3.1159 \(\int \frac{1}{x^{16} \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(15*a*x^15*(a + b*x^4)^(1/4)) + (16*b)/(165*a^2*x^11*(a + b*x^4)^(1/4)) - (64
*b^2)/(385*a^3*x^7*(a + b*x^4)^(1/4)) + (512*b^3)/(1155*a^4*x^3*(a + b*x^4)^(1/4
)) + (2048*b^4*x)/(1155*a^5*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.110963, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

-1/(15*a*x^15*(a + b*x^4)^(1/4)) + (16*b)/(165*a^2*x^11*(a + b*x^4)^(1/4)) - (64
*b^2)/(385*a^3*x^7*(a + b*x^4)^(1/4)) + (512*b^3)/(1155*a^4*x^3*(a + b*x^4)^(1/4
)) + (2048*b^4*x)/(1155*a^5*(a + b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 12.7195, size = 109, normalized size = 0.96 \[ - \frac{1}{15 a x^{15} \sqrt [4]{a + b x^{4}}} + \frac{16 b}{165 a^{2} x^{11} \sqrt [4]{a + b x^{4}}} - \frac{64 b^{2}}{385 a^{3} x^{7} \sqrt [4]{a + b x^{4}}} + \frac{512 b^{3}}{1155 a^{4} x^{3} \sqrt [4]{a + b x^{4}}} + \frac{2048 b^{4} x}{1155 a^{5} \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**16/(b*x**4+a)**(5/4),x)

[Out]

-1/(15*a*x**15*(a + b*x**4)**(1/4)) + 16*b/(165*a**2*x**11*(a + b*x**4)**(1/4))
- 64*b**2/(385*a**3*x**7*(a + b*x**4)**(1/4)) + 512*b**3/(1155*a**4*x**3*(a + b*
x**4)**(1/4)) + 2048*b**4*x/(1155*a**5*(a + b*x**4)**(1/4))

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Mathematica [A]  time = 0.0593613, size = 64, normalized size = 0.56 \[ \frac{-77 a^4+112 a^3 b x^4-192 a^2 b^2 x^8+512 a b^3 x^{12}+2048 b^4 x^{16}}{1155 a^5 x^{15} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

(-77*a^4 + 112*a^3*b*x^4 - 192*a^2*b^2*x^8 + 512*a*b^3*x^12 + 2048*b^4*x^16)/(11
55*a^5*x^15*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.009, size = 61, normalized size = 0.5 \[ -{\frac{-2048\,{b}^{4}{x}^{16}-512\,{b}^{3}{x}^{12}a+192\,{a}^{2}{x}^{8}{b}^{2}-112\,b{x}^{4}{a}^{3}+77\,{a}^{4}}{1155\,{a}^{5}{x}^{15}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^16/(b*x^4+a)^(5/4),x)

[Out]

-1/1155*(-2048*b^4*x^16-512*a*b^3*x^12+192*a^2*b^2*x^8-112*a^3*b*x^4+77*a^4)/x^1
5/(b*x^4+a)^(1/4)/a^5

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Maxima [A]  time = 1.43924, size = 117, normalized size = 1.03 \[ \frac{b^{4} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{5}} + \frac{\frac{1540 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} - \frac{990 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{420 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} - \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^16),x, algorithm="maxima")

[Out]

b^4*x/((b*x^4 + a)^(1/4)*a^5) + 1/1155*(1540*(b*x^4 + a)^(3/4)*b^3/x^3 - 990*(b*
x^4 + a)^(7/4)*b^2/x^7 + 420*(b*x^4 + a)^(11/4)*b/x^11 - 77*(b*x^4 + a)^(15/4)/x
^15)/a^5

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Fricas [A]  time = 0.244309, size = 97, normalized size = 0.85 \[ \frac{{\left (2048 \, b^{4} x^{16} + 512 \, a b^{3} x^{12} - 192 \, a^{2} b^{2} x^{8} + 112 \, a^{3} b x^{4} - 77 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \,{\left (a^{5} b x^{19} + a^{6} x^{15}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^16),x, algorithm="fricas")

[Out]

1/1155*(2048*b^4*x^16 + 512*a*b^3*x^12 - 192*a^2*b^2*x^8 + 112*a^3*b*x^4 - 77*a^
4)*(b*x^4 + a)^(3/4)/(a^5*b*x^19 + a^6*x^15)

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Sympy [A]  time = 51.5855, size = 928, normalized size = 8.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**16/(b*x**4+a)**(5/4),x)

[Out]

-231*a**7*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*
gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4)
+ 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) - 357*a*
*6*b**(71/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*ga
mma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) +
4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) - 261*a**5
*b**(75/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamm
a(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4
096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 585*a**4*b
**(79/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma
(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 40
96*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 9360*a**3*b
**(83/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma
(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 40
96*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 22464*a**2*
b**(87/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamm
a(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4
096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 19968*a*b*
*(91/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(
5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 409
6*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 6144*b**(95/
4)*x**28*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4)
+ 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**
6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^16),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^16), x)