3.1192 \(\int \frac{\sqrt [4]{a-b x^4}}{x^{14}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \]

[Out]

-(a - b*x^4)^(5/4)/(13*a*x^13) - (8*b*(a - b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2
*(a - b*x^4)^(5/4))/(585*a^3*x^5)

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Rubi [A]  time = 0.0672735, 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 \[ -\frac{32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac{8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^14,x]

[Out]

-(a - b*x^4)^(5/4)/(13*a*x^13) - (8*b*(a - b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2
*(a - b*x^4)^(5/4))/(585*a^3*x^5)

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Rubi in Sympy [A]  time = 7.34424, size = 63, normalized size = 0.89 \[ - \frac{\left (a - b x^{4}\right )^{\frac{5}{4}}}{13 a x^{13}} - \frac{8 b \left (a - b x^{4}\right )^{\frac{5}{4}}}{117 a^{2} x^{9}} - \frac{32 b^{2} \left (a - b x^{4}\right )^{\frac{5}{4}}}{585 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/4)/x**14,x)

[Out]

-(a - b*x**4)**(5/4)/(13*a*x**13) - 8*b*(a - b*x**4)**(5/4)/(117*a**2*x**9) - 32
*b**2*(a - b*x**4)**(5/4)/(585*a**3*x**5)

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Mathematica [A]  time = 0.0348209, size = 54, normalized size = 0.76 \[ \frac{\sqrt [4]{a-b x^4} \left (-45 a^3+5 a^2 b x^4+8 a b^2 x^8+32 b^3 x^{12}\right )}{585 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x^14,x]

[Out]

((a - b*x^4)^(1/4)*(-45*a^3 + 5*a^2*b*x^4 + 8*a*b^2*x^8 + 32*b^3*x^12))/(585*a^3
*x^13)

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Maple [A]  time = 0.008, size = 40, normalized size = 0.6 \[ -{\frac{32\,{b}^{2}{x}^{8}+40\,ab{x}^{4}+45\,{a}^{2}}{585\,{x}^{13}{a}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^14,x)

[Out]

-1/585*(-b*x^4+a)^(5/4)*(32*b^2*x^8+40*a*b*x^4+45*a^2)/x^13/a^3

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Maxima [A]  time = 1.41353, size = 74, normalized size = 1.04 \[ -\frac{\frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{130 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{45 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{585 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(-b*x^4 + a)^(5/4)*b^2/x^5 + 130*(-b*x^4 + a)^(9/4)*b/x^9 + 45*(-b*x
^4 + a)^(13/4)/x^13)/a^3

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Fricas [A]  time = 0.236575, size = 68, normalized size = 0.96 \[ \frac{{\left (32 \, b^{3} x^{12} + 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} - 45 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, a^{3} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

1/585*(32*b^3*x^12 + 8*a*b^2*x^8 + 5*a^2*b*x^4 - 45*a^3)*(-b*x^4 + a)^(1/4)/(a^3
*x^13)

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Sympy [A]  time = 16.5758, size = 1100, normalized size = 15.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**4+a)**(1/4)/x**14,x)

[Out]

Piecewise((45*a**5*b**(17/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*
x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-
1/4)) - 95*a**4*b**(21/4)*x**4*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**
4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma
(-1/4)) + 47*a**3*b**(25/4)*x**8*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b
**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gam
ma(-1/4)) - 21*a**2*b**(29/4)*x**12*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**
5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*
gamma(-1/4)) + 56*a*b**(33/4)*x**16*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**
5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*
gamma(-1/4)) - 32*b**(37/4)*x**20*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*
b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*ga
mma(-1/4)), Abs(a/(b*x**4)) > 1), (45*a**5*b**(17/4)*(-a/(b*x**4) + 1)**(1/4)*ex
p(17*I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*
gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 95*a**4*b**(21/4)*x**4*(-a/(b*x*
*4) + 1)**(1/4)*exp(17*I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 12
8*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 47*a**3*b**(25
/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(17*I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**1
2*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)
) - 21*a**2*b**(29/4)*x**12*(-a/(b*x**4) + 1)**(1/4)*exp(17*I*pi/4)*gamma(-13/4)
/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**
6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*exp(17*I*pi
/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/
4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 32*b**(37/4)*x**20*(-a/(b*x**4) + 1)**(1/
4)*exp(17*I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x
**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)), True))

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GIAC/XCAS [A]  time = 0.258182, size = 151, normalized size = 2.13 \[ \frac{\frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )} b^{2}}{x} - \frac{130 \,{\left (b^{2} x^{8} - 2 \, a b x^{4} + a^{2}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{9}} + \frac{45 \,{\left (b^{3} x^{12} - 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} - a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x^{13}}}{585 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^14,x, algorithm="giac")

[Out]

1/585*(117*(-b*x^4 + a)^(1/4)*(b - a/x^4)*b^2/x - 130*(b^2*x^8 - 2*a*b*x^4 + a^2
)*(-b*x^4 + a)^(1/4)*b/x^9 + 45*(b^3*x^12 - 3*a*b^2*x^8 + 3*a^2*b*x^4 - a^3)*(-b
*x^4 + a)^(1/4)/x^13)/a^3