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]

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

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.79, size = 53, normalized size = 1.51 \[ \frac {{\left (2 \, a x^{4} + 3 \, b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} \]

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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giac [B]  time = 0.24, size = 78, normalized size = 2.23 \[ -\frac {2}{3 \, \sqrt {a} b^{2}} + \frac {3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} a + 2 \, \sqrt {a} b}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} + b\right )}^{3} a} \]

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]

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

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maple [A]  time = 0.00, size = 39, normalized size = 1.11 \[ \frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}+3 b \right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^(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 = 0.92, size = 29, normalized size = 0.83 \[ \frac {1}{\sqrt {a + \frac {b}{x^{2}}} b^{2}} - \frac {a}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2}} \]

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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mupad [B]  time = 1.26, size = 27, normalized size = 0.77 \[ \frac {2\,a\,x^2+3\,b}{3\,b^2\,x^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b/x^2)^(5/2)),x)

[Out]

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

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sympy [A]  time = 5.70, size = 94, normalized size = 2.69 \[ \begin {cases} \frac {2 a x^{2}}{3 a b^{2} x^{2} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{3} \sqrt {a + \frac {b}{x^{2}}}} + \frac {3 b}{3 a b^{2} x^{2} \sqrt {a + \frac {b}{x^{2}}} + 3 b^{3} \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{4 a^{\frac {5}{2}} x^{4}} & \text {otherwise} \end {cases} \]

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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