3.1934 \(\int \frac {1}{(a+\frac {b}{x^2})^{3/2} x^5} \, dx\)

Optimal. Leaf size=34 \[ -\frac {a}{b^2 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 34, 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} \[ -\frac {a}{b^2 \sqrt {a+\frac {b}{x^2}}}-\frac {\sqrt {a+\frac {b}{x^2}}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(3/2)*x^5),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.82 \[ \frac {-2 a x^2-b}{b^2 x^2 \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(3/2)*x^5),x]

[Out]

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

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fricas [A]  time = 0.75, size = 37, normalized size = 1.09 \[ -\frac {{\left (2 \, a x^{2} + b\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a b^{2} x^{2} + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 28, normalized size = 0.82 \[ -\frac {\frac {2 \, a x^{2}}{b^{2}} + \frac {1}{b}}{\sqrt {a x^{4} + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 37, normalized size = 1.09 \[ -\frac {\left (a \,x^{2}+b \right ) \left (2 a \,x^{2}+b \right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^(3/2)/x^5,x)

[Out]

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

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maxima [A]  time = 0.89, size = 30, normalized size = 0.88 \[ -\frac {\sqrt {a + \frac {b}{x^{2}}}}{b^{2}} - \frac {a}{\sqrt {a + \frac {b}{x^{2}}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.22, size = 36, normalized size = 1.06 \[ -\frac {x\,\sqrt {a+\frac {b}{x^2}}\,\left (\frac {1}{b}+\frac {2\,a\,x^2}{b^2}\right )}{a\,x^3+b\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b/x^2)^(3/2)),x)

[Out]

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

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sympy [A]  time = 3.38, size = 48, normalized size = 1.41 \[ \begin {cases} - \frac {2 a}{b^{2} \sqrt {a + \frac {b}{x^{2}}}} - \frac {1}{b x^{2} \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{4 a^{\frac {3}{2}} x^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(3/2)/x**5,x)

[Out]

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

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