3.558 \(\int \frac{1}{x^9 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

[Out]

-(a + b*x^3)^(2/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(2/3))/(20*a^2*x^5) - (9*b^2*(a
+ b*x^3)^(2/3))/(40*a^3*x^2)

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Rubi [A]  time = 0.0661638, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(2/3))/(20*a^2*x^5) - (9*b^2*(a
+ b*x^3)^(2/3))/(40*a^3*x^2)

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Rubi in Sympy [A]  time = 6.71274, size = 61, normalized size = 0.9 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{8 a x^{8}} + \frac{3 b \left (a + b x^{3}\right )^{\frac{2}{3}}}{20 a^{2} x^{5}} - \frac{9 b^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{40 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**3+a)**(1/3),x)

[Out]

-(a + b*x**3)**(2/3)/(8*a*x**8) + 3*b*(a + b*x**3)**(2/3)/(20*a**2*x**5) - 9*b**
2*(a + b*x**3)**(2/3)/(40*a**3*x**2)

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Mathematica [A]  time = 0.0314402, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (5 a^2-6 a b x^3+9 b^2 x^6\right )}{40 a^3 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-((a + b*x^3)^(2/3)*(5*a^2 - 6*a*b*x^3 + 9*b^2*x^6))/(40*a^3*x^8)

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Maple [A]  time = 0.008, size = 39, normalized size = 0.6 \[ -{\frac{9\,{b}^{2}{x}^{6}-6\,ab{x}^{3}+5\,{a}^{2}}{40\,{x}^{8}{a}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(b*x^3+a)^(1/3),x)

[Out]

-1/40*(b*x^3+a)^(2/3)*(9*b^2*x^6-6*a*b*x^3+5*a^2)/x^8/a^3

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Maxima [A]  time = 1.44011, size = 70, normalized size = 1.03 \[ -\frac{\frac{20 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{2}}{x^{2}} - \frac{16 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b}{x^{5}} + \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{x^{8}}}{40 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^9),x, algorithm="maxima")

[Out]

-1/40*(20*(b*x^3 + a)^(2/3)*b^2/x^2 - 16*(b*x^3 + a)^(5/3)*b/x^5 + 5*(b*x^3 + a)
^(8/3)/x^8)/a^3

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Fricas [A]  time = 0.236752, size = 51, normalized size = 0.75 \[ -\frac{{\left (9 \, b^{2} x^{6} - 6 \, a b x^{3} + 5 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^9),x, algorithm="fricas")

[Out]

-1/40*(9*b^2*x^6 - 6*a*b*x^3 + 5*a^2)*(b*x^3 + a)^(2/3)/(a^3*x^8)

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Sympy [A]  time = 6.02433, size = 406, normalized size = 5.97 \[ \frac{10 a^{4} b^{\frac{14}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{8 a^{3} b^{\frac{17}{3}} x^{3} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{4 a^{2} b^{\frac{20}{3}} x^{6} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{24 a b^{\frac{23}{3}} x^{9} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{18 b^{\frac{26}{3}} x^{12} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(b*x**3+a)**(1/3),x)

[Out]

10*a**4*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1
/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 8*a**3*b**
(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) +
54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 4*a**2*b**(20/3)
*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**
4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 24*a*b**(23/3)*x**9*(a
/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x
**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 18*b**(26/3)*x**12*(a/(b*x**3)
+ 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma
(1/3) + 27*a**3*b**6*x**12*gamma(1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^9),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^9), x)