3.1954 \(\int \frac {1}{(a+\frac {b}{x^2})^{5/2} x^4} \, dx\)

Optimal. Leaf size=21 \[ -\frac {1}{3 a x^3 \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ -\frac {1}{3 a x^3 \left (a+\frac {b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(5/2)*x^4),x]

[Out]

-1/(3*a*(a + b/x^2)^(3/2)*x^3)

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.33 \[ -\frac {a x^2+b}{3 a x^5 \left (a+\frac {b}{x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^4),x]

[Out]

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

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fricas [B]  time = 0.69, size = 40, normalized size = 1.90 \[ -\frac {x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^4,x, algorithm="fricas")

[Out]

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

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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)^(5/2)/x^4,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 [32,76.6146142203,-62]Warning, choosing
root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [66,99.659021995
5,-30]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters
values [-50,17.713730142,-44]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0
,4]%%%}] at parameters values [45,14.5515509131,62](b+b)/6/a/b^2/sqrt(b)*sign(-sqrt(b)/a*sign(x)+sqrt(a*x^2+b)
/a/sign(x))-1/3/(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)))^3/a

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maple [A]  time = 0.00, size = 29, normalized size = 1.38 \[ -\frac {a \,x^{2}+b}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} a \,x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^(5/2)/x^4,x)

[Out]

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

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maxima [A]  time = 0.92, size = 17, normalized size = 0.81 \[ -\frac {1}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^4,x, algorithm="maxima")

[Out]

-1/3/((a + b/x^2)^(3/2)*a*x^3)

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mupad [B]  time = 1.20, size = 17, normalized size = 0.81 \[ -\frac {1}{3\,a\,x^3\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x^2)^(5/2)),x)

[Out]

-1/(3*a*x^3*(a + b/x^2)^(3/2))

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sympy [B]  time = 1.96, size = 48, normalized size = 2.29 \[ - \frac {1}{3 a^{2} \sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 3 a b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(5/2)/x**4,x)

[Out]

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

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