3.1190 \(\int \frac{\sqrt [4]{a-b x^4}}{x^6} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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Rubi [A]  time = 0.0207531, 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 \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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Rubi in Sympy [A]  time = 2.96076, size = 17, normalized size = 0.77 \[ - \frac{\left (a - b x^{4}\right )^{\frac{5}{4}}}{5 a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/4)/x**6,x)

[Out]

-(a - b*x**4)**(5/4)/(5*a*x**5)

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Mathematica [A]  time = 0.0141624, size = 22, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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Maple [A]  time = 0.006, size = 19, normalized size = 0.9 \[ -{\frac{1}{5\,a{x}^{5}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^6,x)

[Out]

-1/5*(-b*x^4+a)^(5/4)/a/x^5

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Maxima [A]  time = 1.43575, size = 24, normalized size = 1.09 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^6,x, algorithm="maxima")

[Out]

-1/5*(-b*x^4 + a)^(5/4)/(a*x^5)

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Fricas [A]  time = 0.236871, size = 36, normalized size = 1.64 \[ \frac{{\left (b x^{4} - a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^6,x, algorithm="fricas")

[Out]

1/5*(b*x^4 - a)*(-b*x^4 + a)^(1/4)/(a*x^5)

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Sympy [A]  time = 3.60566, size = 162, normalized size = 7.36 \[ \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{4 x^{4} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{4 a \Gamma \left (- \frac{1}{4}\right )} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\\frac{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{9 i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{4 x^{4} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{9 i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{4 a \Gamma \left (- \frac{1}{4}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**4+a)**(1/4)/x**6,x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(4*x**4*gamma(-1/4)) - b
**(5/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(4*a*gamma(-1/4)), Abs(a/(b*x**4)) >
1), (b**(1/4)*(-a/(b*x**4) + 1)**(1/4)*exp(9*I*pi/4)*gamma(-5/4)/(4*x**4*gamma(
-1/4)) - b**(5/4)*(-a/(b*x**4) + 1)**(1/4)*exp(9*I*pi/4)*gamma(-5/4)/(4*a*gamma(
-1/4)), True))

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GIAC/XCAS [A]  time = 0.244835, size = 35, normalized size = 1.59 \[ \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )}}{5 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^6,x, algorithm="giac")

[Out]

1/5*(-b*x^4 + a)^(1/4)*(b - a/x^4)/(a*x)