3.1251 \(\int \frac{1}{x^6 \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{4 b \sqrt [4]{a-b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a-b x^4}}{5 a x^5} \]

[Out]

-(a - b*x^4)^(1/4)/(5*a*x^5) - (4*b*(a - b*x^4)^(1/4))/(5*a^2*x)

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Rubi [A]  time = 0.0439068, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{4 b \sqrt [4]{a-b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a-b x^4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(5*a*x^5) - (4*b*(a - b*x^4)^(1/4))/(5*a^2*x)

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Rubi in Sympy [A]  time = 4.94272, size = 37, normalized size = 0.8 \[ - \frac{\sqrt [4]{a - b x^{4}}}{5 a x^{5}} - \frac{4 b \sqrt [4]{a - b x^{4}}}{5 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-b*x**4+a)**(3/4),x)

[Out]

-(a - b*x**4)**(1/4)/(5*a*x**5) - 4*b*(a - b*x**4)**(1/4)/(5*a**2*x)

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Mathematica [A]  time = 0.0245878, size = 30, normalized size = 0.65 \[ -\frac{\sqrt [4]{a-b x^4} \left (a+4 b x^4\right )}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a - b*x^4)^(3/4)),x]

[Out]

-((a - b*x^4)^(1/4)*(a + 4*b*x^4))/(5*a^2*x^5)

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Maple [A]  time = 0.007, size = 27, normalized size = 0.6 \[ -{\frac{4\,b{x}^{4}+a}{5\,{x}^{5}{a}^{2}}\sqrt [4]{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-b*x^4+a)^(3/4),x)

[Out]

-1/5*(-b*x^4+a)^(1/4)*(4*b*x^4+a)/x^5/a^2

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Maxima [A]  time = 1.44282, size = 49, normalized size = 1.07 \[ -\frac{\frac{5 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x} + \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{x^{5}}}{5 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^6),x, algorithm="maxima")

[Out]

-1/5*(5*(-b*x^4 + a)^(1/4)*b/x + (-b*x^4 + a)^(5/4)/x^5)/a^2

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Fricas [A]  time = 0.225262, size = 35, normalized size = 0.76 \[ -\frac{{\left (4 \, b x^{4} + a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^6),x, algorithm="fricas")

[Out]

-1/5*(4*b*x^4 + a)*(-b*x^4 + a)^(1/4)/(a^2*x^5)

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Sympy [A]  time = 5.14577, size = 286, normalized size = 6.22 \[ \begin{cases} - \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{16 a x^{4} \Gamma \left (\frac{3}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} - 1} \Gamma \left (- \frac{5}{4}\right )}{4 a^{2} \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\- \frac{a^{2} b^{\frac{5}{4}} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{5 i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{3}{4}\right )} - \frac{3 a b^{\frac{9}{4}} x^{4} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{5 i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{3}{4}\right )} + \frac{4 b^{\frac{13}{4}} x^{8} \sqrt [4]{- \frac{a}{b x^{4}} + 1} e^{\frac{5 i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{3}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(16*a*x**4*gamma(3/4))
- b**(5/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(4*a**2*gamma(3/4)), Abs(a/(b*x**
4)) > 1), (-a**2*b**(5/4)*(-a/(b*x**4) + 1)**(1/4)*exp(5*I*pi/4)*gamma(-5/4)/(-1
6*a**3*b*x**4*gamma(3/4) + 16*a**2*b**2*x**8*gamma(3/4)) - 3*a*b**(9/4)*x**4*(-a
/(b*x**4) + 1)**(1/4)*exp(5*I*pi/4)*gamma(-5/4)/(-16*a**3*b*x**4*gamma(3/4) + 16
*a**2*b**2*x**8*gamma(3/4)) + 4*b**(13/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(5*I*
pi/4)*gamma(-5/4)/(-16*a**3*b*x**4*gamma(3/4) + 16*a**2*b**2*x**8*gamma(3/4)), T
rue))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^6),x, algorithm="giac")

[Out]

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