3.1952 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^4} \, dx\)

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 = 0.0299818, 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 \[ -\frac{1}{3 a x^3 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

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)

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Rubi in Sympy [A]  time = 2.67316, size = 19, normalized size = 0.9 \[ - \frac{1}{3 a x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x**4,x)

[Out]

-1/(3*a*x**3*(a + b/x**2)**(3/2))

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Mathematica [A]  time = 0.0141356, 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 = 0.004, size = 29, normalized size = 1.4 \[ -{\frac{a{x}^{2}+b}{3\,a{x}^{5}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^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 = 1.43884, size = 23, normalized size = 1.1 \[ -\frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a x^{3}} \]

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 [A]  time = 0.24439, size = 54, normalized size = 2.57 \[ -\frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \]

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 [A]  time = 9.49625, size = 48, normalized size = 2.29 \[ - \frac{1}{3 a^{2} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{4}}\,{d x} \]

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)