3.1890 \(\int \sqrt{a+\frac{b}{x^2}} x^2 \, dx\)

Optimal. Leaf size=21 \[ \frac{x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}{3 a} \]

[Out]

((a + b/x^2)^(3/2)*x^3)/(3*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0317097, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]*x^2,x]

[Out]

((a + b/x^2)^(3/2)*x^3)/(3*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.69885, size = 15, normalized size = 0.71 \[ \frac{x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(a + b/x**2)**(3/2)/(3*a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0103431, size = 26, normalized size = 1.24 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]*x^2,x]

[Out]

(Sqrt[a + b/x^2]*x*(b + a*x^2))/(3*a)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 27, normalized size = 1.3 \[{\frac{ \left ( a{x}^{2}+b \right ) x}{3\,a}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)*x^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.42735, size = 23, normalized size = 1.1 \[ \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a + b/x^2)^(3/2)*x^3/a

_______________________________________________________________________________________

Fricas [A]  time = 0.237737, size = 36, normalized size = 1.71 \[ \frac{{\left (a x^{3} + b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.62626, size = 41, normalized size = 1.95 \[ \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*x**2*sqrt(a*x**2/b + 1)/3 + b**(3/2)*sqrt(a*x**2/b + 1)/(3*a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226259, size = 36, normalized size = 1.71 \[ \frac{{\left (a x^{2} + b\right )}^{\frac{3}{2}}{\rm sign}\left (x\right )}{3 \, a} - \frac{b^{\frac{3}{2}}{\rm sign}\left (x\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a*x^2 + b)^(3/2)*sign(x)/a - 1/3*b^(3/2)*sign(x)/a