3.1127 \(\int \frac{1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \sqrt [4]{a+b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a+b x^4}}{5 a x^5} \]

[Out]

-(a + b*x^4)^(1/4)/(5*a*x^5) + (4*b*(a + b*x^4)^(1/4))/(5*a^2*x)

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Rubi [A]  time = 0.041665, 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 \[ \frac{4 b \sqrt [4]{a+b x^4}}{5 a^2 x}-\frac{\sqrt [4]{a+b x^4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^4)^(3/4)),x]

[Out]

-(a + b*x^4)^(1/4)/(5*a*x^5) + (4*b*(a + b*x^4)^(1/4))/(5*a^2*x)

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Rubi in Sympy [A]  time = 4.24671, size = 36, normalized size = 0.82 \[ - \frac{\sqrt [4]{a + b x^{4}}}{5 a x^{5}} + \frac{4 b \sqrt [4]{a + b x^{4}}}{5 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**4+a)**(3/4),x)

[Out]

-(a + b*x**4)**(1/4)/(5*a*x**5) + 4*b*(a + b*x**4)**(1/4)/(5*a**2*x)

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Mathematica [A]  time = 0.0225761, size = 29, normalized size = 0.66 \[ -\frac{\left (a-4 b x^4\right ) \sqrt [4]{a+b x^4}}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^4)^(3/4)),x]

[Out]

-((a - 4*b*x^4)*(a + b*x^4)^(1/4))/(5*a^2*x^5)

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Maple [A]  time = 0.007, size = 26, normalized size = 0.6 \[ -{\frac{-4\,b{x}^{4}+a}{5\,{x}^{5}{a}^{2}}\sqrt [4]{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^4+a)^(3/4),x)

[Out]

-1/5*(b*x^4+a)^(1/4)*(-4*b*x^4+a)/x^5/a^2

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Maxima [A]  time = 1.4426, size = 47, normalized size = 1.07 \[ \frac{\frac{5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x} - \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{5}}}{5 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^6),x, algorithm="maxima")

[Out]

1/5*(5*(b*x^4 + a)^(1/4)*b/x - (b*x^4 + a)^(5/4)/x^5)/a^2

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Fricas [A]  time = 0.237305, size = 36, normalized size = 0.82 \[ \frac{{\left (4 \, b x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^6),x, algorithm="fricas")

[Out]

1/5*(4*b*x^4 - a)*(b*x^4 + a)^(1/4)/(a^2*x^5)

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Sympy [A]  time = 4.64174, size = 68, normalized size = 1.55 \[ - \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{5}{4}\right )}{16 a x^{4} \Gamma \left (\frac{3}{4}\right )} + \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{5}{4}\right )}{4 a^{2} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**4+a)**(3/4),x)

[Out]

-b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-5/4)/(16*a*x**4*gamma(3/4)) + b**(5/4)*
(a/(b*x**4) + 1)**(1/4)*gamma(-5/4)/(4*a**2*gamma(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^6), x)