3.1937 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0241494, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.96324, size = 29, normalized size = 0.83 \[ - \frac{x}{a \sqrt{a + \frac{b}{x^{2}}}} + \frac{2 x \sqrt{a + \frac{b}{x^{2}}}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0218164, size = 27, normalized size = 0.77 \[ \frac{a x^2+2 b}{a^2 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(-3/2),x]

[Out]

(2*b + a*x^2)/(a^2*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.006, size = 37, normalized size = 1.1 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( a{x}^{2}+2\,b \right ) }{{x}^{3}{a}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2),x)

[Out]

(a*x^2+b)*(a*x^2+2*b)/a^2/x^3/((a*x^2+b)/x^2)^(3/2)

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Maxima [A]  time = 1.43021, size = 43, normalized size = 1.23 \[ \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a^{2}} + \frac{b}{\sqrt{a + \frac{b}{x^{2}}} a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23137, size = 53, normalized size = 1.51 \[ \frac{{\left (a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{3} x^{2} + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.94914, size = 42, normalized size = 1.2 \[ \frac{x^{2}}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^2)^(-3/2), x)