3.1193 \(\int \frac{\sqrt [4]{a-b x^4}}{x^{18}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac{12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{\left (a-b x^4\right )^{5/4}}{17 a x^{17}} \]

[Out]

-(a - b*x^4)^(5/4)/(17*a*x^17) - (12*b*(a - b*x^4)^(5/4))/(221*a^2*x^13) - (32*b
^2*(a - b*x^4)^(5/4))/(663*a^3*x^9) - (128*b^3*(a - b*x^4)^(5/4))/(3315*a^4*x^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.0941473, 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 \[ -\frac{128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac{12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{\left (a-b x^4\right )^{5/4}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^18,x]

[Out]

-(a - b*x^4)^(5/4)/(17*a*x^17) - (12*b*(a - b*x^4)^(5/4))/(221*a^2*x^13) - (32*b
^2*(a - b*x^4)^(5/4))/(663*a^3*x^9) - (128*b^3*(a - b*x^4)^(5/4))/(3315*a^4*x^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.7006, size = 87, normalized size = 0.91 \[ - \frac{\left (a - b x^{4}\right )^{\frac{5}{4}}}{17 a x^{17}} - \frac{12 b \left (a - b x^{4}\right )^{\frac{5}{4}}}{221 a^{2} x^{13}} - \frac{32 b^{2} \left (a - b x^{4}\right )^{\frac{5}{4}}}{663 a^{3} x^{9}} - \frac{128 b^{3} \left (a - b x^{4}\right )^{\frac{5}{4}}}{3315 a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/4)/x**18,x)

[Out]

-(a - b*x**4)**(5/4)/(17*a*x**17) - 12*b*(a - b*x**4)**(5/4)/(221*a**2*x**13) -
32*b**2*(a - b*x**4)**(5/4)/(663*a**3*x**9) - 128*b**3*(a - b*x**4)**(5/4)/(3315
*a**4*x**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0423197, size = 65, normalized size = 0.68 \[ \frac{\sqrt [4]{a-b x^4} \left (-195 a^4+15 a^3 b x^4+20 a^2 b^2 x^8+32 a b^3 x^{12}+128 b^4 x^{16}\right )}{3315 a^4 x^{17}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x^18,x]

[Out]

((a - b*x^4)^(1/4)*(-195*a^4 + 15*a^3*b*x^4 + 20*a^2*b^2*x^8 + 32*a*b^3*x^12 + 1
28*b^4*x^16))/(3315*a^4*x^17)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 51, normalized size = 0.5 \[ -{\frac{128\,{b}^{3}{x}^{12}+160\,a{b}^{2}{x}^{8}+180\,{a}^{2}b{x}^{4}+195\,{a}^{3}}{3315\,{x}^{17}{a}^{4}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^18,x)

[Out]

-1/3315*(-b*x^4+a)^(5/4)*(128*b^3*x^12+160*a*b^2*x^8+180*a^2*b*x^4+195*a^3)/x^17
/a^4

_______________________________________________________________________________________

Maxima [A]  time = 1.44093, size = 99, normalized size = 1.03 \[ -\frac{\frac{663 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b^{3}}{x^{5}} + \frac{1105 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} b^{2}}{x^{9}} + \frac{765 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}} b}{x^{13}} + \frac{195 \,{\left (-b x^{4} + a\right )}^{\frac{17}{4}}}{x^{17}}}{3315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^18,x, algorithm="maxima")

[Out]

-1/3315*(663*(-b*x^4 + a)^(5/4)*b^3/x^5 + 1105*(-b*x^4 + a)^(9/4)*b^2/x^9 + 765*
(-b*x^4 + a)^(13/4)*b/x^13 + 195*(-b*x^4 + a)^(17/4)/x^17)/a^4

_______________________________________________________________________________________

Fricas [A]  time = 0.236549, size = 82, normalized size = 0.85 \[ \frac{{\left (128 \, b^{4} x^{16} + 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} + 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, a^{4} x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^18,x, algorithm="fricas")

[Out]

