3 \[ -{\frac{{x}^{2}}{2}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ) -\sqrt{a{x}^{4}+b} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b/x^4)^(1/2),x)

[Out]

-1/2*((a*x^4+b)/x^4)^(1/2)*x^2*(b^(1/2)*ln(2*(b^(1/2)*(a*x^4+b)^(1/2)+b)/x^2)-(a
*x^4+b)^(1/2))/(a*x^4+b)^(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(a + b/x^4)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247167, size = 1, normalized size = 0.02 \[ \left [\frac{1}{2} \, x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \frac{1}{4} \, \sqrt{b} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ), \frac{1}{2} \, x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} - \frac{1}{2} \, \sqrt{-b} \arctan \left (\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-b}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)*x,x, algorithm="fricas")

[Out]

[1/2*x^2*sqrt((a*x^4 + b)/x^4) + 1/4*sqrt(b)*log((a*x^4 - 2*sqrt(b)*x^2*sqrt((a*
x^4 + b)/x^4) + 2*b)/x^4), 1/2*x^2*sqrt((a*x^4 + b)/x^4) - 1/2*sqrt(-b)*arctan(x
^2*sqrt((a*x^4 + b)/x^4)/sqrt(-b))]

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Sympy [A]  time = 6.19021, size = 66, normalized size = 1.35 \[ \frac{\sqrt{a} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{\sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2} + \frac{b}{2 \sqrt{a} x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b/x**4)**(1/2),x)

[Out]

sqrt(a)*x**2/(2*sqrt(1 + b/(a*x**4))) - sqrt(b)*asinh(sqrt(b)/(sqrt(a)*x**2))/2
+ b/(2*sqrt(a)*x**2*sqrt(1 + b/(a*x**4)))

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GIAC/XCAS [A]  time = 0.226718, size = 49, normalized size = 1. \[ \frac{b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{2 \, \sqrt{-b}} + \frac{1}{2} \, \sqrt{a x^{4} + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^4)*x,x, algorithm="giac")

[Out]

1/2*b*arctan(sqrt(a*x^4 + b)/sqrt(-b))/sqrt(-b) + 1/2*sqrt(a*x^4 + b)