3.557 \(\int \frac{1}{x^6 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5} \]

[Out]

-(a + b*x^3)^(2/3)/(5*a*x^5) + (3*b*(a + b*x^3)^(2/3))/(10*a^2*x^2)

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Rubi [A]  time = 0.0422746, 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 \[ \frac{3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^3)^(2/3)/(5*a*x^5) + (3*b*(a + b*x^3)^(2/3))/(10*a^2*x^2)

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Rubi in Sympy [A]  time = 4.24118, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{5 a x^{5}} + \frac{3 b \left (a + b x^{3}\right )^{\frac{2}{3}}}{10 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**3)**(2/3)/(5*a*x**5) + 3*b*(a + b*x**3)**(2/3)/(10*a**2*x**2)

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Mathematica [A]  time = 0.0253708, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^3\right )^{2/3} \left (3 b x^3-2 a\right )}{10 a^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)^(2/3)*(-2*a + 3*b*x^3))/(10*a^2*x^5)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-3\,b{x}^{3}+2\,a}{10\,{x}^{5}{a}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^3+a)^(1/3),x)

[Out]

-1/10*(b*x^3+a)^(2/3)*(-3*b*x^3+2*a)/x^5/a^2

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Maxima [A]  time = 1.4373, size = 47, normalized size = 1.07 \[ \frac{\frac{5 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}}}{x^{5}}}{10 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(5*(b*x^3 + a)^(2/3)*b/x^2 - 2*(b*x^3 + a)^(5/3)/x^5)/a^2

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Fricas [A]  time = 0.235654, size = 36, normalized size = 0.82 \[ \frac{{\left (3 \, b x^{3} - 2 \, a\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{10 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(3*b*x^3 - 2*a)*(b*x^3 + a)^(2/3)/(a^2*x^5)

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Sympy [A]  time = 3.27328, size = 70, normalized size = 1.59 \[ - \frac{2 b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{9 a x^{3} \Gamma \left (\frac{1}{3}\right )} + \frac{b^{\frac{5}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{3 a^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b**(2/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-5/3)/(9*a*x**3*gamma(1/3)) + b**(5/3)
*(a/(b*x**3) + 1)**(2/3)*gamma(-5/3)/(3*a**2*gamma(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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