3.1224 \(\int \frac{1}{x^{20} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}} \]

[Out]

-(a - b*x^4)^(3/4)/(19*a*x^19) - (16*b*(a - b*x^4)^(3/4))/(285*a^2*x^15) - (64*b
^2*(a - b*x^4)^(3/4))/(1045*a^3*x^11) - (512*b^3*(a - b*x^4)^(3/4))/(7315*a^4*x^
7) - (2048*b^4*(a - b*x^4)^(3/4))/(21945*a^5*x^3)

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Rubi [A]  time = 0.129507, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^20*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(19*a*x^19) - (16*b*(a - b*x^4)^(3/4))/(285*a^2*x^15) - (64*b
^2*(a - b*x^4)^(3/4))/(1045*a^3*x^11) - (512*b^3*(a - b*x^4)^(3/4))/(7315*a^4*x^
7) - (2048*b^4*(a - b*x^4)^(3/4))/(21945*a^5*x^3)

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Rubi in Sympy [A]  time = 15.2427, size = 110, normalized size = 0.91 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{19 a x^{19}} - \frac{16 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{285 a^{2} x^{15}} - \frac{64 b^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{1045 a^{3} x^{11}} - \frac{512 b^{3} \left (a - b x^{4}\right )^{\frac{3}{4}}}{7315 a^{4} x^{7}} - \frac{2048 b^{4} \left (a - b x^{4}\right )^{\frac{3}{4}}}{21945 a^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**20/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(19*a*x**19) - 16*b*(a - b*x**4)**(3/4)/(285*a**2*x**15) -
64*b**2*(a - b*x**4)**(3/4)/(1045*a**3*x**11) - 512*b**3*(a - b*x**4)**(3/4)/(73
15*a**4*x**7) - 2048*b**4*(a - b*x**4)**(3/4)/(21945*a**5*x**3)

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Mathematica [A]  time = 0.0492633, size = 65, normalized size = 0.54 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (1155 a^4+1232 a^3 b x^4+1344 a^2 b^2 x^8+1536 a b^3 x^{12}+2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^20*(a - b*x^4)^(1/4)),x]

[Out]

-((a - b*x^4)^(3/4)*(1155*a^4 + 1232*a^3*b*x^4 + 1344*a^2*b^2*x^8 + 1536*a*b^3*x
^12 + 2048*b^4*x^16))/(21945*a^5*x^19)

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Maple [A]  time = 0.012, size = 62, normalized size = 0.5 \[ -{\frac{2048\,{b}^{4}{x}^{16}+1536\,{b}^{3}{x}^{12}a+1344\,{b}^{2}{x}^{8}{a}^{2}+1232\,b{x}^{4}{a}^{3}+1155\,{a}^{4}}{21945\,{x}^{19}{a}^{5}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^20/(-b*x^4+a)^(1/4),x)

[Out]

-1/21945*(-b*x^4+a)^(3/4)*(2048*b^4*x^16+1536*a*b^3*x^12+1344*a^2*b^2*x^8+1232*a
^3*b*x^4+1155*a^4)/x^19/a^5

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Maxima [A]  time = 1.43594, size = 123, normalized size = 1.02 \[ -\frac{\frac{7315 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b^{4}}{x^{3}} + \frac{12540 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} b^{3}}{x^{7}} + \frac{11970 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}} b^{2}}{x^{11}} + \frac{5852 \,{\left (-b x^{4} + a\right )}^{\frac{15}{4}} b}{x^{15}} + \frac{1155 \,{\left (-b x^{4} + a\right )}^{\frac{19}{4}}}{x^{19}}}{21945 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^20),x, algorithm="maxima")

[Out]

-1/21945*(7315*(-b*x^4 + a)^(3/4)*b^4/x^3 + 12540*(-b*x^4 + a)^(7/4)*b^3/x^7 + 1
1970*(-b*x^4 + a)^(11/4)*b^2/x^11 + 5852*(-b*x^4 + a)^(15/4)*b/x^15 + 1155*(-b*x
^4 + a)^(19/4)/x^19)/a^5

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Fricas [A]  time = 0.227284, size = 82, normalized size = 0.68 \[ -\frac{{\left (2048 \, b^{4} x^{16} + 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} + 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{21945 \, a^{5} x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^20),x, algorithm="fricas")

[Out]

-1/21945*(2048*b^4*x^16 + 1536*a*b^3*x^12 + 1344*a^2*b^2*x^8 + 1232*a^3*b*x^4 +
1155*a^4)*(-b*x^4 + a)^(3/4)/(a^5*x^19)

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Sympy [A]  time = 42.0738, size = 2176, normalized size = 17.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**20/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((3465*a**8*b**(67/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b
**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24
*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4
)) - 10164*a**7*b**(71/4)*x**4*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b
**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24
*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4
)) + 10038*a**6*b**(75/4)*x**8*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b
**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24
*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4
)) - 3204*a**5*b**(79/4)*x**12*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b
**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24
*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4
)) + 585*a**4*b**(83/4)*x**16*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b*
*16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*
gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)
) - 9360*a**3*b**(87/4)*x**20*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b*
*16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*
gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)
) + 22464*a**2*b**(91/4)*x**24*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b
**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24
*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4
)) - 19968*a*b**(95/4)*x**28*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**
16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*g
amma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4))
+ 6144*b**(99/4)*x**32*(a/(b*x**4) - 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x*
*16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(
1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)), Abs
(a/(b*x**4)) > 1), (-3465*a**8*b**(67/4)*(-a/(b*x**4) + 1)**(3/4)*exp(23*I*pi/4)
*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/
4) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*
a**5*b**20*x**32*gamma(1/4)) + 10164*a**7*b**(71/4)*x**4*(-a/(b*x**4) + 1)**(3/4
)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) - 4096*a**8*b**1
7*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**6*b**19*x**28*ga
mma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) - 10038*a**6*b**(75/4)*x**8*(-a/(b*
x**4) + 1)**(3/4)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4)
- 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**
6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 3204*a**5*b**(79/
4)*x**12*(-a/(b*x**4) + 1)**(3/4)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x
**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma
(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) - 5
85*a**4*b**(83/4)*x**16*(-a/(b*x**4) + 1)**(3/4)*exp(23*I*pi/4)*gamma(-19/4)/(10
24*a**9*b**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b*
*18*x**24*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*
gamma(1/4)) + 9360*a**3*b**(87/4)*x**20*(-a/(b*x**4) + 1)**(3/4)*exp(23*I*pi/4)*
gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4
) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a
**5*b**20*x**32*gamma(1/4)) - 22464*a**2*b**(91/4)*x**24*(-a/(b*x**4) + 1)**(3/4
)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) - 4096*a**8*b**1
7*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**6*b**19*x**28*ga
mma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 19968*a*b**(95/4)*x**28*(-a/(b*x*
*4) + 1)**(3/4)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) -
4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) - 4096*a**6*
b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) - 6144*b**(99/4)*x**3
2*(-a/(b*x**4) + 1)**(3/4)*exp(23*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*ga
mma(1/4) - 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) -
4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{20}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^20),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^20), x)