3.1158 \(\int \frac{1}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(11*a*x^11*(a + b*x^4)^(1/4)) + (12*b)/(77*a^2*x^7*(a + b*x^4)^(1/4)) - (32*b
^2)/(77*a^3*x^3*(a + b*x^4)^(1/4)) - (128*b^3*x)/(77*a^4*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0822382, antiderivative size = 90, 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 \[ -\frac{128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

-1/(11*a*x^11*(a + b*x^4)^(1/4)) + (12*b)/(77*a^2*x^7*(a + b*x^4)^(1/4)) - (32*b
^2)/(77*a^3*x^3*(a + b*x^4)^(1/4)) - (128*b^3*x)/(77*a^4*(a + b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 8.99094, size = 85, normalized size = 0.94 \[ - \frac{1}{11 a x^{11} \sqrt [4]{a + b x^{4}}} + \frac{12 b}{77 a^{2} x^{7} \sqrt [4]{a + b x^{4}}} - \frac{32 b^{2}}{77 a^{3} x^{3} \sqrt [4]{a + b x^{4}}} - \frac{128 b^{3} x}{77 a^{4} \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**12/(b*x**4+a)**(5/4),x)

[Out]

-1/(11*a*x**11*(a + b*x**4)**(1/4)) + 12*b/(77*a**2*x**7*(a + b*x**4)**(1/4)) -
32*b**2/(77*a**3*x**3*(a + b*x**4)**(1/4)) - 128*b**3*x/(77*a**4*(a + b*x**4)**(
1/4))

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Mathematica [A]  time = 0.0487353, size = 53, normalized size = 0.59 \[ -\frac{7 a^3-12 a^2 b x^4+32 a b^2 x^8+128 b^3 x^{12}}{77 a^4 x^{11} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

-(7*a^3 - 12*a^2*b*x^4 + 32*a*b^2*x^8 + 128*b^3*x^12)/(77*a^4*x^11*(a + b*x^4)^(
1/4))

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Maple [A]  time = 0.009, size = 50, normalized size = 0.6 \[ -{\frac{128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}-12\,{a}^{2}b{x}^{4}+7\,{a}^{3}}{77\,{x}^{11}{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^12/(b*x^4+a)^(5/4),x)

[Out]

-1/77*(128*b^3*x^12+32*a*b^2*x^8-12*a^2*b*x^4+7*a^3)/x^11/(b*x^4+a)^(1/4)/a^4

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Maxima [A]  time = 1.44094, size = 96, normalized size = 1.07 \[ -\frac{b^{3} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}} - \frac{\frac{77 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{2}}{x^{3}} - \frac{33 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b}{x^{7}} + \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{x^{11}}}{77 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^12),x, algorithm="maxima")

[Out]

-b^3*x/((b*x^4 + a)^(1/4)*a^4) - 1/77*(77*(b*x^4 + a)^(3/4)*b^2/x^3 - 33*(b*x^4
+ a)^(7/4)*b/x^7 + 7*(b*x^4 + a)^(11/4)/x^11)/a^4

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Fricas [A]  time = 0.250339, size = 82, normalized size = 0.91 \[ -\frac{{\left (128 \, b^{3} x^{12} + 32 \, a b^{2} x^{8} - 12 \, a^{2} b x^{4} + 7 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \,{\left (a^{4} b x^{15} + a^{5} x^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^12),x, algorithm="fricas")

[Out]

-1/77*(128*b^3*x^12 + 32*a*b^2*x^8 - 12*a^2*b*x^4 + 7*a^3)*(b*x^4 + a)^(3/4)/(a^
4*b*x^15 + a^5*x^11)

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Sympy [A]  time = 23.1003, size = 592, normalized size = 6.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**12/(b*x**4+a)**(5/4),x)

[Out]

21*a**5*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma
(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*
a**4*b**12*x**20*gamma(5/4)) + 6*a**4*b**(43/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gam
ma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768
*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 45*a**3*b**(47
/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 7
68*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**1
2*x**20*gamma(5/4)) + 540*a**2*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-11
/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*
b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 864*a*b**(55/4)*x**1
6*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6
*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20
*gamma(5/4)) + 384*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**
7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*
gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^12),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^12), x)