3.793 \(\int \frac{1}{\sqrt{\left (a^2+x^2\right )^3}} \, dx\)

Optimal. Leaf size=25 \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

[Out]

(x*(a^2 + x^2))/(a^2*Sqrt[(a^2 + x^2)^3])

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Rubi [A]  time = 0.0305376, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(a^2 + x^2)^3],x]

[Out]

(x*(a^2 + x^2))/(a^2*Sqrt[(a^2 + x^2)^3])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (a^{2} + x^{2}\right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt((a**2 + x**2)**3), x)

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Mathematica [A]  time = 0.0380825, size = 25, normalized size = 1. \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(a^2 + x^2)^3],x]

[Out]

(x*(a^2 + x^2))/(a^2*Sqrt[(a^2 + x^2)^3])

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Maple [A]  time = 0.006, size = 24, normalized size = 1. \[{\frac{x \left ({a}^{2}+{x}^{2} \right ) }{{a}^{2}}{\frac{1}{\sqrt{ \left ({a}^{2}+{x}^{2} \right ) ^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((a^2+x^2)^3)^(1/2),x)

[Out]

x*(a^2+x^2)/a^2/((a^2+x^2)^3)^(1/2)

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Maxima [A]  time = 0.707499, size = 19, normalized size = 0.76 \[ \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((a^2 + x^2)^3),x, algorithm="maxima")

[Out]

x/(sqrt(a^2 + x^2)*a^2)

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Fricas [A]  time = 0.277155, size = 86, normalized size = 3.44 \[ \frac{a^{4} + 2 \, a^{2} x^{2} + x^{4} + \sqrt{a^{6} + 3 \, a^{4} x^{2} + 3 \, a^{2} x^{4} + x^{6}} x}{a^{6} + 2 \, a^{4} x^{2} + a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((a^2 + x^2)^3),x, algorithm="fricas")

[Out]

(a^4 + 2*a^2*x^2 + x^4 + sqrt(a^6 + 3*a^4*x^2 + 3*a^2*x^4 + x^6)*x)/(a^6 + 2*a^4
*x^2 + a^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (a^{2} + x^{2}\right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt((a**2 + x**2)**3), x)

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GIAC/XCAS [A]  time = 0.264094, size = 19, normalized size = 0.76 \[ \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((a^2 + x^2)^3),x, algorithm="giac")

[Out]

x/(sqrt(a^2 + x^2)*a^2)