3.569 \(\int \frac{1}{x^5 \left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \sqrt [3]{a+b x^3}}{4 a^2 x}-\frac{\sqrt [3]{a+b x^3}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0410695, 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 \[ \frac{3 b \sqrt [3]{a+b x^3}}{4 a^2 x}-\frac{\sqrt [3]{a+b x^3}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^3)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.23445, size = 36, normalized size = 0.82 \[ - \frac{\sqrt [3]{a + b x^{3}}}{4 a x^{4}} + \frac{3 b \sqrt [3]{a + b x^{3}}}{4 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**3+a)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.0218465, size = 29, normalized size = 0.66 \[ -\frac{\left (a-3 b x^3\right ) \sqrt [3]{a+b x^3}}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^3)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 0.6 \[ -{\frac{-3\,b{x}^{3}+a}{4\,{x}^{4}{a}^{2}}\sqrt [3]{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^3+a)^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.43592, size = 47, normalized size = 1.07 \[ \frac{\frac{4 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}{x^{4}}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244261, size = 36, normalized size = 0.82 \[ \frac{{\left (3 \, b x^{3} - a\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{4 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.37566, size = 68, normalized size = 1.55 \[ - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{4}{3}\right )}{9 a x^{3} \Gamma \left (\frac{2}{3}\right )} + \frac{b^{\frac{4}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{4}{3}\right )}{3 a^{2} \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**3+a)**(2/3),x)

[Out]

-b**(1/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-4/3)/(9*a*x**3*gamma(2/3)) + b**(4/3)*(
a/(b*x**3) + 1)**(1/3)*gamma(-4/3)/(3*a**2*gamma(2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^5),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^5), x)