3.1919 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^5} \, dx\)

Optimal. Leaf size=35 \[ \frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0199261, antiderivative size = 35, normalized size of antiderivative = 1., 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} \[ \frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x^5),x]

[Out]

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

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]]

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])

Rubi steps

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

Mathematica [A]  time = 0.0137037, size = 31, normalized size = 0.89 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (2 a x^2-b\right )}{3 b^2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^5),x]

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 1.1 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}-b \right ) }{3\,{b}^{2}{x}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.991083, size = 39, normalized size = 1.11 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{\sqrt{a + \frac{b}{x^{2}}} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52537, size = 69, normalized size = 1.97 \begin{align*} \frac{{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.12436, size = 231, normalized size = 6.6 \begin{align*} \frac{2 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} + \frac{a^{\frac{5}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{4} b x^{5}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{3} b^{2} x^{3}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**(7/2)*b**(3/2)*x**4*sqrt(a*x**2/b + 1)/(3*a**(5/2)*b**3*x**5 + 3*a**(3/2)*b**4*x**3) + a**(5/2)*b**(5/2)*
x**2*sqrt(a*x**2/b + 1)/(3*a**(5/2)*b**3*x**5 + 3*a**(3/2)*b**4*x**3) - a**(3/2)*b**(7/2)*sqrt(a*x**2/b + 1)/(
3*a**(5/2)*b**3*x**5 + 3*a**(3/2)*b**4*x**3) - 2*a**4*b*x**5/(3*a**(5/2)*b**3*x**5 + 3*a**(3/2)*b**4*x**3) - 2
*a**3*b**2*x**3/(3*a**(5/2)*b**3*x**5 + 3*a**(3/2)*b**4*x**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^2)*x^5), x)