3.1220 \(\int \frac{1}{x^4 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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Rubi [A]  time = 0.0219694, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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Rubi in Sympy [A]  time = 3.14442, size = 17, normalized size = 0.77 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a - b*x**4)**(3/4)/(3*a*x**3)

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Mathematica [A]  time = 0.0207317, size = 22, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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Maple [A]  time = 0.006, size = 19, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-b*x^4+a)^(1/4),x)

[Out]

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

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Maxima [A]  time = 1.42792, size = 24, normalized size = 1.09 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Fricas [A]  time = 0.223281, size = 24, normalized size = 1.09 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Sympy [A]  time = 2.72743, size = 80, normalized size = 3.64 \[ \begin{cases} \frac{b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} - 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a \Gamma \left (\frac{1}{4}\right )} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\- \frac{b^{\frac{3}{4}} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} e^{\frac{7 i \pi }{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a \Gamma \left (\frac{1}{4}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((b**(3/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-3/4)/(4*a*gamma(1/4)), Abs(a/
(b*x**4)) > 1), (-b**(3/4)*(-a/(b*x**4) + 1)**(3/4)*exp(7*I*pi/4)*gamma(-3/4)/(4
*a*gamma(1/4)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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