1/3315*(128*b^4*x^16 + 32*a*b^3*x^12 + 20*a^2*b^2*x^8 + 15*a^3*b*x^4 - 195*a^4)*
(-b*x^4 + a)^(1/4)/(a^4*x^17)

_______________________________________________________________________________________

Sympy [A]  time = 32.8537, size = 1770, normalized size = 18.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**4+a)**(1/4)/x**18,x)

[Out]

Piecewise((585*a**7*b**(37/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-17/4)/(-256*a**7*b*
*9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*g
amma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 1800*a**6*b**(41/4)*x**4*(a/(b*
x**4) - 1)**(1/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**1
0*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*ga
mma(-1/4)) + 1830*a**5*b**(45/4)*x**8*(a/(b*x**4) - 1)**(1/4)*gamma(-17/4)/(-256
*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11
*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 636*a**4*b**(49/4)*x**1
2*(a/(b*x**4) - 1)**(1/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a
**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*
x**28*gamma(-1/4)) - 231*a**3*b**(53/4)*x**16*(a/(b*x**4) - 1)**(1/4)*gamma(-17/
4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a*
*5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) + 924*a**2*b**(57
/4)*x**20*(a/(b*x**4) - 1)**(1/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4)
+ 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**
4*b**12*x**28*gamma(-1/4)) - 1056*a*b**(61/4)*x**24*(a/(b*x**4) - 1)**(1/4)*gamm
a(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) -
768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) + 384*b**(6
5/4)*x**28*(a/(b*x**4) - 1)**(1/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4
) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a*
*4*b**12*x**28*gamma(-1/4)), Abs(a/(b*x**4)) > 1), (585*a**7*b**(37/4)*(-a/(b*x*
*4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) +
768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b
**12*x**28*gamma(-1/4)) - 1800*a**6*b**(41/4)*x**4*(-a/(b*x**4) + 1)**(1/4)*exp(
25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20
*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/
4)) + 1830*a**5*b**(45/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17
/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a
**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 636*a**4*b**(4
9/4)*x**12*(-a/(b*x**4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*
x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamm
a(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 231*a**3*b**(53/4)*x**16*(-a/(b*x*
*4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) +
768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b
**12*x**28*gamma(-1/4)) + 924*a**2*b**(57/4)*x**20*(-a/(b*x**4) + 1)**(1/4)*exp(
25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20
*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/
4)) - 1056*a*b**(61/4)*x**24*(-a/(b*x**4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17/4
)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**
5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) + 384*b**(65/4)*x*
*28*(-a/(b*x**4) + 1)**(1/4)*exp(25*I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*g
amma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) - 768*a**5*b**11*x**24*gamma(-1/4)
+ 256*a**4*b**12*x**28*gamma(-1/4)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263567, size = 230, normalized size = 2.4 \[ \frac{\frac{663 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )} b^{3}}{x} - \frac{1105 \,{\left (b^{2} x^{8} - 2 \, a b x^{4} + a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x^{9}} + \frac{765 \,{\left (b^{3} x^{12} - 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} - a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{13}} - \frac{195 \,{\left (b^{4} x^{16} - 4 \, a b^{3} x^{12} + 6 \, a^{2} b^{2} x^{8} - 4 \, a^{3} b x^{4} + a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{17}}}{3315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^18,x, algorithm="giac")

[Out]

1/3315*(663*(-b*x^4 + a)^(1/4)*(b - a/x^4)*b^3/x - 1105*(b^2*x^8 - 2*a*b*x^4 + a
^2)*(-b*x^4 + a)^(1/4)*b^2/x^9 + 765*(b^3*x^12 - 3*a*b^2*x^8 + 3*a^2*b*x^4 - a^3
)*(-b*x^4 + a)^(1/4)*b/x^13 - 195*(b^4*x^16 - 4*a*b^3*x^12 + 6*a^2*b^2*x^8 - 4*a
^3*b*x^4 + a^4)*(-b*x^4 + a)^(1/4)/x^17)/a^4