3.1952 \(\int \frac{1}{(a+\frac{b}{x^2})^{5/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{8 x \sqrt{a+\frac{b}{x^2}}}{3 a^3}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

-x/(3*a*(a + b/x^2)^(3/2)) - (4*x)/(3*a^2*Sqrt[a + b/x^2]) + (8*Sqrt[a + b/x^2]*x)/(3*a^3)

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Rubi [A]  time = 0.0108795, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{8 x \sqrt{a+\frac{b}{x^2}}}{3 a^3}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(-5/2),x]

[Out]

-x/(3*a*(a + b/x^2)^(3/2)) - (4*x)/(3*a^2*Sqrt[a + b/x^2]) + (8*Sqrt[a + b/x^2]*x)/(3*a^3)

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]

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2}} \, dx &=-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}}+\frac{4 \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{8 \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx}{3 a^2}\\ &=-\frac{x}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{4 x}{3 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{8 \sqrt{a+\frac{b}{x^2}} x}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0185666, size = 51, normalized size = 0.88 \[ \frac{3 a^2 x^4+12 a b x^2+8 b^2}{3 a^3 x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(-5/2),x]

[Out]

(8*b^2 + 12*a*b*x^2 + 3*a^2*x^4)/(3*a^3*Sqrt[a + b/x^2]*x*(b + a*x^2))

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Maple [A]  time = 0.004, size = 50, normalized size = 0.9 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,{a}^{2}{x}^{4}+12\,ab{x}^{2}+8\,{b}^{2} \right ) }{3\,{a}^{3}{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)

[Out]

1/3*(a*x^2+b)*(3*a^2*x^4+12*a*b*x^2+8*b^2)/a^3/x^5/((a*x^2+b)/x^2)^(5/2)

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Maxima [A]  time = 0.969985, size = 69, normalized size = 1.19 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{2}}\right )} b x^{2} - b^{2}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a + b/x^2)*x/a^3 + 1/3*(6*(a + b/x^2)*b*x^2 - b^2)/((a + b/x^2)^(3/2)*a^3*x^3)

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Fricas [A]  time = 1.72266, size = 130, normalized size = 2.24 \begin{align*} \frac{{\left (3 \, a^{2} x^{5} + 12 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} 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, algorithm="fricas")

[Out]

1/3*(3*a^2*x^5 + 12*a*b*x^3 + 8*b^2*x)*sqrt((a*x^2 + b)/x^2)/(a^5*x^4 + 2*a^4*b*x^2 + a^3*b^2)

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Sympy [B]  time = 1.9198, size = 163, normalized size = 2.81 \begin{align*} \frac{3 a^{2} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac{12 a b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} + \frac{8 b^{\frac{13}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**2*b**(9/2)*x**4*sqrt(a*x**2/b + 1)/(3*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6) + 12*a*b**(11/2)*x
**2*sqrt(a*x**2/b + 1)/(3*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6) + 8*b**(13/2)*sqrt(a*x**2/b + 1)/(3
*a**5*b**4*x**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^2)^(-5/2), x)