3.1917 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^5} \, dx\)

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 = 0.065121, 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 \[ \frac{a \sqrt{a+\frac{b}{x^2}}}{b^2}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^2} \]

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)

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Rubi in Sympy [A]  time = 6.99349, size = 29, normalized size = 0.83 \[ \frac{a \sqrt{a + \frac{b}{x^{2}}}}{b^{2}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(1/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.037926, 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 = 0.008, size = 39, normalized size = 1.1 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}-b \right ) }{3\,{b}^{2}{x}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(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 = 1.43742, size = 39, normalized size = 1.11 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{\sqrt{a + \frac{b}{x^{2}}} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^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 = 0.234269, size = 42, normalized size = 1.2 \[ \frac{{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^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 [A]  time = 5.97929, size = 231, normalized size = 6.6 \[ \frac{2 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} + \frac{a^{\frac{5}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{4} b x^{5}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} - \frac{2 a^{3} b^{2} x^{3}}{3 a^{\frac{5}{2}} b^{3} x^{5} + 3 a^{\frac{3}{2}} b^{4} x^{3}} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^2)*x^5),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^2)*x^5), x)