3.1953 \(\int \frac{1}{(a+\frac{b}{x^2})^{5/2} x^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0139506, 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, 264} \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(5/2)*x^2),x]

[Out]

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

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 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 \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^2} \, dx &=-\frac{1}{a \left (a+\frac{b}{x^2}\right )^{3/2} x}+\frac{(2 b) \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^4} \, dx}{a}\\ &=-\frac{2 b}{3 a^2 \left (a+\frac{b}{x^2}\right )^{3/2} x^3}-\frac{1}{a \left (a+\frac{b}{x^2}\right )^{3/2} x}\\ \end{align*}

Mathematica [A]  time = 0.0214679, size = 38, normalized size = 0.9 \[ -\frac{x \sqrt{a+\frac{b}{x^2}} \left (3 a x^2+2 b\right )}{3 a^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 0.9 \begin{align*} -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,a{x}^{2}+2\,b \right ) }{3\,{x}^{5}{a}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^(5/2)/x^2,x)

[Out]

-1/3*(a*x^2+b)*(3*a*x^2+2*b)/a^2/x^5/((a*x^2+b)/x^2)^(5/2)

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Maxima [A]  time = 0.988635, size = 45, normalized size = 1.07 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{2}}\right )} x^{2} - b}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7806, size = 108, normalized size = 2.57 \begin{align*} -\frac{{\left (3 \, a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^2,x, algorithm="fricas")

[Out]

-1/3*(3*a*x^3 + 2*b*x)*sqrt((a*x^2 + b)/x^2)/(a^4*x^4 + 2*a^3*b*x^2 + a^2*b^2)

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Sympy [B]  time = 1.7155, size = 105, normalized size = 2.5 \begin{align*} - \frac{3 a x^{2}}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{2 b}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(5/2)/x**2,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^2,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x^2), x)