3.1223 \(\int \frac{1}{x^{16} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}} \]

[Out]

-(a - b*x^4)^(3/4)/(15*a*x^15) - (4*b*(a - b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2
*(a - b*x^4)^(3/4))/(385*a^3*x^7) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^4*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0989685, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^16*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(15*a*x^15) - (4*b*(a - b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2
*(a - b*x^4)^(3/4))/(385*a^3*x^7) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^4*x^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.1863, size = 87, normalized size = 0.91 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{15 a x^{15}} - \frac{4 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{55 a^{2} x^{11}} - \frac{32 b^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{385 a^{3} x^{7}} - \frac{128 b^{3} \left (a - b x^{4}\right )^{\frac{3}{4}}}{1155 a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**16/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(15*a*x**15) - 4*b*(a - b*x**4)**(3/4)/(55*a**2*x**11) - 32
*b**2*(a - b*x**4)**(3/4)/(385*a**3*x**7) - 128*b**3*(a - b*x**4)**(3/4)/(1155*a
**4*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0423219, size = 54, normalized size = 0.56 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (77 a^3+84 a^2 b x^4+96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^16*(a - b*x^4)^(1/4)),x]

[Out]

-((a - b*x^4)^(3/4)*(77*a^3 + 84*a^2*b*x^4 + 96*a*b^2*x^8 + 128*b^3*x^12))/(1155
*a^4*x^15)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 51, normalized size = 0.5 \[ -{\frac{128\,{b}^{3}{x}^{12}+96\,a{b}^{2}{x}^{8}+84\,{a}^{2}b{x}^{4}+77\,{a}^{3}}{1155\,{x}^{15}{a}^{4}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^16/(-b*x^4+a)^(1/4),x)

[Out]

-1/1155*(-b*x^4+a)^(3/4)*(128*b^3*x^12+96*a*b^2*x^8+84*a^2*b*x^4+77*a^3)/x^15/a^
4

_______________________________________________________________________________________

Maxima [A]  time = 1.44231, size = 99, normalized size = 1.03 \[ -\frac{\frac{385 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} + \frac{495 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{315 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} + \frac{77 \,{\left (-b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^16),x, algorithm="maxima")

[Out]

-1/1155*(385*(-b*x^4 + a)^(3/4)*b^3/x^3 + 495*(-b*x^4 + a)^(7/4)*b^2/x^7 + 315*(
-b*x^4 + a)^(11/4)*b/x^11 + 77*(-b*x^4 + a)^(15/4)/x^15)/a^4

_______________________________________________________________________________________

Fricas [A]  time = 0.228846, size = 68, normalized size = 0.71 \[ -\frac{{\left (128 \, b^{3} x^{12} + 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} + 77 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{4} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^16),x, algorithm="fricas")

[Out]

-1/1155*(128*b^3*x^12 + 96*a*b^2*x^8 + 84*a^2*b*x^4 + 77*a^3)*(-b*x^4 + a)^(3/4)
/(a^4*x^15)

_______________________________________________________________________________________

Sympy [A]  time = 24.8791, size = 1452, normalized size = 15.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**16/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((231*a**6*b**(39/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-15/4)/(-256*a**7*b*
*9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gam
ma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a/(b*x**4)
- 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**1
6*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)
) + 225*a**4*b**(47/4)*x**8*(a/(b*x**4) - 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9
*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma
(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 45*a**3*b**(51/4)*x**12*(a/(b*x**4) -
1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*
gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4))
- 540*a**2*b**(55/4)*x**16*(a/(b*x**4) - 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*
x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(
1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 864*a*b**(59/4)*x**20*(a/(b*x**4) - 1)
**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gam
ma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 3
84*b**(63/4)*x**24*(a/(b*x**4) - 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*ga
mma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 2
56*a**4*b**12*x**24*gamma(1/4)), Abs(a/(b*x**4)) > 1), (-231*a**6*b**(39/4)*(-a/
(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4
) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4
*b**12*x**24*gamma(1/4)) + 441*a**5*b**(43/4)*x**4*(-a/(b*x**4) + 1)**(3/4)*exp(
15*I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*
gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4))
- 225*a**4*b**(47/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-15/4)/(
-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**
11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 45*a**3*b**(51/4)*x**12
*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamm
a(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256
*a**4*b**12*x**24*gamma(1/4)) + 540*a**2*b**(55/4)*x**16*(-a/(b*x**4) + 1)**(3/4
)*exp(15*I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*
x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(
1/4)) - 864*a*b**(59/4)*x**20*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-15/
4)/(-256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5
*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 384*b**(63/4)*x**24
*(-a/(b*x**4) + 1)**(3/4)*exp(15*I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*gamm
a(1/4) + 768*a**6*b**10*x**16*gamma(1/4) - 768*a**5*b**11*x**20*gamma(1/4) + 256
*a**4*b**12*x**24*gamma(1/4)), True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^16),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^16), x)