3.1126 \(\int \frac{1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sqrt [4]{a+b x^4}}{a x} \]

[Out]

-((a + b*x^4)^(1/4)/(a*x))

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Rubi [A]  time = 0.0203061, antiderivative size = 19, 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 \[ -\frac{\sqrt [4]{a+b x^4}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^4)^(1/4)/(a*x))

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Rubi in Sympy [A]  time = 2.66739, size = 14, normalized size = 0.74 \[ - \frac{\sqrt [4]{a + b x^{4}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**4)**(1/4)/(a*x)

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Mathematica [A]  time = 0.0156404, size = 19, normalized size = 1. \[ -\frac{\sqrt [4]{a+b x^4}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^4)^(1/4)/(a*x))

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Maple [A]  time = 0.006, size = 18, normalized size = 1. \[ -{\frac{1}{ax}\sqrt [4]{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^4+a)^(3/4),x)

[Out]

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

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Maxima [A]  time = 1.43666, size = 23, normalized size = 1.21 \[ -\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x^4 + a)^(1/4)/(a*x)

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Fricas [A]  time = 0.239482, size = 23, normalized size = 1.21 \[ -\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x^4 + a)^(1/4)/(a*x)

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Sympy [A]  time = 2.41035, size = 31, normalized size = 1.63 \[ \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{1}{4}\right )}{4 a \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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