3.1099 \(\int \frac{1}{x^{16} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=92 \[ \frac{128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]

[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)

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Rubi [A]  time = 0.092293, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]

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)

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Rubi in Sympy [A]  time = 9.71904, size = 85, normalized size = 0.92 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{15 a x^{15}} + \frac{4 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{55 a^{2} x^{11}} - \frac{32 b^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{385 a^{3} x^{7}} + \frac{128 b^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1155 a^{4} x^{3}} \]

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)

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Mathematica [A]  time = 0.0417341, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^4\right )^{3/4} \left (-77 a^3+84 a^2 b x^4-96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]

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)

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Maple [A]  time = 0.01, size = 50, 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

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Maxima [A]  time = 1.44381, size = 93, normalized size = 1.01 \[ \frac{\frac{385 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} - \frac{495 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{315 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} - \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{4}} \]

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

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Fricas [A]  time = 0.236278, size = 66, normalized size = 0.72 \[ \frac{{\left (128 \, b^{3} x^{12} - 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} - 77 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{4} x^{15}} \]

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)

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Sympy [A]  time = 23.7299, size = 692, normalized size = 7.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-231*a**6*b**(39/4)*(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)) - 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**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)*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)*ga
mma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 7
68*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*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)*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))

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

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)