3.2091 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^2)

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Rubi [A]  time = 0.0317186, 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 \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(3/2)*x^3),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^2)

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Rubi in Sympy [A]  time = 2.67723, size = 19, normalized size = 0.9 \[ - \frac{1}{2 a x^{2} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(3/2)/x**3,x)

[Out]

-1/(2*a*x**2*sqrt(a + b/x**4))

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Mathematica [A]  time = 0.00980364, size = 21, normalized size = 1. \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(3/2)*x^3),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^2)

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Maple [A]  time = 0.007, size = 29, normalized size = 1.4 \[ -{\frac{a{x}^{4}+b}{2\,{x}^{6}a} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(3/2)/x^3,x)

[Out]

-1/2*(a*x^4+b)/a/x^6/((a*x^4+b)/x^4)^(3/2)

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Maxima [A]  time = 1.42855, size = 23, normalized size = 1.1 \[ -\frac{1}{2 \, \sqrt{a + \frac{b}{x^{4}}} a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^3),x, algorithm="maxima")

[Out]

-1/2/(sqrt(a + b/x^4)*a*x^2)

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Fricas [A]  time = 0.235513, size = 42, normalized size = 2. \[ -\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{2} x^{4} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.77622, size = 22, normalized size = 1.05 \[ - \frac{1}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(3/2)/x**3,x)

[Out]

-1/(2*a*sqrt(b)*sqrt(a*x**4/b + 1))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^3),x, algorithm="giac")

[Out]

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