3.540 \(\int \frac{\left (a+b x^3\right )^{2/3}}{x^9} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \left (a+b x^3\right )^{5/3}}{40 a^2 x^5}-\frac{\left (a+b x^3\right )^{5/3}}{8 a x^8} \]

[Out]

-(a + b*x^3)^(5/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(5/3))/(40*a^2*x^5)

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Rubi [A]  time = 0.0420266, 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 \[ \frac{3 b \left (a+b x^3\right )^{5/3}}{40 a^2 x^5}-\frac{\left (a+b x^3\right )^{5/3}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x^9,x]

[Out]

-(a + b*x^3)^(5/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(5/3))/(40*a^2*x^5)

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Rubi in Sympy [A]  time = 4.25496, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{3}\right )^{\frac{5}{3}}}{8 a x^{8}} + \frac{3 b \left (a + b x^{3}\right )^{\frac{5}{3}}}{40 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(2/3)/x**9,x)

[Out]

-(a + b*x**3)**(5/3)/(8*a*x**8) + 3*b*(a + b*x**3)**(5/3)/(40*a**2*x**5)

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Mathematica [A]  time = 0.0252886, size = 44, normalized size = 1. \[ \left (\frac{3 b^2}{40 a^2 x^2}-\frac{b}{20 a x^5}-\frac{1}{8 x^8}\right ) \left (a+b x^3\right )^{2/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(2/3)/x^9,x]

[Out]

(-1/(8*x^8) - b/(20*a*x^5) + (3*b^2)/(40*a^2*x^2))*(a + b*x^3)^(2/3)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-3\,b{x}^{3}+5\,a}{40\,{x}^{8}{a}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x^9,x)

[Out]

-1/40*(b*x^3+a)^(5/3)*(-3*b*x^3+5*a)/x^8/a^2

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Maxima [A]  time = 1.44913, size = 47, normalized size = 1.07 \[ \frac{\frac{8 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b}{x^{5}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{x^{8}}}{40 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^9,x, algorithm="maxima")

[Out]

1/40*(8*(b*x^3 + a)^(5/3)*b/x^5 - 5*(b*x^3 + a)^(8/3)/x^8)/a^2

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Fricas [A]  time = 0.254371, size = 51, normalized size = 1.16 \[ \frac{{\left (3 \, b^{2} x^{6} - 2 \, a b x^{3} - 5 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, a^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^9,x, algorithm="fricas")

[Out]

1/40*(3*b^2*x^6 - 2*a*b*x^3 - 5*a^2)*(b*x^3 + a)^(2/3)/(a^2*x^8)

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Sympy [A]  time = 6.09448, size = 110, normalized size = 2.5 \[ - \frac{5 b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{9 x^{6} \Gamma \left (- \frac{2}{3}\right )} - \frac{2 b^{\frac{5}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{9 a x^{3} \Gamma \left (- \frac{2}{3}\right )} + \frac{b^{\frac{8}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{3 a^{2} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(2/3)/x**9,x)

[Out]

-5*b**(2/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3)) - 2*b**(5/3
)*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(9*a*x**3*gamma(-2/3)) + b**(8/3)*(a/(b*x*
*3) + 1)**(2/3)*gamma(-8/3)/(3*a**2*gamma(-2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^9,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^9, x)