3.1097 \(\int \frac{1}{x^8 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7} \]

[Out]

-(a + b*x^4)^(3/4)/(7*a*x^7) + (4*b*(a + b*x^4)^(3/4))/(21*a^2*x^3)

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Rubi [A]  time = 0.0412989, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4 b \left (a+b x^4\right )^{3/4}}{21 a^2 x^3}-\frac{\left (a+b x^4\right )^{3/4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a + b*x^4)^(1/4)),x]

[Out]

-(a + b*x^4)^(3/4)/(7*a*x^7) + (4*b*(a + b*x^4)^(3/4))/(21*a^2*x^3)

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Rubi in Sympy [A]  time = 4.25805, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{7 a x^{7}} + \frac{4 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{21 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**4+a)**(1/4),x)

[Out]

-(a + b*x**4)**(3/4)/(7*a*x**7) + 4*b*(a + b*x**4)**(3/4)/(21*a**2*x**3)

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Mathematica [A]  time = 0.0266402, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^4\right )^{3/4} \left (4 b x^4-3 a\right )}{21 a^2 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a + b*x^4)^(1/4)),x]

[Out]

((a + b*x^4)^(3/4)*(-3*a + 4*b*x^4))/(21*a^2*x^7)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-4\,b{x}^{4}+3\,a}{21\,{a}^{2}{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^4+a)^(1/4),x)

[Out]

-1/21*(b*x^4+a)^(3/4)*(-4*b*x^4+3*a)/a^2/x^7

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Maxima [A]  time = 1.42221, size = 47, normalized size = 1.07 \[ \frac{\frac{7 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b}{x^{3}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{x^{7}}}{21 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^8),x, algorithm="maxima")

[Out]

1/21*(7*(b*x^4 + a)^(3/4)*b/x^3 - 3*(b*x^4 + a)^(7/4)/x^7)/a^2

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Fricas [A]  time = 0.236807, size = 36, normalized size = 0.82 \[ \frac{{\left (4 \, b x^{4} - 3 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^8),x, algorithm="fricas")

[Out]

1/21*(4*b*x^4 - 3*a)*(b*x^4 + a)^(3/4)/(a^2*x^7)

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Sympy [A]  time = 5.08644, size = 70, normalized size = 1.59 \[ - \frac{3 b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{16 a x^{4} \Gamma \left (\frac{1}{4}\right )} + \frac{b^{\frac{7}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{7}{4}\right )}{4 a^{2} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**4+a)**(1/4),x)

[Out]

-3*b**(3/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(16*a*x**4*gamma(1/4)) + b**(7/4
)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(4*a**2*gamma(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^8),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^8), x)