3.1923 \(\int \frac{x^2}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=42 \[ \frac{x^3 \sqrt{a+\frac{b}{x^2}}}{3 a}-\frac{2 b x \sqrt{a+\frac{b}{x^2}}}{3 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0093987, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 191} \[ \frac{x^3 \sqrt{a+\frac{b}{x^2}}}{3 a}-\frac{2 b x \sqrt{a+\frac{b}{x^2}}}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + b/x^2],x]

[Out]

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

Rule 271

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{a+\frac{b}{x^2}}} \, dx &=\frac{\sqrt{a+\frac{b}{x^2}} x^3}{3 a}-\frac{(2 b) \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx}{3 a}\\ &=-\frac{2 b \sqrt{a+\frac{b}{x^2}} x}{3 a^2}+\frac{\sqrt{a+\frac{b}{x^2}} x^3}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.019603, size = 28, normalized size = 0.67 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (a x^2-2 b\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a + b/x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 0.9 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( a{x}^{2}-2\,b \right ) }{3\,{a}^{2}x}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.994553, size = 43, normalized size = 1.02 \begin{align*} \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 3 \, \sqrt{a + \frac{b}{x^{2}}} b x}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47507, size = 63, normalized size = 1.5 \begin{align*} \frac{{\left (a x^{3} - 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.997504, size = 46, normalized size = 1.1 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{a + \frac{b}{x^{2}}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/sqrt(a + b/x^2), x)