3.1892 \(\int \sqrt{a+\frac{b}{x^2}} x^2 \, dx\)

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)

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Rubi [A]  time = 0.0057577, 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, Rules used = {264} \[ \frac{x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}{3 a} \]

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)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{a+\frac{b}{x^2}} x^2 \, dx &=\frac{\left (a+\frac{b}{x^2}\right )^{3/2} x^3}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0049841, 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)

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Maple [A]  time = 0.004, size = 27, normalized size = 1.3 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) x}{3\,a}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00758, size = 23, normalized size = 1.1 \begin{align*} \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)*x^2,x, algorithm="maxima")

[Out]

1/3*(a + b/x^2)^(3/2)*x^3/a

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Fricas [A]  time = 1.48493, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (a x^{3} + b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)*x^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.750546, size = 41, normalized size = 1.95 \begin{align*} \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} \end{align*}

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)

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Giac [A]  time = 1.72396, size = 36, normalized size = 1.71 \begin{align*} \frac{{\left (a x^{2} + b\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, a} - \frac{b^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)*x^2,x, algorithm="giac")

[Out]

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