3.1221 \(\int \frac{1}{x^8 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a-b x^4\right )^{3/4}}{7 a x^7} \]

[Out]

-(a - b*x^4)^(3/4)/(7*a*x^7) - (4*b*(a - b*x^4)^(3/4))/(21*a^2*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0437292, 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 \[ -\frac{4 b \left (a-b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a-b x^4\right )^{3/4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(7*a*x^7) - (4*b*(a - b*x^4)^(3/4))/(21*a^2*x^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.10602, size = 39, normalized size = 0.85 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{7 a x^{7}} - \frac{4 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{21 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(7*a*x**7) - 4*b*(a - b*x**4)**(3/4)/(21*a**2*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0278673, size = 32, normalized size = 0.7 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (3 a+4 b x^4\right )}{21 a^2 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a - b*x^4)^(1/4)),x]

[Out]

-((a - b*x^4)^(3/4)*(3*a + 4*b*x^4))/(21*a^2*x^7)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 29, normalized size = 0.6 \[ -{\frac{4\,b{x}^{4}+3\,a}{21\,{a}^{2}{x}^{7}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(-b*x^4+a)^(1/4),x)

[Out]

-1/21*(-b*x^4+a)^(3/4)*(4*b*x^4+3*a)/a^2/x^7

_______________________________________________________________________________________

Maxima [A]  time = 1.42342, size = 50, normalized size = 1.09 \[ -\frac{\frac{7 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b}{x^{3}} + \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}}}{x^{7}}}{21 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^8),x, algorithm="maxima")

[Out]

-1/21*(7*(-b*x^4 + a)^(3/4)*b/x^3 + 3*(-b*x^4 + a)^(7/4)/x^7)/a^2

_______________________________________________________________________________________

Fricas [A]  time = 0.226293, size = 38, normalized size = 0.83 \[ -\frac{{\left (4 \, b x^{4} + 3 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^8),x, algorithm="fricas")

[Out]

-1/21*(4*b*x^4 + 3*a)*(-b*x^4 + a)^(3/4)/(a^2*x^7)

_______________________________________________________________________________________

Sympy [A]  time = 5.54024, size = 287, normalized size = 6.24 \[ \begin{cases} - \frac{3 b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} - 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{16 a x^{4} \Gamma \left (\frac{1}{4}\right )} - \frac{b^{\frac{7}{4}} \left (\frac{a}{b x^{4}} - 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{4 a^{2} \Gamma \left (\frac{1}{4}\right )} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\- \frac{3 a^{2} b^{\frac{7}{4}} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} e^{\frac{7 i \pi }{4}} \Gamma \left (- \frac{7}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{1}{4}\right )} - \frac{a b^{\frac{11}{4}} x^{4} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} e^{\frac{7 i \pi }{4}} \Gamma \left (- \frac{7}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{1}{4}\right )} + \frac{4 b^{\frac{15}{4}} x^{8} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} e^{\frac{7 i \pi }{4}} \Gamma \left (- \frac{7}{4}\right )}{- 16 a^{3} b x^{4} \Gamma \left (\frac{1}{4}\right ) + 16 a^{2} b^{2} x^{8} \Gamma \left (\frac{1}{4}\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-3*b**(3/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-7/4)/(16*a*x**4*gamma(1/4)
) - b**(7/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-7/4)/(4*a**2*gamma(1/4)), Abs(a/(b*x
**4)) > 1), (-3*a**2*b**(7/4)*(-a/(b*x**4) + 1)**(3/4)*exp(7*I*pi/4)*gamma(-7/4)
/(-16*a**3*b*x**4*gamma(1/4) + 16*a**2*b**2*x**8*gamma(1/4)) - a*b**(11/4)*x**4*
(-a/(b*x**4) + 1)**(3/4)*exp(7*I*pi/4)*gamma(-7/4)/(-16*a**3*b*x**4*gamma(1/4) +
16*a**2*b**2*x**8*gamma(1/4)) + 4*b**(15/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*exp(7
*I*pi/4)*gamma(-7/4)/(-16*a**3*b*x**4*gamma(1/4) + 16*a**2*b**2*x**8*gamma(1/4))
, True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^8),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^8), x)