3.1939 \(\int \frac {1}{(a+\frac {b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac {2 x \sqrt {a+\frac {b}{x^2}}}{a^2}-\frac {x}{a \sqrt {a+\frac {b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {2 x \sqrt {a+\frac {b}{x^2}}}{a^2}-\frac {x}{a \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(-3/2),x]

[Out]

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

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]

Rule 192

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.77 \[ \frac {a x^2+2 b}{a^2 x \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(-3/2),x]

[Out]

(2*b + a*x^2)/(a^2*Sqrt[a + b/x^2]*x)

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fricas [A]  time = 0.79, size = 39, normalized size = 1.11 \[ \frac {{\left (a x^{3} + 2 \, b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a^{3} x^{2} + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-2,[1
,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [11,86.0946290595,-72]Warning, choosing
root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [79,97.172926544
2,77]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters v
alues [35,94.710570252,-15]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4
]%%%}] at parameters values [-46,57.5337947248,-63]b/a^2/(a*(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))+sqrt(
b)*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x)))+(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))/a-sqrt(b)/a^
2*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)/a/sign(x))

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maple [A]  time = 0.00, size = 37, normalized size = 1.06 \[ \frac {\left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 b \right )}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.89, size = 32, normalized size = 0.91 \[ \frac {\sqrt {a + \frac {b}{x^{2}}} x}{a^{2}} + \frac {b}{\sqrt {a + \frac {b}{x^{2}}} a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.19, size = 36, normalized size = 1.03 \[ \frac {a^2\,x^4+3\,a\,b\,x^2+2\,b^2}{a^2\,x^3\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^2)^(3/2),x)

[Out]

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

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sympy [A]  time = 0.99, size = 42, normalized size = 1.20 \[ \frac {x^{2}}{a \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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