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 = 0.0061545, antiderivative size = 21, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0059178, 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]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{a{x}^{2}+b}{3\,a{x}^{5}} \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^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 = 0.968092, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a x^{3}} \end{align*}

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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Fricas [B]  time = 1.70378, size = 84, normalized size = 4. \begin{align*} -\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 )}} \end{align*}

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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Sympy [B]  time = 2.03345, size = 48, normalized size = 2.29 \begin{align*} - \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}} \end{align*}

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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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^{4}}\,{d x} \end{align*}

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]

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