3.85 \(\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{a \sqrt{a+b x^2}} \]

[Out]

x/(a*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.00889905, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x}{a \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 1.27352, size = 12, normalized size = 0.75 \[ \frac{x}{a \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(3/2),x)

[Out]

x/(a*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.01152, size = 16, normalized size = 1. \[ \frac{x}{a \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \[{\frac{x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.35486, size = 19, normalized size = 1.19 \[ \frac{x}{\sqrt{b x^{2} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(sqrt(b*x^2 + a)*a)

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Fricas [A]  time = 0.206676, size = 31, normalized size = 1.94 \[ \frac{\sqrt{b x^{2} + a} x}{a b x^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)*x/(a*b*x^2 + a^2)

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Sympy [A]  time = 1.80627, size = 17, normalized size = 1.06 \[ \frac{x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(3/2),x)

[Out]

x/(a**(3/2)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.224857, size = 19, normalized size = 1.19 \[ \frac{x}{\sqrt{b x^{2} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(sqrt(b*x^2 + a)*a)