3.2081 \(\int \frac{x}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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Rubi [A]  time = 0.0254802, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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Rubi in Sympy [A]  time = 2.54613, size = 15, normalized size = 0.71 \[ \frac{x^{2} \sqrt{a + \frac{b}{x^{4}}}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b/x**4)/(2*a)

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Mathematica [A]  time = 0.018982, size = 21, normalized size = 1. \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^2)/(2*a)

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Maple [A]  time = 0.01, size = 29, normalized size = 1.4 \[{\frac{a{x}^{4}+b}{2\,a{x}^{2}}{\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}} \]

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

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Maxima [A]  time = 1.43566, size = 23, normalized size = 1.1 \[ \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(a + b/x^4)*x^2/a

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Fricas [A]  time = 0.237415, size = 28, normalized size = 1.33 \[ \frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*sqrt((a*x^4 + b)/x^4)/a

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Sympy [A]  time = 2.55025, size = 19, normalized size = 0.9 \[ \frac{\sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*sqrt(a*x**4/b + 1)/(2*a)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(a + b/x^4), x)