3.1951 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0616614, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 b}{3 a^2 x^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{1}{a x \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.58083, size = 37, normalized size = 0.88 \[ - \frac{1}{a x \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{2 b}{3 a^{2} x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.041058, size = 38, normalized size = 0.9 \[ -\frac{x \sqrt{a+\frac{b}{x^2}} \left (3 a x^2+2 b\right )}{3 a^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 39, normalized size = 0.9 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,a{x}^{2}+2\,b \right ) }{3\,{x}^{5}{a}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.44415, size = 45, normalized size = 1.07 \[ -\frac{3 \,{\left (a + \frac{b}{x^{2}}\right )} x^{2} - b}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.254659, size = 70, normalized size = 1.67 \[ -\frac{{\left (3 \, a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.98056, size = 105, normalized size = 2.5 \[ - \frac{3 a x^{2}}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{2 b}{3 a^{3} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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