3.856 \(\int \frac{x}{\left (a+b x^4\right )^{3/2}} \, dx\)

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

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0170324, 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 \[ \frac{x^2}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 2.50186, size = 15, normalized size = 0.71 \[ \frac{x^{2}}{2 a \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**4+a)**(3/2),x)

[Out]

x**2/(2*a*sqrt(a + b*x**4))

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Mathematica [A]  time = 0.0171319, size = 21, normalized size = 1. \[ \frac{x^2}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43703, size = 23, normalized size = 1.1 \[ \frac{x^{2}}{2 \, \sqrt{b x^{4} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)

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Fricas [A]  time = 0.272596, size = 35, normalized size = 1.67 \[ \frac{\sqrt{b x^{4} + a} x^{2}}{2 \,{\left (a b x^{4} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.80735, size = 20, normalized size = 0.95 \[ \frac{x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**4+a)**(3/2),x)

[Out]

x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a))

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GIAC/XCAS [A]  time = 0.230945, size = 23, normalized size = 1.1 \[ \frac{x^{2}}{2 \, \sqrt{b x^{4} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)