3.1948 \(\int \frac{1}{(a+\frac{b}{x^2})^{5/2} x^5} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0206982, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(5/2)*x^5),x]

[Out]

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0126012, size = 37, normalized size = 1.06 \[ \frac{2 a x^2+3 b}{3 b^2 \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 1.1 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}+3\,b \right ) }{3\,{b}^{2}{x}^{4}} \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^5,x)

[Out]

1/3*(a*x^2+b)/x^4*(2*a*x^2+3*b)/b^2/((a*x^2+b)/x^2)^(5/2)

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Maxima [A]  time = 1.00135, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{\sqrt{a + \frac{b}{x^{2}}} b^{2}} - \frac{a}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.53224, size = 109, normalized size = 3.11 \begin{align*} \frac{{\left (2 \, a x^{4} + 3 \, b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.47328, size = 94, normalized size = 2.69 \begin{align*} \begin{cases} \frac{2 a x^{2}}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{3 b}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{5}{2}} x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*x**2/(3*a*b**2*x**2*sqrt(a + b/x**2) + 3*b**3*sqrt(a + b/x**2)) + 3*b/(3*a*b**2*x**2*sqrt(a + b
/x**2) + 3*b**3*sqrt(a + b/x**2)), Ne(b, 0)), (-1/(4*a**(5/2)*x**4), True))

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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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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