3.1222 \(\int \frac{1}{x^{12} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}-\frac{8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac{\left (a-b x^4\right )^{3/4}}{11 a x^{11}} \]

[Out]

-(a - b*x^4)^(3/4)/(11*a*x^11) - (8*b*(a - b*x^4)^(3/4))/(77*a^2*x^7) - (32*b^2*
(a - b*x^4)^(3/4))/(231*a^3*x^3)

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Rubi [A]  time = 0.0699588, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}-\frac{8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac{\left (a-b x^4\right )^{3/4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^12*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(11*a*x^11) - (8*b*(a - b*x^4)^(3/4))/(77*a^2*x^7) - (32*b^2*
(a - b*x^4)^(3/4))/(231*a^3*x^3)

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Rubi in Sympy [A]  time = 7.96001, size = 63, normalized size = 0.89 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{11 a x^{11}} - \frac{8 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{77 a^{2} x^{7}} - \frac{32 b^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{231 a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**12/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(11*a*x**11) - 8*b*(a - b*x**4)**(3/4)/(77*a**2*x**7) - 32*
b**2*(a - b*x**4)**(3/4)/(231*a**3*x**3)

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Mathematica [A]  time = 0.0360806, size = 43, normalized size = 0.61 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (21 a^2+24 a b x^4+32 b^2 x^8\right )}{231 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^12*(a - b*x^4)^(1/4)),x]

[Out]

-((a - b*x^4)^(3/4)*(21*a^2 + 24*a*b*x^4 + 32*b^2*x^8))/(231*a^3*x^11)

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Maple [A]  time = 0.009, size = 40, normalized size = 0.6 \[ -{\frac{32\,{b}^{2}{x}^{8}+24\,ab{x}^{4}+21\,{a}^{2}}{231\,{x}^{11}{a}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^12/(-b*x^4+a)^(1/4),x)

[Out]

-1/231*(-b*x^4+a)^(3/4)*(32*b^2*x^8+24*a*b*x^4+21*a^2)/x^11/a^3

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Maxima [A]  time = 1.43097, size = 74, normalized size = 1.04 \[ -\frac{\frac{77 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b^{2}}{x^{3}} + \frac{66 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} b}{x^{7}} + \frac{21 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}}}{x^{11}}}{231 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^12),x, algorithm="maxima")

[Out]

-1/231*(77*(-b*x^4 + a)^(3/4)*b^2/x^3 + 66*(-b*x^4 + a)^(7/4)*b/x^7 + 21*(-b*x^4
+ a)^(11/4)/x^11)/a^3

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Fricas [A]  time = 0.225881, size = 53, normalized size = 0.75 \[ -\frac{{\left (32 \, b^{2} x^{8} + 24 \, a b x^{4} + 21 \, a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{231 \, a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^12),x, algorithm="fricas")

[Out]

-1/231*(32*b^2*x^8 + 24*a*b*x^4 + 21*a^2)*(-b*x^4 + a)^(3/4)/(a^3*x^11)

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Sympy [A]  time = 12.2911, size = 864, normalized size = 12.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**12/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((21*a**4*b**(19/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*
x**8*gamma(1/4) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)
) - 18*a**3*b**(23/4)*x**4*(a/(b*x**4) - 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x*
*8*gamma(1/4) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4))
+ 5*a**2*b**(27/4)*x**8*(a/(b*x**4) - 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*
gamma(1/4) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) - 4
0*a*b**(31/4)*x**12*(a/(b*x**4) - 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamm
a(1/4) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) + 32*b*
*(35/4)*x**16*(a/(b*x**4) - 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4)
- 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)), Abs(a/(b*x**
4)) > 1), (-21*a**4*b**(19/4)*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-11/
4)/(64*a**5*b**4*x**8*gamma(1/4) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6
*x**16*gamma(1/4)) + 18*a**3*b**(23/4)*x**4*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi
/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) - 128*a**4*b**5*x**12*gamma(1/4)
+ 64*a**3*b**6*x**16*gamma(1/4)) - 5*a**2*b**(27/4)*x**8*(-a/(b*x**4) + 1)**(3/4
)*exp(15*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) - 128*a**4*b**5*x**1
2*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) + 40*a*b**(31/4)*x**12*(-a/(b*x**4
) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4) - 128*a*
*4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)) - 32*b**(35/4)*x**16*(
-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*gamma(1/4
) - 128*a**4*b**5*x**12*gamma(1/4) + 64*a**3*b**6*x**16*gamma(1/4)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^12),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^12), x)